Higher algebra of A_\infty and ΩB As-algebras in Morse theory I
HHIGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I
THIBAUT MAZUIR
Abstract.
Elaborating on works by Abouzaid and Mescher, we prove that for a Morse function ona smooth compact manifold, its Morse cochain complex can be endowed with an Ω BAs -algebra struc-ture by counting moduli spaces of perturbed Morse gradient trees. This rich structure descends to itsalready known A ∞ -algebra structure. We then introduce the notion of Ω BAs -morphism between two Ω BAs -algebras and prove that given two Morse functions, one can construct an Ω BAs -morphismbetween their associated Ω BAs -algebras by counting moduli spaces of two-colored perturbed Morsegradient trees. This morphism induces a standard A ∞ -morphism between the induced A ∞ -algebras.We work with integer coefficients, and provide to this extent a detailed account on the sign con-ventions for A ∞ (resp. Ω BAs )-algebras and A ∞ (resp. Ω BAs )-morphisms, using polytopes (resp.moduli spaces) which explicitly realize the dg-operadic objects encoding them. Our proofs alsoinvolve building at the level of polytopes an explicit functor from the category of Ω BAs -algebras tothe category of A ∞ -algebras, drawing from a result by Markl and Shnider. This paper is adressedto people acquainted with either symplectic topology or algebraic operads, and written in a wayto be hopefully understood by both communities. It comes in particular with a detailed surveyon operads, A ∞ -algebras and A ∞ -morphisms, the associahedra and the multiplihedra, as well assome details on the usual techniques used in symplectic topology to define algebraic structures ongeometrical (co)chain complexes. It moreover lays the basis for a second article in which we solvethe problem of finding a satisfactory homotopic notion of higher morphisms between A ∞ -algebrasand between Ω BAs -algebras, and show how this higher algebra of A ∞ and Ω BAs -algebras naturallyarises in the context of Morse theory.
The associahedron K and the multiplihedron J ... a r X i v : . [ m a t h . S G ] F e b THIBAUT MAZUIR
Contents
81. Operadic algebra 82.
Poly -operads 143. Moduli spaces of metric trees 174. Signs and polytopes for A ∞ -algebras and A ∞ -morphisms 275. Signs and moduli spaces for Ω BAs -algebras and Ω BAs -morphisms 38 A ∞ and Ω BAs -algebra structures on the Morse cochains 552. A ∞ and Ω BAs -morphisms between the Morse cochains 623. Transversality 694. Signs, orientations and gluing 72 µ Y is a quasi-isomorphism 882. More on the Ω BAs viewpoint 903. Quilted disks 904. Towards higher algebra 91References 93 ... and their Ω BAs -cell decompositions
IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 3
Introduction
Goals of articles I and II . — As the title suggests, our main goal lies at the crossroads ofalgebraic operads and Morse theory and deals with two questions. First, understand and define thecorrect homotopic notion of higher morphisms between A ∞ -algebras, which would give a satisfactorydescription of the higher algebra of A ∞ -algebras. Secondly, elaborating on the work of Abouzaidand Mescher on perturbed Morse gradient trees, realize these higher morphisms through modulispaces in Morse theory. These two articles may be of interest to the algebraic operads and to thesymplectic topology communities and are written with a sufficient level of detail in order to behopefully understandable by both. Operads . — The theory of algebraic operads aims at defining an appropriate framework to studyalgebraic structures and their homotopy theory. Introduced in works by Adams, Boardman, May orVogt in the sixties, in order to study problems in algebraic topology, the theory experienced a lossof interest in the following two decades until its renaissance in the nineties. Its target then shiftedfrom topology to algebra, for instance in the works of Ginzburg, Kontsevich, Markl and Tamarkin,through the study of questions in deformation theory and quantum field theory. At the same time,the first steps towards the construction of Fukaya categories in symplectic topology were undertaken.The paradigm is to see the collection of operations encoding a specific algebraic structure, aswell as the relations they have to satisfy, as an algebraic entity on its own, which can be studiedsystematically : this algebraic entity is called an operad . Put differently, an operad P encodes a category of P -algebras . For instance, the operad denoted As will define the classical category ofassociative algebras. This viewpoint can turn out to be extremely efficient to study some aspectsof these P -algebras. For instance, Vallette shows in [Val20] that whenever the operad P is Koszul(which is the case for the operads Ass , Com and
Lie ), one can define a satisfactory notion of P -algebras up to homotopy : these algebras are in particular modeled on an operad denoted P ∞ . Afamous instance is that of the operad A ∞ , encoding algebras which are associative up to homotopy.It was introduced for the first time in the seminal paper of Stasheff [Sta63] and arises in variousfields of mathematics, from string topology to symplectic topology. The operad A ∞ originates infact from a family of polytopes, known as the associahedra . This remarkable result comes with richcombinatorics and allows for non-trivial algebraic constructions. For example in [MTTV19], theauthors define the natural tensor product of two A ∞ -algebras, by constructing a polytopal diagonalfor the associahedra. This construction recovers in particular simple formulae for the A ∞ operationson the tensor product, known as Loday’s magical formulae . Algebraic structures in symplectic topology . — A symplectic manifold corresponds to the dataof a smooth manifold M together with a closed non-degenerate 2-form ω on M . The purpose of symplectic topology is the study of the geometrical properties of symplectic manifolds ( M, ω ) , andof the way they are preserved under smooth transformations preserving the symplectic structure.As algebraic topology seeks to associate algebraic invariants to topological spaces, in the hopeof distinguishing them and understanding some of their topological properties, the same modusoperandi can be applied to the study of symplectic manifolds. This point view was prompted by theseminal work of Gromov [Gro85] on moduli spaces of pseudo-holomorphic curves . In technical terms, THIBAUT MAZUIR by counting the points of 0-dimensional moduli spaces of pseudo-holomorphic curves, one will be ableto define algebraic operations stemming from the geometry of the underlying symplectic manifolds.The most famous example is that of the
Fukaya category
Fuk( M ) of a symplectic manifold M (withadditional technical assumptions), which is an A ∞ -category whose higher multiplications are definedby counting moduli spaces of pseudo-holomorphic disks with Lagrangian boundary conditions and n +1 marked points on their boundary. We refer for instance to [Smi15] and [Aur14] for introductionsto the subject.In fact, there is a tight link between algebraic operads and algebraic structures arising fromsymplectic topology. Numerous operadic objects originate from moduli spaces of curves. This is forinstance the case of the operad A ∞ : writing M n, for the moduli space of disks with n + 1 markedpoints on their boundary, where n points are seen as incoming, and 1 as outgoing, the moduli space M n, can be compactified and topologized in such a way that its image under the cellular chainsfunctor yields the operad A ∞ . See [Sei08] for instance. It is for this reason that the operations ofthe Fukaya category, defined by realizing these disks in the realm of symplectic geometry, fit into an A ∞ -category structure. This way of constructing algebraic structures is summarized in the followingdiagram Set of operations in sym-plectic topology (e.g. onthe Fukaya category)Moduli spaces of pseudo-holomorphic curves (e.g. disks withLagrangian boundary conditions)Compactified moduli spacesof curves (e.g. M n, ) Operadic object (e.g. operad A ∞ ) EncodesCounting 0-dimensionalmoduli spacesFloer theory Functor C ∗ cell .It is worth mentioning the work of Bottman which fully realizes this motto. He is currently developingan algebraic model for the notion of ( A ∞ , -categories, using moduli spaces of witch curves. The goalis to prove that one can then define an ( A ∞ , -category Symp whose objects would be symplecticmanifolds (with suitable technical assumptions), and such that the space of morphisms between twosymplectic manifolds M and N would be the Fukaya category Fuk( M − × N ) . We refer to his recentpapers [Bot19a] and [Bot19b] for more details. Morse theory corresponds to the study of manifolds endowed with a
Morse function , i.e. afunction whose critical points are non-degenerate. Symplectic topology, or more precisely Floertheory, is sometimes presented as an infinite-dimensional analogue of Morse theory. In this vein,given a smooth compact manifold M , Fukaya constructed in [Fuk97] an A ∞ -category whose objectsare functions f i on M , whose spaces of morphisms between two functions f i and f j (such that f i − f j is Morse) are the Morse cochain complexes C ∗ ( f i − f j ) , and whose higher multiplications aredefined by counting moduli spaces of Morse ribbon trees. Adapting this construction to the case ofa single Morse function f on M , Abouzaid defines in [Abo11] an A ∞ -algebra structure on the Morsecochains C ∗ ( f ) by counting moduli spaces of perturbed Morse gradient ribbon trees . His work wassubsequently continued by Mescher in [Mes18]. Outline of the paper and main results . — Our first part aims firstly at giving a concise and self-contained introduction to the theory of algebraic operads. The theory is subsequently specialized tothe case of A ∞ -algebras and of A ∞ -morphisms between them, with recollections on basic results on IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 5 their homotopy theory. We introduce in particular the convenient setting of operadic bimodules todefine the operadic bimodule A ∞ − Morph encoding A ∞ -morphisms between A ∞ -algebras. We thenrecall how the operad A ∞ (resp. the operadic bimodule A ∞ − Morph ) can be realized using familiesof polytopes, known as the associahedra (resp. multiplihedra). Seeing the associahedra as modulispaces of metric stable ribbon trees, they come with a thinner cell decomposition that realizes theoperad Ω BAs . Likewise, seeing the multiplihedra as moduli spaces of two-colored metric stableribbon trees, they come with a thinner cell decomposition that yields a new operadic bimodule : theoperadic bimodule Ω BAs − Morph , encoding Ω BAs -morphisms between Ω BAs -algebras.
Definition 16.
The operadic bimodule Ω BAs − Morph is the quasi-free (Ω BAs, Ω BAs ) -operadicbimodule freely generated by the set of two-colored stable ribbon trees Ω BAs − Morph := F Ω BAs, Ω BAs ( , , , , · · · , sCRT n , · · · ) , where a two-colored stable ribbon tree t g with e ( t ) internal edges and whose gauge crosses j verticeshas degree | t g | := j − e ( t ) − . The differential of a two-colored stable ribbon tree t g is given by thesigned sum of all two-colored stable ribbon trees obtained from t g under the rule prescribed by thetop dimensional strata in the boundary of CT n ( t g ) .The Ω BAs framework provides another template to study algebras which are homotopy-associative,together with morphisms between them which preserve the product up to homotopy. This is followedby a comprehensive survey on the A ∞ and Ω BAs sign conventions. In the A ∞ case, we show howthe two usual sign conventions for A ∞ -algebras and A ∞ -morphisms are naturally induced by theshifted bar construction viewpoint. Using the Loday realizations of the associahedra [MTTV19] andthe Forcey-Loday realizations of the multiplihedra [MV], we give a complete proof of the followingtwo folklore propositions : Propositions 5 and 6.
The Loday associahedra and the Forcey-Loday multiplihedra contain theusual sign conventions for A ∞ -algebras and A ∞ -morphisms between them. On the Ω BAs side, we start by recalling the formulation of the operad Ω BAs by Markl andShnider [MS06]. We then proceed to study the moduli spaces of stable two-colored metric ribbontrees CT n ( t g ) and compute the signs arising in the top dimensional strata of their boundary in propo-sitions 9 to 13. This extensive study allows us to complete the definition of the operadic bimodule Ω BAs − Morph by making explicit the signs for the action-composition maps and the differential. Wefinally give an alternative and more geometric construction of the morphism of operads A ∞ → Ω BAs defined in [MS06], and build a morphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph using the same ideas.
Propositions 2 , 3 and 4.
There exist a morphism of operads A ∞ → A ∞ − Morph and a morphismof operadic bimodules A ∞ − Morph → Ω BAs − Morph that induce a functor
ΩBAs − alg −→ A ∞ − alg . In the second part of this paper, we consider a Morse function f on a smooth closed manifold M and adapt the constructions of Abouzaid [Abo11], using the terminology of Mescher [Mes18], toperform two constructions on the Morse cochains C ∗ ( f ) . Firstly, we introduce the notion of smoothchoices of perturbation data X n on the moduli spaces T n that we use to define the moduli spaces ofperturbed Morse gradient trees T X t t ( y ; x , . . . , x n ) . THIBAUT MAZUIR
Theorems 7 and 8.
Under some generic assumptions on the choices of perturbation data { X n } n (cid:62) ,the moduli spaces T X t t ( y ; x , . . . , x n ) are orientable manifolds. If they have dimension 0, they arecompact. If they have dimension 1, they can be compactified to compact manifolds with boundary,whose boundary is modeled on the boundary of the moduli spaces T n ( t ) . We then show that under a generic choice of perturbation data { X n } n (cid:62) the Morse cochains C ∗ ( f ) can be endowed with an Ω BAs -algebra structure, by counting 0-dimensional moduli spaces of Morsegradient ribbon trees.
Theorem 9.
Defining for every n and t ∈ sRT n the operations m t as m t : C ∗ ( f ) ⊗ · · · ⊗ C ∗ ( f ) −→ C ∗ ( f ) x ⊗ · · · ⊗ x n (cid:55)−→ (cid:88) | y | = (cid:80) ni =1 | x i |− e ( t ) T X t ( y ; x , · · · , x n ) · y , they endow the Morse cochains C ∗ ( f ) with an Ω BAs -algebra structure.
Given now two Morse functions f and g , we can perform the same constructions in Morse theoryusing this time the moduli spaces CT n as blueprints. The counterparts of theorems 7 and 8 stillhold. Moreover, given generic choices of perturbation data X f and X g , one can construct an Ω BAs -morphism between the Ω BAs -algebras C ∗ ( f ) and C ∗ ( g ) by counting 0-dimensional moduli spacesof two-colored Morse gradient trees. Theorem 12.
Let ( Y n ) n (cid:62) be a generic choice of perturbation data on the moduli spaces CT n .Defining for every n and t g ∈ sCRT n the operations µ t g as µ Y t g : C ∗ ( f ) ⊗ · · · ⊗ C ∗ ( f ) −→ C ∗ ( g ) x ⊗ · · · ⊗ x n (cid:55)−→ (cid:88) | y | = (cid:80) ni =1 | x i | + | t g | CT Y t g ( y ; x , · · · , x n ) · y . they fit into an Ω BAs -morphism µ Y : ( C ∗ ( f ) , m X f t ) → ( C ∗ ( g ) , m X g t ) . We recover in particular the already known A ∞ -algebra structure on the Morse cochains, as wellas an A ∞ -morphism between two A ∞ -algebras, using the previously defined functor ΩBAs − alg → A ∞ − alg . These constructions are followed by a section dedicated to the proof of theorems 7 and 10using standard transversality arguments. We show in our last section on signs and orientations,that we have in fact defined a twisted Ω BAs -algebra structure on the Morse cochains, and a twisted Ω BAs -morphism between two Morse cochains complexes : when the manifold M is odd-dimensional,the word "twisted" can be dropped. Definition 39. A twisted A ∞ -algebra is a dg- Z -module A endowed with two different differentials ∂ and ∂ , and a sequence of degree − n operations m n : A ⊗ n → A such that [ ∂, m n ] = − (cid:88) i + i + i = n (cid:54) i (cid:54) n − ( − i + i i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) , where [ ∂, · ] denotes the bracket for the maps ( A ⊗ n , ∂ ) → ( A, ∂ ) . A twisted Ω BAs -algebra and a twisted Ω BAs -morphism are defined similarly.
IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 7
This last section gives us the opportunity to recall in detail the basic method to compute the relationssatisfied by algebraic operations defined in the context of Morse theory or symplectic topology :counting the points on the boundary of an oriented 1-dimensional manifold. We also introduce theviewpoint of signed short exact sequences of vector bundles, to compute the signs contained in theorientations of our moduli spaces. We moreover pay a particular attention tFinally, the third and last part is composed of a series of developments on the algebraic andgeometric constructions performed in the first two parts. We show in particular that :
Proposition 21.
The twisted Ω BAs -morphism µ Y : ( C ∗ ( f ) , m X f t ) −→ ( C ∗ ( g ) , m X g t ) constructed intheorem 12 is a quasi-isomorphism. In the last section we complete some preparatory work for our second article [Maz21], where we willdefine a notion of higher homotopies, or equivalently higher morphisms, between A ∞ and Ω BAs -algebras, and realize them in Morse theory using moduli spaces of perturbed Morse gradient trees.
Acknowledgements . My first thanks go to my advisor Alexandru Oancea, for his continuous helpand support through the settling of this series of papers. I also express my gratitude to Bruno Vallettefor his constant reachability and his suggestions and ideas on the algebra underlying this work. Ispecially thank Jean-Michel Fischer and Guillaume Laplante-Anfossi who repeatedly took the timeto offer explanations on higher algebra and ∞ -categories. I finally adress my thanks to FlorianBertuol, Thomas Massoni, Amiel Peiffer-Smadja and Victor Roca Lucio for useful discussions. THIBAUT MAZUIR
Part 1
Algebra Operadic algebra
This first section is devoted to some basic recollections on operadic algebra, and the particularcase of the operad A ∞ . The specialist already acquainted with these notions will only have to readsections 1.3 and 1.5, which introduce the notion of operadic bimodule and the ( A ∞ , A ∞ ) -operadicbimodule A ∞ − Morph . All the signs of this section are worked out in section 4.2, and will temporarilybe written ± here.We let in the rest of this section C be one the two following monoidal categories : the category ofdifferential graded Z -modules with cohomological convention ( dg − Z − mod , ⊗ ) and the category ofpolytopes ( Poly , × ) , introduced in detail in subsection 2.1.2. We will write ⊗ for the tensor producton C , and I for its identity element.Sections 1.1 and 1.2 are derived from [LV12]. Apart from the operadic bimodule viewpoint, mostof the material presented in sections 1.4 and 1.5 is inspired from [LV12] and [Val14].1.1. Operads.
Definition.
Definition 1. A C -operad P is the data of a collection of objects { P n } n (cid:62) of C together with a unitelement e ∈ P and with compositions P k ⊗ P i ⊗ · · · ⊗ P i k −→ c i ,...,ik P i + ··· + i k which are unital and associative. The objects P n are to be thought as spaces encoding arity n operations while the compositions c i ,...,i k define how to compose these operations together.Operads can be defined in an equivalent fashion using partial compositions instead of total com-positions. An operad is then the data of a collection of objects { P n } n (cid:62) together with a unit element e ∈ P and with partial composition maps ◦ i : P k ⊗ P h −→ P h + k − , (cid:54) i (cid:54) k which are unital and associative. Finally a morphism of operads P → Q is a sequence of maps P n → Q n compatible with the compositions and preserving the identity.1.1.2. Schur functors.
There is a third equivalent definition of operads using the notion of Schurfunctors. Call any collection P = { P n } of objects of C a N -module . To each N -module one canassociate its Schur functor , which is the endofunctor S P : C → C defined as C (cid:55)−→ ∞ (cid:77) n =1 P n ⊗ C ⊗ n . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 9
Given two N -modules P and Q , composing their Schur functors gives the following formula S P ◦ S Q : C −→ ∞ (cid:77) n =1 ( P k ⊗ (cid:77) i + ··· + i k = n Q i ⊗ · · · ⊗ Q i k ) ⊗ C ⊗ n . In other words, there is a N -module associated to the composition of the Schur functors of two N -modules, and it is given by P ◦ Q = { P k ⊗ (cid:77) i + ··· + i k = n Q i ⊗ · · · ⊗ Q i k } n (cid:62) . The category (End( C ) , ◦ , Id C ) , endowed with composition of endofunctors, is a monoidal category.In particular, there is a well-defined notion of monoid in End( C ) . A monoid structure on an endo-functor F : C → C is the data of natural transformations µ F : F ◦ F → F and e : Id C → F , whichsatisfy the usual commutative diagrams for monoids. This viewpoint yields the following equivalentdefinition of an operad. Albeit tedious, it will prove useful in the following section when consideringoperadic modules. Definition 2. A C -operad is the data of a N -module P = { P n } of C together with a monoid structureon its Schur functor S P .1.2. P -algebras. Let A be a dg- Z -module and n (cid:62) . Define the graded Z -module Hom( A ⊗ n , A ) i of i -graded maps A ⊗ n → A , and endow it with the differential [ ∂, f ] = ∂f − ( − | f | f ∂ . The N -module Hom( A ) := Hom( A ⊗ n , A ) in dg- Z -modules can then naturally be endowed with an operadstructure, where composition maps are defined as one expects. Let P be a ( dg − Z − mod ) -operad.A structure of P -algebra on A is defined to be the datum of a morphism of operads P −→ Hom( A ) , that is of a way to interpret each operation of P n in Hom( A ⊗ n , A ) , such that abstract compositionin P coincides with actual composition in Hom( A ) .A morphism of P -algebras between A and B is then simply a dg-map f : A → B , which commuteswith every operation of P n interpreted in A and B . In other words, for every m n ∈ P n , m Bn ◦ f ⊗ n = f ◦ m An . Operadic bimodules.
Definition with Schur functors.
Let now ( D , ⊗ D , I ) be any monoidal category, and ( A, µ A ) and ( B, µ B ) be two monoids in D . Reproducing the diagrams of usual algebra, one can define thenotion of an ( A, B ) -bimodule in D . It is simply the data of an object R of D , together with actionmaps λ : A ⊗ R → R and µ : R ⊗ B → R which are compatible with the product on A and B , acttrivially under their identity elements and satisfy the obvious associativity conditions.Take for instance D to be the category dg − Z − mod . A monoid in D is then a unital associativedifferential graded algebra, and the notion of bimodules in the previous paragraph then coincideswith the usual notion of bimodules over dg-algebras. Definition 3.
Given P and Q two operads seen as their Schur functors S P and S Q , let R = { R n } be a N -module of C seen as its Schur functor S R . A ( P, Q ) -operadic bimodule structure on R is a ( S P , S Q ) -bimodule structure λ : S P ◦ S R → S R and µ : S R ◦ S Q → S R on S R in (End( C ) , ◦ , Id C ) . Operadic bimodules with operations.
This definition is of course of no use for actual compu-tations. Unraveling the definitions, we get an equivalent definition for ( P, Q ) -operadic bimodules. Definition 4. A ( P, Q ) -operadic bimodule structure on R is the data of action-composition maps R k ⊗ Q i ⊗ · · · ⊗ Q i k −→ µ i ,...,ik R i + ··· + i k ,P h ⊗ R j ⊗ · · · ⊗ R j h −→ λ j ,...,jh R j + ··· + j h , which are compatible with one another, with identities, and with compositions in P and Q .Note that the action of Q on R can be reduced to partial action-composition maps ◦ i : R k ⊗ Q h −→ R h + k − (cid:54) i (cid:54) k , as Q has an identity. This cannot be done for the action of P on R , as R does not necessarily havean identity.1.3.3. The (Hom( B ) , Hom( A )) -operadic bimodule Hom(
A, B ) . Let A and B be two dg- Z -modules.We have seen that they each determine an operad, Hom( A ) and Hom( B ) respectively. Then the N -module Hom(
A, B ) := { Hom( A ⊗ n , B ) } n (cid:62) in dg- Z -modules is a (Hom( B ) , Hom( A )) -operadicbimodule where the action-composition maps are defined as one could expect.1.4. The operad A ∞ . Suspension of a dg- Z -module. Let A be a graded Z -module. We define sA to be the graded Z -module ( sA ) i := A i − . In other words, | sa | = | a | − . It is merely a notation that gives a convenientway to handle certain degrees. Note for instance that a degree − n map A ⊗ n → A is simply a mapof degree +1 ( sA ) ⊗ n → sA . This will be used thoroughly in the rest of this part.1.4.2. A ∞ -algebras. Let A be a dg- Z -module with differential m . Recall that we are working in thecohomological framework hence m has degree +1 . A structure of A ∞ -algebra on A is the data ofa collection of maps of degree − n m n : A ⊗ n −→ A , n (cid:62) , extending m and which satisfy the following equations, called the A ∞ -equations [ m , m n ] = (cid:88) i + i + i = n (cid:54) i (cid:54) n − ± m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) . We refer to section 4.2 for the signs. Representing m n as n , this equation reads as [ m , n ] = (cid:88) h + k = n +12 (cid:54) h (cid:54) n − (cid:54) i (cid:54) k ± ki h . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 11
We have in particular that [ m , m ] = 0 , [ m , m ] = m (id ⊗ m − m ⊗ id) . Defining H ∗ ( A ) to be the cohomology of A relative to m , the last two equations show that m descends to an associative product on H ∗ ( A ) . An A ∞ -algebra is simply a correct notion of a dg-algebra whose product is associative up to homotopy. Indeed to define such a notion, we have tokeep track of all the higher homotopies coming with the fact that the product is associative up tohomotopy : these higher homotopies are exactly the m n .1.4.3. The operad A ∞ . The A ∞ -algebra structure defined previously is actually governed by thefollowing operad : Definition 5.
The operad A ∞ is the quasi-free dg − Z − mod -operad generated in arity n by oneoperation m n of degree − n and whose differential is defined by ∂ ( m n ) = (cid:88) i + i + i = n (cid:54) i (cid:54) n − ± m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) . This is often written as A ∞ = F ( , , , · · · ) where ∂ ( n ) = (cid:88) h + k = n +12 (cid:54) h (cid:54) n − (cid:54) i (cid:54) k ± ki h . Recall that quasi-free means that the operad is freely generated by the operations n as a gradedobject, with the additional data of a differential on its generating operations that is non-canonical.We then check that an A ∞ -algebra structure on a dg- Z -module A amounts simply to a morphismof operads A ∞ → Hom( A ) .1.4.4. The bar construction. A ∞ -algebras can also be defined using the bar construction . Define thereduced tensor coalgebra of a graded Z -module V to be T V := V ⊕ V ⊗ ⊕ · · · endowed with the coassociative comultiplication ∆ T V ( v . . . v n ) := n − (cid:88) i =1 v . . . v i ⊗ v i +1 . . . v n . Then, we have a correspondence (cid:26) collections of morphisms of degree − nm n : A ⊗ n → A , n (cid:62) (cid:27) ←→ (cid:26) collections of morphisms of degree +1 b n : ( sA ) ⊗ n → sA , n (cid:62) (cid:27) (cid:108) (cid:8) coderivations D of degree +1 of T ( sA ) (cid:9) . Indeed, to each map family of maps b n : ( sA ) ⊗ n → sA of degree +1 associate a map D : T ( sA ) → T ( sA ) of degree +1 whose restriction to the ( sA ) ⊗ n summand is given by (cid:88) i + i + i = n ± id ⊗ i ⊗ b i ⊗ id ⊗ i . Then the map D is a coderivation of T ( sA ) .There is a second correspondence collections of morphisms of degree − nm n : A ⊗ n → A , n (cid:62) , satisfying the A ∞ -equations ←→ (cid:26) coderivations D of degree +1 of T ( sA ) such that D = 0 (cid:27) . Hence, the following proposition
Proposition 1.
There is a one-to-one correspondence between A ∞ -algebra structures on A andcoderivations D : T ( sA ) → T ( sA ) of degree +1 which square to 0. A ∞ -morphisms. dg-morphisms between A ∞ -algebras. Using the definition of section 1.2, a morphism betweentwo A ∞ -algebras A and B is simply a dg-morphism f : A → B which is compatible with all the m n . This notion of morphism is however not satisfactory from an homotopy-theoretic point of view.Indeed, an A ∞ -algebra being an algebra whose product is associative up to homotopy, the correcthomotopy notion of a morphism between two A ∞ -algebras would be that of a map which preservesthe product m up to homotopy, i.e. of a dg-morphism f : A → B together with higher coherenthomotopies, the first one satisfying [ ∂, f ] = f m A − m B ( f ⊗ f ) . A ∞ -morphisms. Definition 6. An A ∞ -morphism between two A ∞ -algebras A and B is a dg-coalgebra morphism F : ( T ( sA ) , D A ) → ( T ( sB ) , D B ) between their bar constructions.As previously, we have a one-to-one correspondence (cid:26) collections of morphisms of degree − nf n : A ⊗ n → B , n (cid:62) , (cid:27) ←→ (cid:26) morphisms of graded coalgebras F : T ( sA ) → T ( sB ) (cid:27) . The component of F mapping ( sA ) ⊗ n to ( sB ) ⊗ s is given by (cid:88) i + ··· + i s = n ± f i ⊗ · · · ⊗ f i s . A coalgebra morphism preserves the differential if and only if for all n (cid:62) , (cid:88) i + i + i = n ± f i +1+ i (id ⊗ i ⊗ m Ai ⊗ id ⊗ i ) = (cid:88) i + ··· + i s = n ± m Bs ( f i ⊗ · · · ⊗ f i s ) . ( (cid:63) )This yields the following equivalent definition : Definition 7. An A ∞ -morphism between two A ∞ -algebras A and B is a family of maps f n : A ⊗ n → B of degree − n satisfying equations (cid:63) . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 13
See section 4.2 for signs. We check that we recover in particular [ ∂, f ] = f m A − m B ( f ⊗ f ) . As a result, an A ∞ -morphism of A ∞ -algebras induces a morphism of associative algebras on thelevel of cohomology. An A ∞ -quasi-isomorphism is then defined to be an A ∞ -morphism inducing anisomorphism in cohomology.1.5.3. Composing A ∞ -morphisms. Given two coalgebra morphisms F : T V → T W and G : T W → T Z , the family of morphisms associated to G ◦ F is given by ( G ◦ F ) n := (cid:88) i + ··· + i s = n ± g s ( f i ⊗ · · · ⊗ f i s ) . Hence, the composition of two A ∞ -morphisms f : A → B and g : B → C is defined to be ( g ◦ f ) n := (cid:88) i + ··· + i s = n ± g s ( f i ⊗ · · · ⊗ f i s ) . The ( A ∞ , A ∞ )-operadic bimodule encoding A ∞ -morphisms. In fact there is an ( A ∞ , A ∞ )-operadic bimodule encoding the notion of A ∞ -morphisms of A ∞ -algebras. Definition 8.
The operadic bimodule A ∞ − Morph is the ( A ∞ , A ∞ ) -operadic bimodule generatedin arity n by one operation f n of degree − n and whose differential is defined by ∂ ( f n ) = (cid:88) i + i + i = ni (cid:62) ± f i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) + (cid:88) i + ··· + i s = ns (cid:62) ± m s ( f i ⊗ · · · ⊗ f i s ) . Representing the generating operations of the operad A ∞ acting on the right in blue n andthe ones of the operad A ∞ acting on the left in red n , we represent f n by n . This operadicbimodule can then be written as A ∞ − Morph = F A ∞ ,A ∞ ( , , , , · · · ) , with differential defined as ∂ ( n ) = (cid:88) h + k = n +11 (cid:54) i (cid:54) kh (cid:62) ± ki h + (cid:88) i + ··· + i s = ns (cid:62) ± i s i . Consider A and B two A ∞ -algebras, which we can see as two morphisms of operads A ∞ → Hom( A ) and A ∞ → Hom( B ) . Recall from subsection 1.3.3 that Hom(
A, B ) is a (Hom( B ) , Hom( A )) -operadic bimodule. The two previous morphisms of operads make Hom(
A, B ) into an ( A ∞ , A ∞ )-operadic bimodule. An A ∞ -morphism between A and B is then simply a morphism of ( A ∞ , A ∞ )-operadic bimodules A ∞ − Morph −→ Hom(
A, B ) . It is in that sense that A ∞ − Morph is the ( A ∞ , A ∞ )-operadic bimodule encoding the notion of A ∞ -morphisms of A ∞ -algebras. The framework of two-colored operads.
In fact, our choice of notation n reveals that thenatural framework to work with the operad A ∞ and the operadic bimodule A ∞ − Morph is providedby a two-colored operad. We won’t dwell on this notion, but simply mention that this approachwould amount to define the quasi-free two-colored operad A ∞ := F ( , , , · · · , , , , · · · , , , , , · · · ) , where the differential on the generating operations is given by the previous formulae.1.6. Homotopy theory of A ∞ -algebras. A ∞ -algebras with A ∞ -morphisms between them providea framework that behaves well with respect to homotopy-theoretic constructions. They are thecorrect notion to consider when studying homotopy theory of associative algebras. This is becausethe two-colored operad A ∞ is a resolution A ∞ ˜ −→ As , of the two-colored operad encoding associative algebras with morphisms of algebras, and a quasi-free object in the model category of two-colored operads in dg- Z -modules, hence a fibrant-cofibrantobject in this model category. We illustrate these statements with two fundamental theorems. Werefer moreover to [Mar06] for a more general version of theorem 1. Theorem 1 (Homotopy transfer theorem [Val14]) . Let ( A, ∂ A ) and ( H, ∂ H ) be two cochain com-plexes. Suppose that H is a homotopy retract of A , that is that they fit into a diagram ( A, ∂ A ) ( H, ∂ H ) , h pi where id A − ip = [ ∂, h ] and pi = id H . Then if ( A, ∂ A ) is endowed with an associative algebrastructure, H can be made into an A ∞ -algebra such that i and p extend to A ∞ -morphisms, that arethen A ∞ -quasi-isomorphisms. Theorem 2 (Fundamental theorem of A ∞ -quasi-isomorphisms [LH02]) . For every A ∞ -quasi-isomor-phism f : A → B there exists an A ∞ -quasi-isomorphism B → A which inverts f on the level ofcohomology. Poly -operads
We recall in the first section the monoidal category
Poly defined in [MTTV19], which yieldsthe good framework to handle operadic calculus in a category whose objects are polytopes. Wethen introduce in sections 2.2 and 2.3 the two main combinatorial objects of this article : the associahedra and the multiplihedra , which are polytopes that respectively encode A ∞ -algebras and A ∞ -morphisms between them. Explicit realizations of the associahedra and the multiplihedra willbe given in sections 4.3 and 4.4.2.1. Three monoidal categories and their operadic algebra.
IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 15
Differential graded Z -modules and CW-complexes. Consider dg − Z − mod to be the categorywith objects differential graded Z -modules with cohomological convention, and morphisms the mor-phisms of dg- Z -modules. It is a monoidal category with the classical tensor product of dg- Z -modulesand unit the underlying field seen as a dg- Z -module concentrated in degree 0.Likewise, define CW to be the category whose objects are finite CW-complexes and whose mor-phisms are CW-maps between CW-complexes. This category is again a monoidal category withproduct the usual cartesian product and unit the point ∗ . The cellular chain functor C cell ∗ : CW → dg − Z − mod is then strong monoidal, i.e. it satisfies C cell ∗ ( P × Q ) = C cell ∗ ( P ) ⊗ C cell ∗ ( Q ) . To be consistent with the cohomological degree convention on A ∞ -algebras, we will actually workwith the strong monoidal functor C cell −∗ : CW −→ dg − Z − mod , where C cell −∗ ( P ) is simply the Z -module C cell ∗ ( P ) taken with its opposite grading.2.1.2. The category of polytopes ( [MTTV19] ). Define a polytope to be the convex hull of a finitenumber of points in a Euclidean space R n . A polytopal complex is then a finite collection P ofpolytopes satisfying three conditions :(i) ∅ ∈ P ,(ii) if P ∈ P then all the faces of P are also in P ,(iii) if P and Q are two polytopes of P then the intersection P ∩ Q belongs to P .The realisation of a polytopal complex is simply |P| := (cid:91) P ∈P P .
Given P a polytope, we say in particular that a polytopal complex Q is a polytopal subdivision of P if |Q| = P . Every polytope P comes with a polytopal complex L ( P ) consisting of all its faces,which realizes a polytopal subdivision of P .Following [MTTV19], we then define the category Poly as :
Objects.
Polytopes.
Morphisms.
A continuous map f : P → Q which is a homeomorphism P → |D| where D is a polytopal subcomplex of L ( Q ) and f − ( D ) is a polytopal subdivision of P . Such a mapwill be called a polytopal map .This is a monoidal category with product the usual cartesian product and unit the polytope reducedto a point ∗ . It is in fact a monoidal subcategory of CW . Note that the category Poly is usuallydefined with morphisms being affine maps : the larger class of morphisms introduced above allowsfor the flexibility necessary to define
Poly -operads.2.1.3.
From operadic algebra in
Poly to operadic algebra in dg − Z − mod . Let { X n } be a Poly -operad, that is a collection of polytopes X n together with polytopal maps ◦ i : X k × X h −→ X h + k − , satisfying the compatibility conditions of partial compositions. Then, the functor C cell −∗ yields a new dg − Z − mod -operad { P n } defined by P n := C cell −∗ ( X n ) and whose partial compositions are ◦ i : C cell −∗ ( X k ) ⊗ C cell −∗ ( X h ) −→ C cell −∗ ( X k × X h ) −→ C cell −∗ ( X h + k − ) . In the same way, let { X n } and { Y n } be two Poly -operads, and { Z n } be a ( { X n } , { Y n } ) -operadicbimodule, that is a collection of polytopes { Z n } together with polytopal action-composition maps X s × Z i × · · · × Z i s −→ Z i + ··· + i s ,Z k × Y h −→ ◦ i Z h + k − , which are compatible with the composition maps of { X n } and { Y n } . Then, the functor C cell −∗ yieldsa new operadic-bimodule in dg − Z − mod as follows. Denote P n = C cell −∗ ( X n ) and Q n = C cell −∗ ( Y n ) .These are both operads in dg − Z − mod . Defining R n := C cell −∗ ( Z n ) , this is a ( P, Q ) -operadic bimod-ule with action-composition maps defined by C cell −∗ ( X s ) ⊗ C cell −∗ ( Z i ) ⊗ · · · ⊗ C cell −∗ ( Z i s ) −→ C cell −∗ ( X s × Z i × · · · × Z i s ) −→ C cell −∗ ( Z i + ··· + i s ) ,C cell −∗ ( Z k ) ⊗ C cell −∗ ( Y h ) −→ C cell −∗ ( Z k × Y h ) −→ C cell −∗ ( Z h + k − ) . The associahedra.
The dg − Z − mod -operad A ∞ actually stems from a Poly -operad :
Theorem 3 ([MTTV19]) . There exists a collection of polytopes, called the associahedra and denoted { K n } , endowed with a structure of operad in the category Poly and whose image under the functor C cell −∗ yields the operad A ∞ . We refer to section 4.3 in the appendix for a detailed construction and a proof that A ∞ ( n ) = C cell −∗ ( K n ) , and only list noteworthy properties of these polytopes in the following paragraphs.As A ∞ ( n ) = C cell −∗ ( K n ) , we know that K n has to have a unique cell [ K n ] of dimension n − whoseimage under ∂ cell is the A ∞ -equation, that is such that ∂ cell [ K n ] = (cid:88) ± ◦ i ([ K k ] ⊗ [ K h ]) . In fact, these polytopes are constructed such that the boundary of K n is exactly ∂K n = (cid:91) h + k = n +12 (cid:54) h (cid:54) n − (cid:91) (cid:54) i (cid:54) k K k × i K h , where × i is in fact the standard × cartesian product, and such that partial compositions are thensimply polytopal inclusions of K k × K h in the boundary of K h + k − .The first three associahedra K , K and K are represented in figure 3, labeling their cells by theoperations they define in A ∞ when seen in C cell −∗ ( K n ) .2.3. The multiplihedra.
Just like the operad A ∞ , the dg − Z − mod -operadic bimodule A ∞ − Morph is the image under the functor C cell −∗ of a Poly -operadic bimodule :
IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 17
Figure 3.
The associahedra K , K and K Theorem 4 ([MV]) . There exists a collection of polytopes, called the multiplihedra and denoted { J n } , endowed with a structure of ( { K n } , { K n } ) -operadic bimodule, i.e. with polytopal action-composition maps K s × J i × · · · × J i s µ −→ J i + ··· + i s ,J k × K h −→ ◦ i J h + k − , whose image under the functor C cell −∗ yields the ( A ∞ , A ∞ ) -operadic bimodule A ∞ − Morph . We refer this time to section 4.4 for details and conclude again by listing the main noteworthyproperties of the J n . Knowing that A ∞ − Morph( n ) = C cell −∗ ( J n ) , we know that J n has to have aunique n − -dimensional cell [ J n ] whose image under ∂ cell is the A ∞ -equation for A ∞ -morphisms,that is such that ∂ cell [ J n ] = (cid:88) ± ◦ i ([ J k ] ⊗ [ K h ]) + (cid:88) ± µ ([ K s ] ⊗ [ J i ] ⊗ · · · ⊗ [ J i s ]) . In fact, the polytopes J n have the following properties(i) the boundary of J n is exactly ∂J n = (cid:91) h + k = n +1 h (cid:62) (cid:91) (cid:54) i (cid:54) k J k × i K h ∪ (cid:91) i + ··· + i s = ns (cid:62) K s × J i × · · · × J i s , where × k is the standard cartesian product × ,(ii) action-compositions are polytopal inclusions of faces in the boundary of J n .The first three polytopes J , J and J are represented in figure 4, labeling their cells by theoperations they define in A ∞ − Morph .3.
Moduli spaces of metric trees
The associahedra and the multiplihedra are the polytopes governing the structures of A ∞ -algebrasand A ∞ -morphisms between them. We show in this section that these polytopes can in fact berealized as geometric moduli spaces : the associahedra are the compactified moduli spaces of stable Figure 4.
The multiplihedra J , J and J metric ribbon trees T n , while the multiplihedra are the compactified moduli spaces of stable two-colored metric ribbon trees CT n .These moduli spaces will come with two cell decompositions : their A ∞ -cell decomposition,corresponding to the cell decomposition of the associahedra (resp. multiplihedra), and a thinnerdecomposition, called the Ω BAs -cell decomposition. This second cell decomposition recovers theoperad Ω BAs in the case of T n , and an (Ω BAs, Ω BAs ) -operadic bimodule denoted Ω BAs − Morph in the case of CT n . They are respectively related to the operad A ∞ and the operadic bimodule A ∞ − Morph by a morphism of operads A ∞ → Ω BAs and a morphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph (propositions 2, 3 and 4).3.1.
The associahedra and metric ribbon trees.
We refer to section 2 of [MW10] and section 7of [Abo11] for the moduli space viewpoint on the associahedra.3.1.1.
Definitions.
We begin by giving the definitions of the trees we will need in the rest of thesection. The best way to understand them is with the examples depicted in figure 5.
Definition 9. (i) A (rooted) ribbon tree , is the data of a tree together with a cyclic orderingon the edges at each vertex of the tree and a distinguished vertex adjacent to an externaledge called the root . This external edge is then called the outgoing edge , while all the otherexternal edges are called the incoming edges . For a ribbon tree t , we will write E ( t ) for theset of its internal edges, E ( t ) for the set of all its edges, and e ( t ) for its number of internaledges.(ii) A metric ribbon tree is the data of a ribbon tree, together with a length l e ∈ ]0 , + ∞ [ for eachof its internal edges e . The external edges are thought as having length equal to + ∞ .(iii) A ribbon tree is called stable if all its inner vertices are at least trivalent. It is called binary if all its inner vertices are trivalent. We denote sRT n the set of all stable ribbon trees, and BRT n the set of all binary ribbon trees. Note in particular that for a binary tree t ∈ BRT n we have that e ( t ) = n − .3.1.2. Moduli spaces of stable metric ribbon trees.
IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 19
A ribbon tree l l A metric ribbon tree l l A stable metricribbon tree l l A binary metricribbon tree
Figure 5
Definition 10.
Define T n to be moduli space of stable metric ribbon trees with n incoming edges .For each stable ribbon tree type t , we define moreover T n ( t ) ⊂ T n to be the moduli space T n ( t ) := { stable metric ribbon trees of type t } . We then have that T n = (cid:91) t ∈ sRT n T n ( t ) . Writing e ( t ) the number of internal edges for a ribbon tree of type t , each T n ( t ) is naturally topolo-gized as ]0 , + ∞ [ e ( t ) , and they form a stratification of T n . This is illustrated in figures 6 and 7.Interpreting a length in ]0 , + ∞ [ e ( t ) which goes towards 0 as the contraction of the correspondingedge of t , the strata T n ( t ) can in fact be consistently glued together. With this observation, onecan prove that the space T n is in fact itself homeomorphic to R n − . Allowing lengths of internaledges to go to + ∞ , this moduli space can be compactified into a ( n − -dimensional CW-complex T n , where T n is seen as its unique ( n − -dimensional stratum. The codimension 1 stratum of thisCW-complex is given by (cid:91) h + k = n +12 (cid:54) h (cid:54) n − (cid:91) (cid:54) i (cid:54) k T k × i T h , where × i is the standard cartesian product × , and the i means that the outgoing edge of a tree in T h connects to the i -th incoming edge of a tree in T k . It corresponds to metric trees with one internaledge of infinite length. More generally, the codimension m stratum is given by metric trees with m internal edges of infinite lengths. Theorem 5.
The moduli space T n is isomorphic as a CW-complex to the associahedron K n . This was first noticed in section 1.4. of Boardman-Vogt [BV73]. See two examples on figure 7.3.1.3.
The second cell decomposition of T n . In fact the previous compactification can be obtainedby first compactifying each cell T n ( t ) individually and then gluing consistently all compactificationstogether. For t ∈ RT n , the stratum T n ( t ) is homeomorphic to ]0 , + ∞ [ e ( t ) and its compactification in T n is homeomorphic to [0 , + ∞ ] e ( t ) . A length equal to 0 simply corresponds to collapsing one edgeof t and a length equal to + ∞ is interpreted as breaking this edge. This is illustrated in the instanceof a cell of T ( t ) in figure 6. Definition 11. A broken ribbon tree is a ribbon tree some of whose internal edges may be broken.Equivalently, it is the datum of a finite collection of (unbroken) ribbon trees together with a wayof arranging this collection into a new tree (with broken edges). A broken ribbon tree is said to be stable if every unbroken ribbon tree forming it is stable. l l l l l l l l Figure 6.
Compactification of a stratum of T The viewpoint introduced in the previous paragraph yields a new cell decomposition of T n , anexample of which is given in figure 7. Its cells are indexed by broken stable ribbon trees, a brokenstable ribbon tree with i finite internal edges labeling an i -dimensional cell. l l l l l l l l l l l l Figure 7.
The compactified moduli spaces T and T with their celldecomposition by broken stable ribbon tree type3.1.4. The operad Ω BAs . Endowing the T n with this new cell decomposition, the maps T k × T h −→ ◦ i T h + k − are then cellular maps, and hence form a new operad in CW . Taking its image under the functor C cell −∗ yields an operad in dg − Z − mod : the operad Ω BAs . We refer to section 5.1 for a completedescription of this operad and its sign conventions.
Definition 12.
The operad Ω BAs is the quasi-free operad freely generated by the set of stableribbon trees, where a stable ribbon tree t has degree | t | := − e ( t ) . Its differential on a stable ribbontree t is given by the signed sum of all stable ribbon trees obtained from t by breaking or collapsingexactly one of its internal edges. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 21
In other words, it is the quasi-free operad Ω BAs := F ( , , , , · · · , sRT n , · · · ) where for instance | | = − ,∂ ( ) = ± ± ± ± . As the choice of notation Ω BAs suggests, this dg − Z − mod -operad is in fact the bar-cobar con-struction of the operad As , usually denoted Ω BAs . To put it shortly, the classical cobar-bar ad-junction for standard algebras and coalgebras
Ω : conilpotent dg − coalgebras (cid:11) augmented dg − algebras : B , admits a counterpart in the realm of operads and cooperads
Ω : coaugmented dg − cooperads (cid:11) augmented dg − operads : B , and the previously obtained operad is exactly equal to Ω BAs . We refer the curious reader to thededicated section in Loday-Vallette [LV12], for more details on that matter.3.1.5.
From the operad A ∞ to the operad Ω BAs . The dg − Z − mod -operads A ∞ and Ω BAs are infact related by the following proposition :
Proposition 2 ([MS06]) . There exists a morphism of operads A ∞ → Ω BAs given on the generatingoperations of A ∞ by m n (cid:55)−→ (cid:88) t ∈ BRT n ± m t . This morphism stems from the image under the functor C cell −∗ of the identity map id : T n −→ T n refining the cell decomposition on T n . The formula on m n then simply corresponds to associatingto the n − -dimensional cell of T n with the A ∞ -cell decomposition, the signed sum of all n − -dimensional cells of T n with the Ω BAs -cell decomposition.This geometric construction of the morphism A ∞ → Ω BAs is an adaptation of the algebraicconstruction by Markl and Shnider in [MS06] and is detailed in subsection 5.1.4. We moreover pointout that the morphism A ∞ → Ω BAs will be crucial in the rest of this paper. It implies indeed thatin order to construct a structure of A ∞ -algebra on a cochain complex, it is enough to endow it witha structure of Ω BAs -algebra.3.2.
The multiplihedra and two-colored metric ribbon trees.
We have seen in the previoussection that the polytopes K n can be realized as the compactified moduli spaces of stable metricribbon trees. So can the polytopes J n : they are the compactified moduli spaces of stable two-coloredmetric ribbon trees. Two-colored metric ribbon trees.
Definition 13. A stable two-colored metric ribbon tree or stable gauged metric ribbon tree is definedto be a stable metric ribbon tree together with a length λ ∈ R . This length is to be thought ofas a gauge drawn over the metric tree, at distance λ from its root, where the positive direction ispointing down.The gauge divides the tree into two parts, each of which we think of as being colored in a differentcolor. See an instance on figure 8. This definition, despite being visual, will prove difficult tomanipulate when trying to compactify moduli spaces of stable two-colored metric ribbon trees. Wethus proceed to give an equivalent definition, which will provide a natural way of compactifying thesemoduli spaces. The equivalence between the two definitions is depicted on an example in figure 8. Definition 14. (i)
A two-colored ribbon tree is defined to be a ribbon tree together with adistinguished subset of vertices E col ( T ) called the colored vertices . This set is such that,either there is exactly one colored vertex in every non-self crossing path from an incomingedge to the root and none in the path from the outgoing edge to the root, or there is nocolored vertex in any non-self crossing path from an incoming edge to the root and exactlyone in the path from the outgoing edge to the root. These colored vertices are to be thoughtas the intersection points of the gauge with the ribbon tree.(ii) A two-colored ribbon tree is called stable if all its non-colored vertices are at least trivalent.We denote sCRT n the set of all stable two-colored ribbon trees, and CBRT n the set of alltwo-colored binary ribbon trees.(iii) A two-colored metric ribbon tree is the data of a length for all internal edges l e ∈ ]0 , + ∞ [ ,such that the lengths of all non self-crossing paths from a colored vertex to the root are allequal. λl l l l Figure 8.
An example of a stable two-colored metric ribbon tree with the twodefinitions : here l = l = − λ and l = l + l These two definitions of two-colored metric ribbon trees are easily seen to be equivalent, byviewing the colored vertices as the intersection points between the gauge and the edges. In therest of the paper, the notations t c and t g will both stand for a two-colored stable ribbon tree, seenrespectively from the colored vertices and from the gauged viewpoint. The symbol t will then denotethe underlying stable ribbon tree.3.2.2. Moduli spaces of stable two-colored metric ribbon trees.
The results presented in this subsectioncan be found in section 7 of Mau-Woodward [MW10], where they are formulated in the two-coloredviewpoint.
IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 23
Definition 15.
For n (cid:62) , we define CT n to be the moduli space of stable two-colored metric ribbontrees . It has a cell decomposition by stable two-colored ribbon tree type, CT n = (cid:91) t c ∈ sCRT n CT n ( t c ) . We also denote CT := { } the space whose only element is the unique two-colored ribbon tree ofarity 1.The space CT n is homeomorphic to R n − : T n is homeomorphic to R n − and, using the gaugedescription, the datum of a gauge adds a factor R . Allowing again internal edges of metric treesto go to + ∞ by using the second definition for two-colored metric ribbon trees, this moduli space CT n can be compactified into a ( n − -dimensional CW-complex CT n . It has one n − dimensionalstratum given by CT n . Its codimension 1 stratum is given by (cid:91) i + ··· + i s = n T s × CT i × · · · × CT i s ∪ (cid:91) i + i + i = n CT i +1+ i × T i . Two sequences of stable two-colored metric ribbon trees converging in the compactification CT arerepresented in figure 9. l l l l −→ + ∞ l l l l l l = l −→ + ∞ l Figure 9.
Two sequences of stable two-colored metric ribbon trees converging inthe compactification CT Theorem 6 ([MW10]) . The moduli space CT n is isomorphic as a CW-complex to the multiplihedron J n . This theorem is illustrated in figure 11.3.2.3.
The second cell decomposition of CT n . As for T n , the compactified moduli space CT n can beendowed with a thinner cell decomposition. This subsection sums up some of the main results ofsection 5.2, where we provide an extensive study of the strata of this thinner cell decomposition.Let t g be a gauged stable ribbon tree. Writing again e ( t ) for the number of internal edges of theunderlying stable ribbon tree, the stratum CT n ( t g ) is a polyhedral cone in R e ( t )+1 . For instance, CT ( ) = { ( λ, l , l ) such that l > l > < − λ < l , l } . Denote j the number of vertices v of t crossed by the gauge as depicted below v . l λ λ = − llλ l λλl Figure 10.
Compactification of a stratum of CT There is for instance one vertex intersected by the gauge in . The stratum CT n ( t g ) then hasdimension e ( t ) + 1 − j , but is not naturally isomorphic to ]0 , + ∞ [ e ( t )+1 − j , in the sense that itscompactification will not coincide with a ( e ( t ) + 1 − j ) -dimensional cube.Switching now to the colored vertices viewpoint, the polyhedral cones CT n ( t c ) can be compact-ified, by allowing lengths of internal edges to go towards 0 or + ∞ . The compactification CT n issimply obtained by gluing the previous compactifications. See an instance of the compactificationof CT ( ) = { ( λ, l ) such that l > − λ > l } in figure 10.This yields a new cell decomposition of CT n , where each cell is labeled by a broken two-coloredstable ribbon tree. A two-colored stable ribbon tree t g with e ( t ) internal edges and whose gaugecrosses j vertices labels a e ( t ) + 1 − j -dimensional cell. The dimension of a cell labeled by a brokentwo-colored tree can then simply be obtained by adding the dimensions associated to each of thepieces of the broken tree. The cell decompositions for CT and CT are represented in figure 11. λ λ λl λlλlλlλl λl Figure 11.
The compactified moduli spaces CT and CT with their cell decompo-sition by stable two-colored ribbon tree type IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 25
Endowing the moduli spaces T n with their Ω BAs -cell decomposition and the moduli spaces CT n with this new cell decomposition, the maps T s × CT i × · · · × CT i s −→ CT i + ··· + i s , CT k × T h −→ ◦ i CT h + k − , are cellular : the N -module {CT n } is a ( {T n } , {T n } ) -operadic bimodule for this new cell decompo-sition.3.2.4. The operadic bimodule Ω BAs − Morph . The functor C cell −∗ sends the previous operadic bimod-ule in CW to an (Ω BAs, Ω BAs ) -operadic bimodule in dg − Z − mod , that we will denote Ω BAs − Morph . We refer to section 5.3 for a complete description of Ω BAs − Morph and explicit signcomputations.
Definition 16.
The operadic bimodule Ω BAs − Morph is the quasi-free (Ω BAs, Ω BAs ) -operadicbimodule freely generated by the set of two-colored stable ribbon trees. A two-colored stable ribbontree t g with e ( t ) internal edges and whose gauge crosses j vertices has degree | t g | := j − e ( t ) − .The differential of a two-colored stable ribbon tree t c is given by the signed sum of all two-coloredstable ribbon trees obtained from t c under the rule prescribed by the top dimensional strata in theboundary of CT n ( t c ) .Before giving tedious written details for the differential rule, we refer the reader to figure 10 andto the upcoming example. Consider the following two-colored stable ribbon tree . Whichcodimension 1 phenomena can happen ?(i) The gauge can be moved to cross exactly one vertex of : these situations are givenby , and .(ii) An internal edge can break above the gauge : and .(iii) Both internal edges can break below the gauge : .Note that unlike for CT ( ) , no internal edge can collapse in this example : that would be acodimension 2 phenomenon. These two examples list all four possible codimension 1 phenomenathat can happen : the gauge moves to cross exactly one additional vertex of the underlying stableribbon tree (gauge-vertex) ; an internal edge located above the gauge or intersecting it breaks or,when the gauge is below the root, the outgoing edge breaks between the gauge and the root (above-break) ; edges (internal or incoming) that are possibly intersecting the gauge, break below it, suchthat there is exactly one edge breaking in each non-self crossing path from an incoming edge to theroot (below-break) ; an internal edge that does not intersect the gauge collapses (int-collapse).In other words, we constructed the quasi-free (Ω BAs, Ω BAs ) -operadic bimodule Ω BAs − Morph := F Ω BAs, Ω BAs ( , , , , · · · , sCRT n , · · · ) , where for instance | | = − ,∂ ( ) = ± ± ± ± ± ± . Note that the symbol used here is the same as the one used for the only arity 2 generatingoperation of A ∞ − Morph . It will however be clear from the context what stands for in the restof this paper.3.2.5.
From A ∞ − Morph to Ω BAs − Morph . The morphism of operads A ∞ → Ω BAs makes the (Ω BAs, Ω BAs ) -operadic bimodule Ω BAs − Morph into an ( A ∞ , A ∞ ) -operadic bimodule. Proposition 3.
There exists a morphism of ( A ∞ , A ∞ ) -operadic bimodules A ∞ − Morph −→ Ω BAs − Morph given on the generating operations of A ∞ − Morph by f n (cid:55)−→ (cid:88) t g ∈ CBRT n ± f t g . As in subsection 3.1.5, this morphism stems again from the image under the functor C cell −∗ of theidentity morphism on CT n refining its cell decomposition. The formula for f n is obtained by sendingthe n − -dimensional cell of CT n appearing in the A ∞ − Morph -cell decomposition, to the signedsum of all n − -dimensional cells CT n appearing in the Ω BAs − Morph -cell decomposition. Werefer to subsection 5.3.5 for a complete proof and the details on signs.As a result, to construct an A ∞ -morphism between two A ∞ -algebras whose A ∞ -algebra structurecomes from an Ω BAs -algebra structure, it is enough to construct an Ω BAs -morphism between them.In other words, defining A ∞ − alg the category of A ∞ -algebras with A ∞ -morphisms between themand ΩBAs − alg the category of Ω BAs -algebras with Ω BAs -morphisms between them.
Proposition 4.
The morphism of operads A ∞ → Ω BAs and the morphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph induce a functor
ΩBAs − alg −→ A ∞ − alg . Résumé.
The moduli space of stable metric ribbon trees T n can be compactified by allowinglengths of internal edges to go towards + ∞ . This compactification comes with two cell decompo-sitions. The first one, by considering the moduli spaces T n as ( n − -dimensional strata, yields aCW-complex isomorphic to the associahedron K n . Its realization under the functor C cell −∗ then yieldsthe operad A ∞ . The second one is obtained by considering the stratification of T n by strata labeledby stable ribbon tree types. It is sent under the functor C cell −∗ to the operad Ω BAs . These twooperads in dg − Z − mod are then related by a morphism of operads A ∞ → Ω BAs .The moduli space of stable two-colored metric ribbon trees CT n can be compactified by allowinglengths to go towards + ∞ . There are again two cell decompositions for this compactification. Con-sidering the moduli spaces CT n as ( n − -dimensional strata yields a first CW-complex isomorphic tothe multiplihedron J n . Its image under C cell −∗ is the ( A ∞ , A ∞ ) -operadic bimodule A ∞ − Morph . Like-wise, considering the stratification of CT n by strata labeled by two-colored stable ribbon tree types,we obtain a second cell decomposition. The functor C cell −∗ sends it to the (Ω BAs, Ω BAs ) -operadicbimodule Ω BAs − Morph . The morphism of operads A ∞ → Ω BAs makes Ω BAs − Morph into a
IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 27 ( A ∞ , A ∞ ) -operadic bimodule. It is related to A ∞ − Morph by a morphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph , inducing itself a functor
ΩBAs − alg → A ∞ − alg .4. Signs and polytopes for A ∞ -algebras and A ∞ -morphisms The goal of this section is twofold : work out all the signs written as ± in the A ∞ -equationsin section 1 and provide explicit realizations for the associahedra and multiplihedra as polytopes.We begin by introducing the basic Koszul sign rules to work in a graded algebraic framework,and explain how to compute signs by comparing orientations on the boundary of a manifold withboundary. We then recall two equivalent sign conventions for A ∞ -algebras and A ∞ -morphisms andshow how they naturally ensue from the bar construction viewpoint. We subsequently detail explicitpolytopal realizations of the associahedra and the multiplihedra, introduced in [MTTV19] and [MV],and conclude by showing that these polytopes contain indeed the A ∞ -sign conventions previouslydefined.4.1. Basic conventions for signs and orientations.
Koszul sign rule.
All formulae in this section will be written using the Koszul sign rule thatwe briefly recall. We will work exclusively with cohomological conventions.Given A and B two dg Z -modules, the differential on A ⊗ B is defined as ∂ A ⊗ B ( a ⊗ b ) = ∂ A a ⊗ b + ( − | a | a ⊗ ∂ B b . Given A and B two dg Z -modules, we consider the graded Z -module Hom(
A, B ) whose degree r component is given by all maps A → B of degree r . We endow it with the differential ∂ Hom(
A,B ) ( f ) := ∂ B ◦ f − ( − | f | f ◦ ∂ A =: [ ∂, f ] . Given f : A → A (cid:48) and g : B → B (cid:48) two graded maps between dg- Z -modules, we set ( f ⊗ g )( a ⊗ b ) = ( − | g || a | f ( a ) ⊗ g ( b ) . Finally, given f : A → A (cid:48) , f (cid:48) : A (cid:48) → A (cid:48)(cid:48) , g : B → B (cid:48) and g (cid:48) : B (cid:48) → B (cid:48)(cid:48) , we define ( f (cid:48) ⊗ g (cid:48) ) ◦ ( f ⊗ g ) = ( − | g (cid:48) || f | ( f (cid:48) ◦ f ) ⊗ ( g (cid:48) ◦ g ) . We check in particular that with this sign rule, the differential on a tensor product A ⊗ · · · ⊗ A n isgiven by ∂ A ⊗···⊗ A n = n (cid:88) i =1 id A ⊗ · · · ⊗ ∂ A i ⊗ · · · ⊗ id A n . Orientation of the boundary of a manifold with boundary.
Let ( M, ∂M ) be an oriented n -manifold with boundary. We choose to orient its boundary ∂M as follows : given x ∈ ∂M , a basis e , . . . , e n − of T x ( ∂M ) , and an outward pointing vector ν ∈ T x M , the basis e , . . . , e n − is positivelyoriented if and only if the basis ν, e , . . . , e n − is a positively oriented basis of T x M . Note that inthe particular case when the manifold with boundary is a half-space inside the Euclidean space R n ,defined by an inequality n (cid:88) i =1 a i x i (cid:54) C , the vector ( a , . . . , a n ) is outward-pointing.We recover under this convention the classical singular and cubical differentials. Take X a topo-logical space. Given a singular simplex σ : ∆ n → X , its differential is classically defined as ∂ sing ( σ ) := n (cid:88) i =0 ( − i σ i , where σ i stands for the restriction [0 < · · · < ˆ i < · · · < n ] (cid:44) → ∆ n → X . Realizing ∆ n as a polytopein R n and orienting it with the canonical orientation of R n , we check that its boundary reads exactlyas ∂ ∆ n = n (cid:91) i =0 ( − i ∆ n − i , where ∆ n − i is the ( n − -simplex corresponding to the face [0 < · · · < ˆ i < · · · < n ] . The sign ( − i means that the orientation of ∆ n − i induced by its canonical identification with ∆ n − and itsorientation as the boundary of ∆ n , differ by a ( − i sign.Similarly, given a singular cube σ : I n → X , its differential is ∂ cub σ := n (cid:88) i =1 ( − i ( σ i, − σ i, ) , where σ i, denotes the singular cube I n − → X obtained from σ by setting its i -th entry to , and σ i, is defined similarly. We check again that considering I n ⊂ R n as a polytope of R n , its boundaryreads as ∂I n = n (cid:91) i =1 ( − i ( I n − i, ∪ − I n − i, ) , where I n − i, is the face of I n obtained by setting the i -th coordinate equal to 0, and I n − i, is definedlikewise.4.1.3. Coorientations.
Our convention for orienting the boundary of an oriented manifold withboundary ( M, ∂M ) can in fact be rephrased as follows : the boundary ∂M is cooriented by theoutward pointing vector field ν .More generally consider an oriented manifold N and a submanifold S ⊂ N . A coorientation of S is defined to be an orientation of the normal bundle to S . Given any complement bundle ν S to T S in T N | S , T N | S = ν S ⊕ T S , this orientation induces in turn an orientation on ν S , the normal bundle being canonically isomorphicto ν S . The manifold S is then orientable if and only if it is coorientable. This can be proven usingthe first Stiefel-Whitney class for instance. Given a coorientation for S , the induced orientation on S is set to be the one whose concatenation with that of ν S , in the order ( ν S , T S ) , gives the orientationon T N | S . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 29
Signs for A ∞ -algebras and A ∞ -morphisms using the bar construction. There existvarious conventions on signs for A ∞ -algebras and A ∞ -morphisms between them, which can seeminexplicable when met out of context. The goal of this section is twofold : to give a comprehensiveaccount of the two sign conventions coming from the bar construction, and to state our choice ofsigns for the rest of the paper. The eager reader can straightaway jump to subsection 4.2.4, whereour choice of signs is given.4.2.1. A ∞ -algebras. We will first be interested in the following two sign conventions for A ∞ -algebras: [ m , m n ] = − (cid:88) i + i + i = n (cid:54) i (cid:54) n − ( − i i + i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) , (A) [ m , m n ] = − (cid:88) i + i + i = n (cid:54) i (cid:54) n − ( − i + i i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) , (B)which can we rewritten as (cid:88) i + i + i = n ( − i i + i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) = 0 , (A) (cid:88) i + i + i = n ( − i + i i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) = 0 . (B)First, note that these two sign conventions are equivalent in the following sense : given a sequenceof operations m n : A ⊗ n → A satisfying equations (A), we check that the operations m (cid:48) n := ( − n ) m n satisfy equations (B). This sign change does not come out of the blue, and appears in the followingproof that these equations come indeed from the bar construction.Introduce the suspension and desuspension maps s : A −→ sA w : sA → Aa (cid:55)−→ sa sa (cid:55)−→ a , which are respectively of degree − and +1 . We check that with the Koszul sign rule, w ⊗ n ◦ s ⊗ n = ( − n )id A ⊗ n . Then, note that a degree − n map m n : A ⊗ n → A yields a degree +1 map b n := sm n w ⊗ n :( sA ) ⊗ n → sA . Consider now a collection of degree − n maps m n : A ⊗ n → A , and the associateddegree +1 maps b n : ( sA ) ⊗ n → sA . Denoting D the unique coderivation on T ( sA ) associated to the b n , the equation D = 0 is then equivalent to the equations (cid:88) i + i + i = n b i +1+ i (id ⊗ i ⊗ b i ⊗ id ⊗ i ) = 0 . There are now two ways to unravel the signs from these equations.
The first way consists in simply replacing the b i by their definition. It leads to the (A) signconventions : (cid:88) i + i + i = n b i +1+ i (id ⊗ i ⊗ b i ⊗ id ⊗ i )= (cid:88) i + i + i = n sm i +1+ i ( w ⊗ i ⊗ w ⊗ w ⊗ i )(id ⊗ i ⊗ sm i w ⊗ i ⊗ id ⊗ i )= (cid:88) i + i + i = n ( − i sm i +1+ i ( w ⊗ i ⊗ m i w ⊗ i ⊗ w ⊗ i )= (cid:88) i + i + i = n ( − i + i i sm i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i )( w ⊗ i ⊗ w ⊗ i ⊗ w ⊗ i )= s (cid:32) (cid:88) i + i + i = n ( − i i + i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) (cid:33) w ⊗ n . The second way consists in first composing and post-composing by w and s ⊗ n and then replacingthe b i by their definition. It leads to the (B) sign conventions and makes the ( − n ) sign changeappear: (cid:88) i + i + i = n wb i +1+ i (id ⊗ i ⊗ b i ⊗ id ⊗ i ) s ⊗ n = (cid:88) i + i + i = n wb i +1+ i (id ⊗ i ⊗ b i ⊗ id ⊗ i )( s ⊗ i ⊗ s ⊗ i ⊗ s ⊗ i )= (cid:88) i + i + i = n ( − i wb i +1+ i ( s ⊗ i ⊗ b i s ⊗ i ⊗ s ⊗ i )= (cid:88) i + i + i = n ( − i wsm i +1+ i w ⊗ i +1+ i ( s ⊗ i ⊗ sm i w ⊗ i s ⊗ i ⊗ s ⊗ i )= (cid:88) i + i + i = n ( − i m i +1+ i w ⊗ i +1+ i ( s ⊗ i ⊗ ( − i ) sm i ⊗ s ⊗ i )= (cid:88) i + i + i = n ( − i + i i m i +1+ i w ⊗ i +1+ i s ⊗ i +1+ i (id ⊗ i ⊗ ( − i ) m i ⊗ id ⊗ i )= (cid:88) i + i + i = n ( − i + i i ( − i i ) m i +1+ i (id ⊗ i ⊗ ( − i ) m i ⊗ id ⊗ i ) . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 31 A ∞ -morphisms. We now dwell into the two sign conventions for A ∞ -morphisms that arecoming with the bar construction viewpoint. They are as follows : [ m , f n ] = (cid:88) i + i + i = ni (cid:62) ( − i i + i f i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) (A) − (cid:88) i + ··· + i s = ns (cid:62) ( − (cid:15) A m s ( f i ⊗ · · · ⊗ f i s ) , [ m , f n ] = (cid:88) i + i + i = ni (cid:62) ( − i + i i f i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) (B) − (cid:88) i + ··· + i s = ns (cid:62) ( − (cid:15) B m s ( f i ⊗ · · · ⊗ f i s ) , which can we rewritten as (cid:88) i + i + i = n ( − i i + i f i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) = (cid:88) i + ··· + i s = n ( − (cid:15) A m s ( f i ⊗ · · · ⊗ f i s ) , (A) (cid:88) i + i + i = n ( − i + i i f i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) = (cid:88) i + ··· + i s = n ( − (cid:15) B m s ( f i ⊗ · · · ⊗ f i s ) , (B)where (cid:15) A = s (cid:88) u =1 i u (cid:32) (cid:88) u Composition of A ∞ -morphisms. Let f n : A ⊗ n → B and g n : B ⊗ n → C be two A ∞ -morphismsunder conventions (A). The arity n component of their composition g ◦ f is defined as (cid:88) i + ··· + i s = n ( − (cid:15) A g s ( f i ⊗ · · · ⊗ f i s ) , (A)where (cid:15) A is as previously.Let f n : A ⊗ n → B and g n : B ⊗ n → C be two A ∞ -morphisms under conventions (B). The arity n component of their composition g ◦ f is this time defined as (cid:88) i + ··· + i s = n ( − (cid:15) B g s ( f i ⊗ · · · ⊗ f i s ) , (B)where (cid:15) B is as previously.We check that in each case, this newly defined morphism satisfies the A ∞ -equations, respectivelyunder the sign conventions (A) and (B). This can again be proven using the bar construction andapplying the previous transformations.4.2.4. Choice of convention in this paper. We will work in the rest of this paper under the set ofconventions (B). The operations m n of an A ∞ -algebra will satisfy equations [ m , m n ] = − (cid:88) i + i + i = n (cid:54) i (cid:54) n − ( − i + i i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) , an A ∞ -morphism between two A ∞ -algebras will satisfy equations [ m , f n ] = (cid:88) i + i + i = ni (cid:62) ( − i + i i f i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) − (cid:88) i + ··· + i s = ns (cid:62) ( − (cid:15) B m s ( f i ⊗ · · · ⊗ f i s ) , and two A ∞ -morphisms will be composed as (cid:88) i + ··· + i s = n ( − (cid:15) B g s ( f i ⊗ · · · ⊗ f i s ) , where (cid:15) B = (cid:80) su =1 ( s − u )(1 − i u ) .This choice of conventions will be accounted for in the next two sections : the signs are the oneswhich arise naturally from the realizations of the associahedra and the multiplihedra à la Loday. Wealso point out that a choice of convention for the signs on A ∞ -algebras completely determines theconventions on A ∞ -morphisms and their composition.4.3. Loday associahedra and signs. A ∞ -structures were introduced for the first time in twoseminal papers by Stasheff on homotopy associative H-spaces [Sta63]. In the first paper of theseries, he defined cell complexes K n ⊂ I n − which govern A n -structures on topological spaces, andhence realize the associahedra as cell complexes. The associahedra were later realized as polytopesby Loday in [Lod04]. They were recently endowed with an operad structure in the category Poly byMasuda, Thomas, Tonks and Vallette in [MTTV19], using the notion of weighted Loday realizations.Following [MTTV19], we explain the construction of these realizations. We then show that thesign convention (B) for A ∞ -algebras is contained in these realizations : this gives a more geometricexplanation of these signs, which does not come from a ( − n ) twist after reading the signs on the IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 33 bar construction. This also provides an explicit proof with signs of the statement in [MTTV19], thatthese polytopes are sent to the operad A ∞ by the functor C cell −∗ (proposition 5). These realizationsmoreover achieve the first step towards constructing the morphism of operads of Markl-Shnider A ∞ → Ω BAs .4.3.1. Realizations of the associahedra à la Loday. Definition 17 ([MTTV19]) . Given n (cid:62) , define a weight ω to be a list of n positive integers ( ω , . . . , ω n ) . The Loday realization of weight ω of K n is defined as the common intersection in R n − of the hyperplane of equation H ω : n − (cid:88) i =1 x i = (cid:88) (cid:54) k The Loday realizations K (1 , and K (1 , , : the lighter grey depicts H ω ,while the darker grey stands for K ω .The Loday realizations K (1 , and K (1 , , are represented in figure 12. The polytope K ω beingdefined as an intersection of half-spaces inside the ( n − -dimensional space H ω , it has dimension n − . In fact, denoting n the weight of length n whose entries are all equal to 1, it is one of themain results of [MTTV19] that the collection of polytopes ( K n ) n (cid:62) can be made into an operad inthe category Poly . The goal of this section is to show the following proposition : Proposition 5. The Loday associahedra contain sign conventions (B) for A ∞ -algebras. That is, after orienting each polytope K n := K n the boundary of K n reads as ∂K n = − (cid:91) i + i + i = n (cid:54) i (cid:54) n − ( − i + i i K i +1+ i × K i , where K i +1+ i × K i is sent to m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) under the functor C cell −∗ . The signsmean that after comparing the product orientation on K i +1+ i × K i induced by the orientations of K i +1+ i and K i , to the orientation of the boundary of K n , they differ by the sign − ( − i + i i .We explain now how to obtain the set-theoretic decomposition of the boundary ∂K n = (cid:91) i + i + i = n (cid:54) i (cid:54) n − K i +1+ i × K i , and inspect the signs in the next section.The top dimensional strata in the boundary of some K ω are obtained by allowing exactly one ofthe inequalities x i +1 + · · · + x i + i − (cid:62) (cid:88) i +1 (cid:54) k The directing hyperplane H ω of the affine hyperplane H ω has basis e ωj = (1 , , · · · , , − j +1 , , · · · , , where − is in the j + 1 -th spot, and we add a superscript ω for later use. We choose this basis as apositively oriented basis for H ω : this defines our orientation of K ω . Choosing any ( a , . . . , a n − ) ∈ H ω , the basis e ωj parametrizes H ω under the map ( y , . . . , y n − ) (cid:55)−→ ( n − (cid:88) j =1 y j + a , − y + a , . . . , − y n − + a n − ) . Hence in the coordinates of the basis e ωj , the half-space H ω ∩ D i ,i ,i reads aswhen i = 0 : − y i − − · · · − y n − (cid:54) C , when i (cid:62) : y i + · · · + y i + i − (cid:54) C , where C denotes some constant that we are not interested in. Hence, in the basis e ωj , an outwardpointing vector for the boundary H ω ∩ H i ,i ,i iswhen i = 0 : ν := (0 , . . . , , − i − , . . . , − n − ) , when i (cid:62) : ν := (0 , . . . , , i , . . . , i + i − , , . . . , . We have chosen orienting bases for the directing hyperplanes H ω , and computed all outwardpointing vectors for the boundaries in these bases. It only remains to study the image of thesebases under the maps θ . We write e ωj for the orienting basis of K ω and e (cid:101) ωj for the one of K (cid:101) ω . Wedistinguish two cases. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 35 When i = 0 , the map θ reads as θ ( x , . . . , x i , y , . . . , y i − ) = ( y , . . . , y i − , x , . . . , x i ) , and we compute that : θ ( e ωj ) = − e ωi − + e ωj + i − θ ( e (cid:101) ωj ) = e ωj . The determinant then has value det e ωj (cid:16) ν, θ ( e ωj ) , θ ( e (cid:101) ωj ) (cid:17) = − i ( − i i . Thus, we recover the − ( − i + i i K i +1+ i × K i oriented component of the boundary.When i (cid:62) , the map θ now reads as θ ( x , . . . , x i , y , . . . , y i − ) = ( x , . . . , x i , y , . . . , y i − , x i +1 , . . . , x i + i ) , and we compute that : j (cid:54) i − , θ ( e ωj ) = e ωj j (cid:62) i , θ ( e ωj ) = e ωj + i − θ ( e (cid:101) ωj ) = e ωj + i − e ωi . This time, det e ωj (cid:16) ν, θ ( e ωj ) , θ ( e (cid:101) ωj ) (cid:17) = − ( i − − i + i i . We find again the − ( − i + i i K i +1+ i × K i oriented component of the boundary, which concludesthe proof of proposition 5.4.4. Forcey-Loday multiplihedra and signs. While Stasheff gives an explicit construction ofthe cell complexes realizing the associahedra in I n − , analogous realizations of the multiplihedra arenot known to this day. The multiplihedra can however be realized as polytopes, using a method à laLoday. This was first proven in Forcey [For08] and later adapted in an upcoming paper by Masudaand Vallette [MV], which uses again the notion of weighted Loday realizations.The goal of this section is to show that the sign convention (B) for A ∞ -morphisms is naturallycontained in the weighted Loday realizations of Masuda-Vallette. In this regard, we lay out theexplicit construction of [MV], and follow the same lines of proof as in the previous section. This alsoprovides a proof with signs that these polytopes are sent to the operadic bimodule A ∞ − Morph bythe functor C cell −∗ (proposition 6).4.4.1. Forcey-Loday realizations of the multiplihedra. Definition 18 ([MV]) . Given n (cid:62) , choose a weight ω = ( ω , . . . , ω n ) . The Forcey-Loday realization of weight ω of J n is defined as the intersection in R n − of the half-spaces of equation D i ,i ,i : x i +1 + · · · + x i + i − (cid:62) (cid:88) i +1 (cid:54) k The Forcey-Loday realizations J (1 , , and J (1 , , , The Forcey-Loday realizations J (1 , , and J (1 , , , are depicted in figure 13. The polytope J ω being an intersection of half-spaces in R n − , it has dimension n − . Setting J n := J n , Masuda andVallette prove in [MV] that the collection of polytopes { J n } n (cid:62) can be made into a ( { K n } , { K n } ) -operadic bimodule in the category Poly . Proposition 6. The Forcey-Loday realizations contain sign conventions (B) for A ∞ -morphisms. More precisely our goal is to prove that, after orienting the K n as before and choosing an orien-tation for the J n , the boundary of J n reads as ∂J n = (cid:91) i + i + i = ni (cid:62) ( − i + i i J i +1+ i × K i ∪ − (cid:91) i + ··· + i s = ns (cid:62) ( − (cid:15) B K s × J i × · · · × J i s , where (cid:15) B is as in subsection 4.2.4 ; K i +1+ i × K i is sent to f i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) while K s × J i × · · · × J i s is sent to m s ( f i ⊗ · · · ⊗ f i s ) by the functor C cell −∗ .We conclude this section with a proof of the set-theoretic equality for the boundary ∂J n = (cid:91) i + i + i = ni (cid:62) J i +1+ i × K i ∪ (cid:91) i + ··· + i s = ns (cid:62) K s × J i × · · · × J i s , and postpone the processing of signs to the next subsection. The top dimensional strata in theboundary of a J ω are obtained by allowing exactly one of the inequalities x i +1 + · · · + x i + i − (cid:62) (cid:88) i +1 (cid:54) k We set the orientation on R n − , and hence on J ω ,to be such that the vectors f ωj := (0 , , · · · , , − j , , · · · , , define a positively oriented basis of R n − . In the coordinates of the basis f ωj , the half-space D i ,i ,i reads as z i +1 + · · · + z i + i − (cid:54) − (cid:88) i +1 (cid:54) k This section completes section 3 by explicitly describing the two families of moduli spaces of metrictrees T n ( t ) and CT n ( t g ) , working out the induced signs for Ω BAs -algebras and Ω BAs -morphismsand eventually constructing the morphisms of propositionsé 2 and 3.More precisely, we begin by recalling the definition of the operad Ω BAs from Markl-Shnider,using the formalism of orientations on broken stable ribbon trees. This establishes a direct link tothe moduli spaces T n ( t ) . Using the fact that the dual decomposition on the associahedron coincideswith its Ω BAs decomposition, we give a new proof of the morphism of operads A ∞ → Ω BAs , thatrelies uniquely on polytopes and not on sign computations. We then attend to the definition of theoperadic bimodule Ω BAs − Morph . This goes through a long and comprehensive study of the signsensuing from orientations of the codimension 1 strata of the compactified moduli spaces CT n ( t g ) .We finally define the morphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph , using againsolely the realizations of the multiplihedra from [MV]. This is an opportunity to state a MacLane’scoherence theorem encoded by the multiplihedra, while the classical MacLane’s coherence theoremon monoidal categories is encoded by the associahedra (see subsection 5.3.4).5.1. The operad Ω BAs . Definition of the operad Ω BAs . The definition of the operad Ω BAs that we now lay outis the one given by Markl and Shnider in [MS06]. We only expose the material necessary to ourconstruction, and refer to their paper for further details and proofs. In the rest of the section, thenotation t stands for a stable ribbon tree, and the notation t br denotes a broken stable ribbon tree.Observe that a stable ribbon tree is a broken stable ribbon tree with 0 broken edge. As a result,all constructions performed for broken stable ribbon trees in the upcoming subsections will hold inparticular for stable ribbon trees. Definition 19 ([MS06]) . Given a broken stable ribbon tree t br , an ordering of t br is defined to bean ordering of its i finite internal edges e , . . . , e i . Two orderings are said to be equivalent if onepasses from one ordering to the other by an even permutation. An orientation of t br is then definedto be an equivalence class of orderings, and written ω := e ∧ · · · ∧ e i . Each tree t br has exactly twoorientations. Given an orientation ω of t br we will write − ω for the second orientation on t br , calledits opposite orientation . Definition 20 ([MS06]) . Consider the Z -module freely generated by the pairs ( t br , ω ) where t br isa broken stable ribbon tree and ω an orientation of t br . We define the arity n space of operations Ω BAs ( n ) ∗ to be the quotient of this Z -module under the relation ( t br , − ω ) = − ( t br , ω ) . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 39 A pair ( t br , ω ) where t br has i finite internal edges, is defined to have degree − i . The partialcompositions are then ( t br , ω ) ◦ k ( t (cid:48) br , ω (cid:48) ) = ( t br ◦ k t (cid:48) br , ω ∧ ω (cid:48) ) , where the tree t br ◦ k t (cid:48) br is the broken ribbon tree obtained by grafting t (cid:48) br to the k -th incoming edgeof t br , and the edge resulting from the grafting is broken. The differential ∂ Ω BAs on Ω BAs ( n ) ∗ isfinally set to send an element ( t br , e ∧ · · · ∧ e i ) to i (cid:88) j =1 ( − j (( t br /e j , e ∧ · · · ∧ ˆ e j ∧ · · · ∧ e i ) − (( t br ) j , e ∧ · · · ∧ ˆ e j ∧ · · · ∧ e i )) , where t br /e j is the tree obtained from t by collapsing the edge e j and ( t br ) j is the tree obtainedfrom t br by breaking the edge e j . It can be checked that the collection of dg- Z -modules Ω BAs ( n ) ∗ defines indeed an operad in dg − Z − mod .Choosing a distinguished orientation for every stable ribbon tree t ∈ sRT , this definition of theoperad Ω BAs yields the definition as the quasi-free operad F ( , , , , · · · , sRT n , · · · ) , given in subsection 3.1.4. Our definition with the pairs ( t, ω ) , albeit more tedious at first sight,allows however for easier computations of signs.5.1.2. Canonical orientations for the binary ribbon trees ( [MS06] ). For a fixed n (cid:62) , the set ofbinary ribbon trees BRT n can be endowed with a partial order that Tamari introduced in histhesis [Tam54]. Definition 21. The Tamari order on BRT n is the partial order generated by the covering relations t t t t > t t t t where t , t , t and t are binary ribbon trees.The left-hand side in the above covering relation will be called a right-leaning configuration , andthe right-hand side a left-leaning configuration . Hence given two trees t and t (cid:48) in BRT n , the inequality t (cid:62) t (cid:48) holds if and only one can pass from t to t (cid:48) by successive transformations of a right-leaningconfiguration into a left-leaning configuration. For example in the case of BRT , we obtain theHasse diagram in figure 14.The Tamari poset has a unique maximal element and a unique minimal element, respectively givenby the right-leaning and left-leaning combs, denoted t max and t min . Given moreover a binary ribbontree t , its immediate neighbours are by definition the trees obtained from t by either transformingexactly one right-leaning configuration of t into a left-leaning configuration, or transforming exactlyone left-leaning configuration of t into a right-leaning configuration. e e e ∧ e e e − e ∧ e e e e ∧ e = e e − e ∧ e e e e ∧ e e e − e ∧ e Figure 14. On the left, the Hasse diagram of the Tamari poset, where the maximalelement is written at the top. On the right, all the canonical orientations for BRT computed going down the Tamari poset.The canonical orientation on the maximal binary tree is defined as e e n − ω can := e ∧ · · · ∧ e n − . Using the Tamari order, we can now build inductively canonical orientations on all binary trees. Westart at the maximal binary ribbon tree, and use the following rule on the covering relations t t t t e ω = · · · ∧ e ∧ · · · −→ t t t t e − ω = · · · ∧ ( − e ) ∧ · · · , to define the orientations of its immediate neighbours. We then repeat this rule while going downthe Tamari poset until the minimal binary tree is reached. This process is consistent (see subsec-tion 5.3.4), i.e. it does not depend on the path taken in the Tamari poset from the maximal binarytree to the binary tree whose orientation is being defined. A full example for BRT is illustrated infigure 14. Definition 22 ([MS06]) . The orientations obtained under this process are called the canonicalorientations and written ω can .5.1.3. The moduli spaces T n realize the operad Ω BAs . We explained in subsection 3.1.3 that thecompactified moduli space T n comes with a thin cell decomposition, which is labeled by all brokenstable ribbon trees with n incoming edges. Consider then a cell T n ( t br ) ⊂ T n , where t br is a brokenstable ribbon tree. An ordering of its finite internal edges e , . . . , e i induces an isomorphism T n ( t br ) ˜ −→ [0 , + ∞ ] i , where the length l e j is seen as the j -th coordinate in [0 , + ∞ ] i . This ordering induces in particularan orientation on T n ( t br ) , by taking the image of the canonical orientation of ]0 , + ∞ [ i under the IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 41 isomorphism. We check that two orderings of t br define the same orientation on T n ( t br ) if and onlyif they are equivalent : in other words, an orientation of t br amounts to an orientation of T n ( t br ) .Consider now the Z -module freely generated by the pairs ( T n ( t br ) , choice of orientation ω on the cell T n ( t br )) , where t br is a broken stable ribbon tree. The complex C cell −∗ ( T n ) can simply be defined to be thequotient of this Z -module under the relation − ( T n ( t br ) , ω ) = ( T n ( t br ) , − ω ) . The differential of an element ( T n ( t br ) , ω ) is moreover given by the classical cubical differential on [0 , + ∞ ] i . Defining the cell chain complex in this way, it becomes tautological that : Proposition 7. The functor C cell −∗ sends the operad T n to the operad Ω BAs . What’s more, it can be easily seen that given a binary ribbon tree t , the cells labeled by theimmediate neighbours to the tree t in the Tamari order are exactly the cells having a codimension1 stratum in common with the cell T n ( t ) .5.1.4. The morphism of operads A ∞ → Ω BAs . The moduli space T n endowed with its A ∞ -celldecomposition is isomorphic to the Loday realization K n of the associahedron. In fact, tediouscomputations show that under this isomorphism, the Ω BAs -decomposition is sent to the dual sub-division of K n . See appendix C of [LV12] and an illustration in figure 7 for instance. The goal ofthis section is to prove the following proposition : Proposition 8. The map id : ( T n ) A ∞ → ( T n ) Ω BAs is sent under the functor C cell −∗ to the morphismof operads A ∞ → Ω BAs acting as m n (cid:55)−→ (cid:88) t ∈ BRT n ( t, ω can ) . For this purpose, we will work with the Loday realizations of the associahedra. We will show thattaking the restriction of the orientation of K n chosen in section 4.3 to the top dimensional cells ofits dual subdivision yields the canonical orientations on these cells in the T n viewpoint.We begin by proving this statement for the cell labeled by the right-leaning comb t max . Con-sider the orientation on the cell T n ( t max ) induced by the canonical ordering e , . . . , e n − under theisomorphism T n ( t max ) ˜ −→ [0 , + ∞ ] n − . The face of T n ( t max ) associated to the breaking of the i -th edge corresponds to the face H i,n − i, when seen in the Loday polytope. An outward-pointing vector for the face H i,n − i, is moreover ν i := (0 , . . . , , i , . . . , n − ) , where coordinates are taken in the basis e ωj . The orientation defined by the canonical basis of [0 , + ∞ ] n − being exactly the one defined by the ordered list of the outwarding-point vectors to the + ∞ boundary, it is sent to the orientation of the basis ( ν , . . . , ν n − ) in the Loday polytope. Wethen check that det e ωj ( ν j ) = 1 . l e l e l e l f l f l e l f l f l e l e = l f = 0 v e v e v e = − v f v e = v f Figure 15. Gluing the cells T n ( t max ) and T n ( t ) along their common boundary :on this diagram, a vector of the form v e is the vector orienting the axis associated tothe length l e Hence the orientation of K n and the one induced by the canonical orientation are the same for thecell T n ( t max ) .As explained in the previous subsection, the cells labeled by the immediate neighbours of theright-leaning comb t max in the Tamari order are exactly the cells having a codimension 1 stratumin common with this cell. Choose an immediate neighbour t , and write e for the edge that has beencollapsed to obtain the common codimension 1 stratum. We detail the process to obtain the inducedorientation on T n ( t ) following figure 15. Gluing the cells T n ( t max ) and T n ( t ) along their commonboundary, we obtain a new copy of [0 , + ∞ ] n − which can be divided into two halves t max and t .We then orient the total space [0 , + ∞ ] n − as the t max half. Reading the induced orientation on the t half, it is the one obtained from the t max half by reversing the axis associated to the edge e . Byconstruction, this orientation is exactly the one obtained by restricting the global orientation on K n to an orientation on T n ( t ) .Finally, going down the Tamari order, we can read the induced orientation on the top dimensionalcells one immediate neighbour after another. And the rule to do this step-by-step process is exactlythe one given in 5.1.2 on the covering relations. Hence, by construction, the global orientation on K n restricts to the canonical orientations on binary trees, which concludes the proof of proposition 2.5.2. The moduli spaces CT n ( t br,g ) . We give a detailed definition of the moduli spaces of gaugedstable metric ribbon trees CT n ( t g ) , introduced in part 3.2. Building on these explicit realizations,we then thoroughly compute the signs appearing in the codimension 1 strata of the compactifiedmoduli spaces CT n ( t g ) . This yields in particular the signs which will appear in subsection 5.3.1, inthe definition of the differential on the operadic bimodule Ω BAs − Morph .5.2.1. Definition. In the rest of the section, we will write t br,g for a broken gauged stable ribbontree, and t g for an unbroken gauged stable ribbon tree. Definition 23. We set to be the unique stable gauged tree of arity 1, and will call it the trivialgauged tree . We define the underlying broken stable ribbon tree t br of a t br,g to be the ribbon tree IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 43 obtained by first deleting all the in t br,g , and then forgetting all the remaining gauges of t br,g .We refer moreover to a gauge in t br,g which is associated to a non-trivial gauged tree, as a non-trivialgauge of t br,g . Figure 16. An instance of association t br,g (cid:55)→ t br We now define the moduli spaces CT n ( t br,g ) in three steps. Consider a gauged stable ribbon tree t g whose gauge does not intersect any of its vertices. Locally at any vertex directly adjacent to thegauge, the intersection between the gauge and the edges of t corresponds to one of the following twocases v v (cid:48) . Write r for the root, the unique vertex adjacent to the outgoing edge. For a vertex v , we denote d ( r, v ) the distance separating it from the root : the sum of the lengths of the edges appearing inthe unique non self-crossing path going from r to v . Associating lengths l e > to all edges of t , wethen associate the following inequalities to the two above cases − λ > d ( r, v ) − λ < d ( r, v (cid:48) ) . Note that this set of inequalities amounts to seeing the gauge as going towards −∞ when going up,and towards + ∞ as going down. The moduli space CT n ( t g ) is then defined as CT n ( t g ) := (cid:8) ( λ, { l e } e ∈ E ( t ) ) , λ ∈ R , l e > , − λ > d ( r, v ) , − λ < d ( r, v (cid:48) ) (cid:9) , where the set of inequalities on λ is prescribed by the gauged tree t g .Consider now a gauged stable ribbon tree t g whose gauge may intersect some of its vertices. Tothe two previous local pictures, one has to add the case v (cid:48)(cid:48) to which we associate the equality − λ = d ( r, v (cid:48)(cid:48) ) . The moduli space CT n ( t g ) is this time defined as CT n ( t g ) := (cid:8) ( λ, { l e } e ∈ E ( t ) ) , λ ∈ R , l e > , − λ > d ( r, v ) , − λ < d ( r, v (cid:48) ) , − λ = d ( r, v (cid:48)(cid:48) ) (cid:9) , where the set of equalities and inequalities on λ is prescribed by the gauged tree t g .Finally, consider a gauged broken stable ribbon tree t br,g , whose gauges may intersect some of itsvertices. We order the non-trivial unbroken gauged ribbon trees appearing in t br,g from left to right,as t , br t ,i br t g t s, br t s,i s br t sg (cid:124) (cid:123)(cid:122) (cid:125) t br where t , br , . . . , t ,i br , . . . , t s, br , . . . , t s,i s br and t br are broken stable ribbon trees, and the non-trivial un-broken gauged ribbon trees are represented in the picture as gauged corollae t g , . . . , t sg for the sake ofreadability. We write moreover r , . . . , r s and λ , . . . , λ s for their respective roots and gauges. Themoduli space CT n ( t br,g ) is this time defined as CT n ( t br,g ) := (cid:26) ( λ , . . . , λ s , { l e } e ∈ E ( t br ) ) , λ i ∈ R , l e > , − λ i > d ( r i , v ) , − λ i < d ( r i , v (cid:48) ) , − λ i = d ( r i , v (cid:48)(cid:48) ) (cid:27) , where the set of equalities and inequalities on λ i is prescribed by the unbroken gauged tree t ig .5.2.2. Orienting the moduli spaces CT n ( t br,g ) . Definition 24. Define an orientation on a broken gauged stable ribbon tree t br,g , to be an orientation e ∧ · · · ∧ e i on t br .We now explain how to orient the moduli spaces CT n ( t br,g ) , following the previous three stepsapproach. Begin with a gauged stable ribbon tree t g whose gauge does not intersect any of itsvertices. An orientation ω on t g identifies CT n ( t g ) with a polyhedral cone CT n ( t g ) ⊂ ] − ∞ , + ∞ [ × ]0 , + ∞ [ e ( t ) , defined by the inequalities − λ > d ( r, v ) and − λ < d ( r, v (cid:48) ) . This polyhedral cone has dimension e ( t ) + 1 , and we choose to orient it as an open subset of ] − ∞ , + ∞ [ × ]0 , + ∞ [ e ( t ) endowed with itscanonical orientation.Consider now a gauged stable ribbon tree t g whose gauge may intersect some of its vertices. Thistime, an orientation ω on t g identifies CT n ( t g ) with a polyhedral cone CT n ( t g ) ⊂ ] − ∞ , + ∞ [ × ]0 , + ∞ [ e ( t ) , defined by the inequalities − λ > d ( r, v ) and − λ < d ( r, v (cid:48) ) , to which we add the equalities − λ = d ( r, v (cid:48)(cid:48) ) . If there are exactly j gauge-vertex intersections in the gauged tree t g , this polyhedral conehas dimension e ( t ) + 1 − j . Order now the j intersections from left to right v v j , and consider the tree t (cid:48) g obtained by replacing these intersections by v v j . One can see t g as lying in the boundary of t (cid:48) g , by allowing the inequalities − λ > d ( r, v k ) to becomeequalities − λ = d ( r, v k ) for k = 1 , . . . , j . This determines in particular j vectors ν k corresponding tothe outwarding-pointing vectors to the boundary of the half-space − λ (cid:62) d ( r, v k ) . We finally chooseto coorient (and hence orient) CT n ( t g ) inside ] − ∞ , + ∞ [ × ]0 , + ∞ [ e ( t ) with the vectors ( ν , . . . , ν j ) .Lastly, consider a gauged broken stable ribbon tree t br,g , whose gauges may intersect some of itsvertices. Suppose there are exactly s non-trivial unbroken gauged trees t g , . . . , t sg appearing in t br,g ,which are ordered from left to right as previously. Suppose also that in each tree t ig , there are j i gauge-vertex intersections. An orientation ω on t br,g identifies CT n ( t br,g ) with a polyhedral cone CT n ( t br,g ) ⊂ ] − ∞ , + ∞ [ s × ]0 , + ∞ [ e ( t br ) , IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 45 defined by the set of equalities and inequalities on the λ i , and where the factor ] − ∞ , + ∞ [ s cor-responds to ( λ , . . . , λ s ) . This polyhedral cone has dimension e ( t br ) + s − (cid:80) si =1 j i . Now, as in theprevious paragraph, order all gauge-vertex intersections from left to right in every tree t ig , and con-struct a new tree t (cid:48) br,g . Seeing CT n ( t br,g ) as lying in the boundary of CT n ( t (cid:48) br,g ) , this determines againa collection of outward-pointing vectors ν i, , . . . , ν i,j i for i = 1 , . . . , s . We then coorient CT n ( t br,g ) inside ] − ∞ , + ∞ [ s × ]0 , + ∞ [ e ( t br ) with the vectors ( ν , , . . . , ν ,j , . . . , ν s, , . . . , ν s,j s ) . Definition 25. We define CT n ( t br,g , ω ) to be the moduli space CT n ( t br,g ) endowed with the previousorientation.We moreover insist on the fact that for a given broken stable ribbon tree type t br all gauged trees t br,g whose underlying ribbon tree is t br form polyhedral cones ⊂ ] − ∞ , + ∞ [ s × ]0 , + ∞ [ e ( t br ) , and thecollection of these polyhedral cones is a partition of ] − ∞ , + ∞ [ s × ]0 , + ∞ [ e ( t br ) . This is illustratedin figure 17. λl λlλlλl Figure 17 Compactification. Recall from section 3.2 that each broken gauged ribbon tree t br,g can beseen as a broken two-colored ribbon tree t br,c . Using the two-colored metric trees viewpoint, thecompactification of CT n ( t br,c ) is defined by allowing lengths of internal edges to go towards or + ∞ ,where combinatorics are induced by the equalities defined by the colored vertices. The compactifi-cation rule for gauged metric trees is then simply defined by transporting the compactification rulefrom the two-colored viewpoint to the gauged viewpoint. We do not give further details here, as wewon’t need them in our upcoming computations.For a gauged stable ribbon tree t g , the compactified moduli space CT n ( t g ) has codimension1 strata given by the four components (int-collapse), (gauge-vertex), (above-break) and (below-break). Choose an orientation ω for t g . As for the moduli spaces T n ( t, ω ) , the question is now todetermine which signs appear in the boundary of the compactification of the oriented moduli space CT n ( t g , ω ) . We will inspect this matter in the four upcoming sections, computing the signs for eachboundary component. Note that this time the compactification is much more elaborate than thecubical compactification of the T n ( t, ω ) , and as a result we will not be able to write nice and elegantformulae. We will rather give recipes to compute the signs in each case.5.2.4. The (int-collapse) boundary component. Consider a gauged stable ribbon tree t g . The (int-collapse) boundary corresponds to the collapsing of an internal edge that does not intersect thegauge of the tree t . Choosing an ordering ω = e ∧ · · · ∧ e i , suppose that it is the p -th edge of t which collapses. Write moreover ( t/e p ) g for the resulting gauged tree, and ω p := e ∧ · · · ∧ (cid:98) e p ∧ · · · ∧ e i forthe induced ordering on the edges of t/e p .We begin by considering the case of a gauged tree t g whose gauge does not intersect any of itsvertices. Suppose first that the collapsing edge is located above the gauge. A neighbordhood of theboundary can then be parametrized as ] − , × CT n (( t/e p ) g , ω p ) −→ CT n ( t g , ω )( δ, λ, l , . . . , (cid:98) l p , . . . , l i ) (cid:55)−→ ( λ, l , . . . , l p := − δ, . . . , l i ) . This map has sign ( − p +1 , and the component CT n (( t/e p ) g , ω p ) consequently bears a ( − p +1 signin the boundary of CT n ( t g , ω ) .Suppose next that the collapsing edge is located below the gauge. We define a parametrizationof a neighborhood of the boundary ] − , × CT n (( t/e p ) g , ω p ) −→ CT n ( t g , ω ) as follows : λ is sent to λ + δ ; if the edge e q is located directly below a gauge-edge intersection e q , then we send l q to l q − δ ; for all the other edges e q of ( t/e p ) , we send l q to l q ; finally, we set l p := − δ .We check again that this map has sign ( − p +1 . Hence, in general, for a gauged tree t g whose gaugedoes not intersect any of its vertices, the component CT n (( t/e p ) g , ω p ) bears a ( − p +1 sign in theboundary of CT n ( t g , ω ) .Move on to the case of a gauged stable ribbon tree t g whose gauge may intersect some of itsvertices. Order the j gauge-vertex intersections from left to right as depicted in subsection 5.2.2.We are going to distinguish three cases, but will eventually end up with the same sign in each case.Suppose to begin with that the collapsing edge e p is located above the gauge, and is not adjacentto a gauge-vertex intersection. Then, denoting ( t/e p ) (cid:48) g the tree obtained via the same process as t (cid:48) g ,we check that the first parametrization introduced in this section Φ : ] − , × CT n (( t/e p ) (cid:48) g , ω p ) −→ CT n ( t (cid:48) g , ω ) , restricts to a parametrization of a neighborhood of the boundary φ : ] − , × CT n (( t/e p ) g , ω p ) −→ CT n ( t g , ω ) . We also check that Φ sends the outward-pointing vectors ν ( t/e p ) k associated to the gauge-vertex inter-sections in ( t/e p ) g , to the outward-pointing vectors ν tk associated to the gauge-vertex intersections in t g . Computing the sign of φ amounts to computing the sign of Φ and then exchanging the direction δ with the outward-pointing vectors ν t , . . . , ν tj . The total sign is hence ( − p +1+ j .Suppose, as second case, that the collapsing edge e p is located above the gauge, and directlyadjacent to a gauge-vertex intersection. v k e p . We cannot use the trees ( t/e p ) (cid:48) g and t (cid:48) g as in the last paragraph, as the gauge would then cut the edge e p in the gauged tree t (cid:48) g . A small change is required. We form the tree t (cid:48)(cid:48) g as the tree t (cid:48) g , but instead IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 47 of moving the gauge up at the vertex v k , we move it down. The tree ( t/e p ) (cid:48)(cid:48) g is defined similarly.Applying the same argument as previously, we compute again a ( − p +1+ j sign for the boundary.Finally, suppose that the collapsing edge e p is located below the gauge. It may this time bedirectly adjacent to a gauge-vertex intersection. Introducing again the trees ( t/e p ) (cid:48) g and t (cid:48) g , andusing this time the second parametrization introduced in this section, we find a ( − p +1+ j sign forthe boundary. Note that there is a small adjustment to make in the proof for the outward-pointingvectors. Indeed, the outward-pointing vector ν ( t/e p ) k gets again sent to the outward-pointing vector ν tk , except if the edge e p is located in the non-self crossing path going from the vertex v k intersectedby the gauge to the root. For such an intersection, the vector ν ( t/e p ) k is sent to ν tk − e p by the map Φ , where e p is the positive direction for the length l p . Though the vector ν tk − e p is not equal to ν tk , it is still outward-pointing to the half-space − λ (cid:62) d ( r, v k ) . As a result, Φ( ν ( t/e p )1 ) , . . . , Φ( ν ( t/e p ) j ) defines indeed the same coorientation of CT n ( t g , ω ) as ν t , . . . , ν tj . Proposition 9. For a gauged stable ribbon tree t g whose gauge intersects j vertices, the boundarycomponent CT n (( t/e p ) g , ω p ) corresponding to the collapsing of the p -th edge of t bears a ( − p +1+ j sign in the boundary of CT n ( t g , ω ) . The (gauge-vertex) boundary component. Consider a gauged stable ribbon tree t g whose gaugemay intersect some of its vertices. We order the gauge-vertex intersections from left to right asdepicted in subsection 5.2.2. The (gauge-vertex) boundary corresponds to the gauge crossing exactlyone additional vertex of t . We suppose that this intersection takes place between the k -th and k + 1 -th intersections of t g . We write moreover t g for the resulting gauged tree, and introduce again thetree t (cid:48) g of subsection 5.2.2. Proposition 10. Suppose the crossing results from a move . Then the boundary component CT n ( t g , ω ) has sign ( − j + k in the boundary of CT n ( t g , ω ) . Indeed the orientation induced on CT n ( t g , ω ) in the boundary of CT n ( t g , ω ) , is defined by the coori-entation ( ν , . . . , ν k , (cid:98) ν, ν k +1 , . . . , ν j , ν ) inside CT n ( t (cid:48) g , ω ) . The orientation defined by ω on CT n ( t g , ω ) ,is the one defined by the coorientation ( ν , . . . , ν k , ν, ν k +1 , . . . , ν j ) inside CT n ( t (cid:48) g , ω ) . Hence, thesetwo orientations differ by a ( − j + k sign. Proposition 11. Suppose the crossing results from a move . Then the boundary component CT n ( t g , ω ) has sign ( − j + k +1 in the boundary of CT n ( t g , ω ) . Again the orientation induced on CT n ( t g , ω ) in the boundary of CT n ( t g , ω ) , is defined by thecoorientation ( ν , . . . , ν k , (cid:98) ν, ν k +1 , . . . , ν j , − ν ) inside CT n ( t (cid:48) g , ω ) . The orientation defined by ω on CT n ( t g , ω ) , is the one defined by the coorientation ( ν , . . . , ν k , ν, ν k +1 , . . . , ν j ) inside CT n ( t (cid:48) g , ω ) .Hence, these two orientations differ by a ( − j + k +1 sign. The (above-break) boundary component. The (above-break) boundary corresponds either tothe breaking of an internal edge of t , that is located above the gauge or intersects the gauge, or,when the gauge is below the root, to the outgoing edge breaking between the gauge and the root.Choosing an ordering ω = e ∧ · · · ∧ e i , suppose that it is the p -th edge of t which breaks and writemoreover ( t p ) g for the resulting broken gauged tree.We begin by considering the case of a gauged tree t g whose gauge does not intersect any of itsvertices. Suppose first that the breaking edge does not intersect the gauge. A neighborhood of theboundary can then be parametrized as ]0 , + ∞ ] × CT n (( t p ) g , ω p ) −→ CT n ( t g , ω )( δ, λ, l , . . . , (cid:98) l p , . . . , l i ) (cid:55)−→ ( λ, l , . . . , l p := δ, . . . , l i ) . This map has sign ( − p . In the case when the breaking edge does intersect the gauge, a neighbor-dhood of the boundary can be parametrized as ]0 , + ∞ ] × CT n (( t p ) g , ω p ) −→ CT n ( t g , ω )( δ, λ, l , . . . , (cid:98) l p , . . . , l i ) (cid:55)−→ ( λ, l , . . . , l p := δ − λ, . . . , l i ) , where we set this time l p := δ − λ in order for the inequality − λ < d ( r, v (cid:48) ) to hold in this case. Thisparametrization again has sign ( − p .The case of a gauged tree t g whose gauge may intersect some of its vertices is treated as insubsection 5.2.4. We check again that the parametrization maps Φ introduced in the previousparagraph, restrict to parametrizations of a neighborhood of the boundary ]0 , + ∞ ] × CT n (( t p ) g , ω p ) −→ CT n ( t g , ω ) , and that Φ sends moreover the coorientation of CT n (( t p ) g , ω p ) to the coorientation of CT n ( t g , ω ) .These coorientations introduce as previously an additional ( − j sign.Finally, suppose that the gauge of t g intersects its outgoing edge and compute the sign of the(above-break) boundary component corresponding to the gauge going towards + ∞ . A parametriza-tion of a neighborhood of the boundary is simply given by ]0 , + ∞ ] × CT n (( t ) g , ω p ) −→ CT n ( t g , ω )( δ, l , . . . , l i ) (cid:55)−→ ( λ := δ, l , . . . , l i ) . This map has sign . Proposition 12. For a gauged stable ribbon tree t g whose gauge intersects j vertices, the boundarycomponent CT n (( t p ) g , ω p ) corresponding to the breaking of the p -th edge of t bears a ( − p + j sign inthe boundary of CT n ( t g , ω ) , where we set e for the outgoing edge of t . The (below-break) boundary component. The (below-break) boundary corresponds to the break-ing of edges of t that are located below the gauge or intersect it, such that there is exactly one edgebreaking in each non-self crossing path from an incoming edge to the root. Write ( t br ) g for theresulting broken gauged tree. Consider now an ordering ω = e ∧ · · · ∧ e i of t g . We order againfrom left to right the s non-trivial unbroken gauged trees t g , . . . , t sg of ( t br ) g , and denote moreover e j , . . . , e j s the internal edges of t whose breaking produce the trees t g , . . . , t sg . Beware that we donot necessarily have that j < · · · < j s . We assume in the next paragraphs that j = 1 , . . . , j s = s , IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 49 and will explain how to deal with the general case at the end of this section. We set to this extent ω br := e s +1 ∧ · · · ∧ e i .We introduce two more pieces of notation. We will denote E ∞ the set of incoming edges of t which are crossed by the gauge and correspond to the trivial gauged trees in ( t br ) g . In other words,the set of edges which are breaking in the (below-break) boundary component associated to ( t br ) g is E ∞ ∪ { e j , . . . , e j s } . For an edge e , internal or external, we will moreover write w e for the vertexadjacent to e which is closest to the root r of t , and set w u := w e u for u = 1 , . . . , s .Start by considering the case of a gauged tree t g whose gauge does not intersect any of its vertices.Suppose first that among the breaking internal edges, none of them intersects the gauge. We definea parametrization of a neighbourhood of the boundary ]0 , + ∞ ] × CT n (( t br ) g , ω br ) −→ CT n ( t g , ω ) by sending ( δ, λ , . . . , λ s , l s +1 , . . . , l i ) to the element of CT n ( t g , ω ) whose entries are defined as λ := − δ + s (cid:88) u =1 ( λ u − d ( r, w u )) − (cid:88) e ∈E ∞ d ( r, w e ) ,l v := δ + (cid:88) u =1 ,...,su (cid:54) = v ( − λ u + d ( r, w u )) + (cid:88) e ∈E ∞ d ( r, w e ) for v = e , . . . , e s ,l k := l k for k = s + 1 , . . . , i . We compute that this map has sign − .Suppose now that among the breaking internal edges of t g , some of them may intersect the gauge.We denote N ∩ ⊂ { , . . . , s } for the set of indices corresponding to the breaking internal edgeswhich intersect the gauge, and N ∅ ⊂ { , . . . , s } for the set of indices corresponding to the breakingof internal edges which do not intersect the gauge. We define this time a parametrization of aneighbourhood of the boundary ]0 , + ∞ ] × CT n (( t br ) g , ω br ) −→ CT n ( t g , ω ) by sending ( δ, λ , . . . , λ s , l s +1 , . . . , l i ) to the element of CT n ( t g , ω ) whose entries are set to be λ := − δ + (cid:88) u ∈N ∅ ( λ u − d ( r, w u )) − (cid:88) u ∈N ∩ d ( r, w u ) − (cid:88) e ∈E ∞ d ( r, w e ) ,l v := δ + (cid:88) u ∈N ∅ u (cid:54) = v ( − λ u + d ( r, w u )) + (cid:88) u ∈N ∩ d ( r, w u ) + (cid:88) e ∈E ∞ d ( r, w e ) for v ∈ N ∅ ,l v := δ + λ v + (cid:88) u ∈N ∅ ( − λ u + d ( r, w u )) + (cid:88) u ∈N ∩ u (cid:54) = v d ( r, w u ) + (cid:88) e ∈E ∞ d ( r, w e ) for v ∈ N ∩ ,l k := l k for k = s + 1 , . . . , i . We compute that this map has again sign − .Consider now the case of a gauged tree t g whose gauge intersects j of its vertices. We check as inthe previous sections that the parametrization maps introduced in the previous paragraphs, restrict to parametrizations of a neighborhood of the boundary ]0 , + ∞ ] × CT n (( t br ) g , ω br ) −→ CT n ( t g , ω ) , and that these maps send moreover the coorientation of CT n (( t br ) g , ω br ) to the coorientation of CT n ( t g , ω ) . These coorientations introduce an additional ( − j sign.We have thus computed the sign of the (below-break) boundary when j = 1 , . . . , j s = s . Now,consider the general case where we dot no necessarily have that j = 1 , . . . , j s = s . We denote ε ( j , . . . , j s ; ω ) the sign obtained after modifying ω by moving e j k to the k -th spot in ω , and write ω for the newly obtained orientation on t g . Twisting the orientation on CT n ( t g , ω ) by ( − ε ( j ,...,j s ; ω ) amounts to identifying it with CT n ( t g , ω ) . We can apply the previous constructions and find thedesired sign for the associated (below-break) component. Proposition 13. For a gauged stable ribbon tree t g whose gauge intersects j vertices, the boundarycomponent CT n (( t br ) g , ω br ) corresponding to the breaking of the internal edges e j , . . . , e j s of t bearsa ( − ε ( j ,...,j s ; ω )+1+ j sign in the boundary of CT n ( t g , ω ) . The operadic bimodule Ω BAs − Morph . Definition of the operadic bimodule Ω BAs − Morph . We choose to define the operadic bimod-ule Ω BAs − Morph with the formalism of orientations on gauged trees, so that it be compatible withthe definition of Markl-Shnider for the operad Ω BAs . As before, t br,g will stand for a broken gaugedstable ribbon tree, while t g will denote an unbroken gauged stable ribbon tree. We also respectivelywrite t br and t for the underlying stable ribbon trees. Definition 26 (Spaces of operations and action-composition maps) . Consider the Z -module freelygenerated by the pairs ( t br,g , ω ) . We define the arity n space of operations Ω BAs − Morph( n ) ∗ tobe the quotient of this Z -module under the relation ( t br,g , − ω ) = − ( t br,g , ω ) . An element ( t br,g , ω ) where t br,g has e ( t br ) finite internal edges and g non-trivial gauges whichintersect j vertices of t br is defined to have degree j − ( e ( t br ) + g ) . The operad Ω BAs then acts on Ω BAs − Morph as follows ( t br,g , ω ) ◦ i ( t (cid:48) br , ω (cid:48) ) = ( t br,g ◦ i t (cid:48) br , ω ∧ ω (cid:48) ) ,µ (( t br , ω ) , ( t br,g , ω ) , . . . , ( t sbr,g , ω s )) = ( − † ( µ ( t br , t br,g . . . , t sbr,g ) , ω ∧ ω ∧ · · · ∧ ω s ) , where the tree t br,g ◦ i t (cid:48) br is the gauged broken ribbon tree obtained by grafting t (cid:48) br to the i -th incomingedge of t br,g and µ ( t br , t br,g . . . , t sbr,g ) is the gauged broken ribbon tree defined by grafting each t jbr,g tothe j -th incoming edge of t br . Writing g i for the number of non-trivial gauges and j i for the numberof gauge-vertex intersections of t ibr,g , i = 1 , . . . , s , and setting t br := t br and g = j = 0 , † := s (cid:88) i =1 g i i − (cid:88) l =0 e ( t lbr ) + s (cid:88) i =1 j i i − (cid:88) l =0 ( e ( t lbr ) + g l − j l ) , or equivalently † = s (cid:88) i =1 g i (cid:32) | t br | + i − (cid:88) l =1 | t lbr | (cid:33) + s (cid:88) i =1 j i (cid:32) | t br | + i − (cid:88) l =1 | t lbr,g | (cid:33) . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 51 Choosing a distinguished orientation for every gauged stable ribbon tree t g ∈ sCRT , this definitionof the operadic bimodule Ω BAs − Morph amounts to defining it as the free operadic bimodule ingraded Z -modules F Ω BAs, Ω BAs ( , , , , · · · , sCRT n , · · · ) . It remains to define a differential on the generating operations ( t g , ω ) to recover the definition givenin subsection 3.2.4. Definition 27 (Differential) . The differential of a gauged stable ribbon tree ( t g , ω ) is defined as thesigned sum of all codimension 1 contributions ∂ ( t g , ω ) = (cid:88) ± ( int − collapse ) + (cid:88) ± ( gauge − vertex ) + (cid:88) ± ( above − break ) + (cid:88) ± ( below − break ) , where the signs are as computed in propositions 9 to 13.For instance, choosing the ordering e ∧ e on e e , the signs in the computation of subsection 3.2.4 are ∂ (cid:32) , e ∧ e (cid:33) = (cid:32) , e ∧ e (cid:33) − (cid:32) , e ∧ e (cid:33) − (cid:32) , e ∧ e (cid:33) + , e − , e − (cid:32) , ∅ (cid:33) . The moduli spaces CT n realize the operadic bimodule Ω BAs − Morph . We only have to checkthat the signs for the action-composition maps of Ω BAs − Morph are indeed the ones containedin the moduli spaces CT n , to conclude that the moduli spaces CT n endowed with their thin celldecomposition realize the operadic bimodule Ω BAs − Morph under the functor C cell −∗ .The computation for ◦ i is straighforward. Consider now the map µ : T ( t br , ω ) × CT ( t br,g , ω ) × · · · × CT ( t sbr,g , ω s ) −→ CT ( µ ( t br , t br,g . . . , t sbr,g ) , ω ∧ ω ∧ · · · ∧ ω s )( L ω , (Λ , L ω ) , . . . , (Λ s , L ω s )) (cid:55)−→ (Λ , . . . , Λ s , L ω , L ω , . . . , L ω s ) , where L ω i stands for the list of lengths of t ibr according to the ordering ω i , and Λ i := ( λ i, , . . . , λ i,g i ) stands for the list of non-trivial gauges of t ibr,g . We compute that, in the absence of gauge vertexintersections, this map has sign ( − (cid:80) si =1 g i (cid:80) i − l =0 e ( t lbr ) . Assuming that there are some gauge-vertex intersections, the combinatorics of coorientations intro-duce an additional sign ( − (cid:80) si =1 j i (cid:80) i − l =0 ( e ( t lbr )+ g l − j l ) . In total, we recover the sign ( − † , which concludes the proof. Canonical orientations for the gauged binary ribbon trees. For a fixed n (cid:62) , the set of gaugedbinary ribbon trees CBRT n can be endowed with a partial order, inspired by the Tamari order on BRT n . It was introduced by Masuda and Vallette in [MV]. Definition 28 ([MV]) . The Tamari order on CBRT n is the partial order generated by the coveringrelations t t t > t t t (A)where t , t and t are binary ribbon trees, t g t g t g t > t g t g t g t (B.1)where t g , t g , t g are gauged binary ribbon trees and t is a binary ribbon tree, and t t t t g > t t t t g (B.2)where t , t , t are binary ribbon trees and t g is a gauged binary ribbon tree.For example in the case of CBRT , we obtain the Hasse diagram in figure 18. This Tamari-like poset has a unique maximal element and a unique minimal element, respectively given by theright-leaning comb whose gauge intersects the outgoing edge, and the left-leaning comb whose gaugeintersects all incoming edges. e e e e e e e − e e − e e − e Figure 18. On the left, the Hasse diagram of the poset CBRT , where the maximalelement is written at the top. On the right, all the canonical orientations for CBRT computed going down the poset. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 53 The canonical orientation on the maximal gauged binary tree is defined as e e n − ω can := e ∧ · · · ∧ e n − . Using this Tamari-like order, we can now build inductively canonical orientations on all gaugedbinary trees. We start at the maximal gauged binary tree, and transport the orientation ω can to itsimmediate neighbours as follows : the immediate neighbours of t maxg obtained under the coveringrelation (A) are endowed with the orientation ω can , while the ones obtained under the coveringrelations (B) are endowed with the orientation − ω can . We then repeat this operation while goingdown the poset until the minimal gauged binary tree is reached. This process is consistent (see nextsection), i.e. it does not depend on the path taken in the poset from t maxg to the gauged binary treewhose orientation is being defined. A full example for CBRT is illustrated in figure 18. Definition 29. The such obtained orientations will again be called the canonical orientations andwritten ω can . They coincide in fact with the canonical orientations on the underlying binary trees.5.3.4. MacLane’s coherence. We stated in subsections 5.1.2 and 5.3.3 that our process of transform-ing orientations is consistent, i.e. it does not depend on the path taken in the Tamari poset fromthe maximal tree to the tree whose orientation is being defined. In fact, our rules to transformorientations under the covering relations enable us to transport the orientation ω of any (gauged)tree t ( g ) to any (gauged) tree t (cid:48) ( g ) , along a path in the Tamari poset. The following result then holds :for a given oriented (gauged) tree ( t ( g ) , ω ) , any two paths in the Tamari poset from t ( g ) to t (cid:48) ( g ) yieldthe same orientation on t (cid:48) ( g ) .As pointed out by Markl and Shnider in [MS06], an adaptation of the proof of MacLane’s coherencetheorem shows that it is enough to prove that the diagram described by K commutes to concludethat this statement holds for BRT n . And this is the case as shown in figure 14. In the case of CBRT n , an adaptation of these arguments shows this time that it is enough to prove that thediagrams described by K and J commute in order to conclude. This is again the case.A conceptual explanation for these two "coherence theorems" can be given as follows. In thecase of BRT n , a path between two trees t and t (cid:48) in the Tamari poset corresponds to a path in the1-skeleton of K n . The faces of the 2-skeleton of K n consist moreover of the products K × · · · × K × K × K × · · · × K × K × K × · · · × K ,K × · · · × K × K × K × · · · × K . The first type of face corresponds to a square diagram that tautologically commutes, while thesecond type of face corresponds to the K diagram. Given now two paths from t to t (cid:48) , they delineatea family of faces in the 2-skeleton of K n . Translating this into algebra, as all faces translate intocommuting diagrams, the two paths produce the same orientation.5.3.5. The morphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph . The moduli space CT n endowed with its A ∞ -cell decomposition is isomorphic to the Forcey-Loday realization J n ofthe multiplihedron. Forcey shows in [For08] that under this isomorphism, the Ω BAs -decompositionis sent to the dual subdivision of J n . This is illustrated on figure 11 for instance. The goal of thissection is again to show that : Proposition 14. The map id : ( CT n ) A ∞ → ( CT n ) Ω BAs is sent under the functor C cell −∗ to themorphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph acting as f n (cid:55)−→ (cid:88) t g ∈ CBRT n ( t g , ω can ) . We prove that taking the restriction of the orientation of J n chosen in section 4.4 to the topdimensional cells of its dual subdivision, yields the canonical orientations on these cells in the CT n viewpoint. We follow in this regard the exact same line of proof as in subsection 5.1.4.This statement is at first shown for the maximal gauged binary tree t maxg , the right-leaning combwhose gauge crosses the outgoing edge. The orientation on the cell CT n ( t maxg ) induced by thecanonical orientation e ∧ · · · ∧ e n − defines an isomorphism CT n ( t maxg ) ˜ −→ [0 , + ∞ ] × [0 , + ∞ ] n − , where the factor [0 , + ∞ ] corresponds to the gauge λ , and the factor [0 , + ∞ ] n − to the lengths of theinner edges. The face of CT n ( t maxg ) associated to the gauge going to + ∞ corresponds to the face H ,n, when seen in the Forcey-Loday polytope, while the face associated to the breaking of the i -thedge corresponds to the face H i,n − i, . An outward-pointing vector for the face H i,n − i, is moreover ν i := (0 , . . . , , i +1 , . . . , n − ) , where coordinates are taken in the basis f ωj . The orientation defined by the canonical basis of [0 , + ∞ ] × [0 , + ∞ ] n − is exactly the one defined by the ordered list of the outward-pointing vectors tothe + ∞ boundary. This orientation is thus sent to the orientation defined by the basis ( ν , . . . , ν n − ) in the Forcey-Loday polytope. It remains to check that det f ωj ( ν j ) = 1 . As a result, the orientation induced by J n and the one defined by the canonical orientation coincidefor the cell CT n ( t maxg ) .The rest of the proof is a mere adaptation of the proof of subsection 5.1.4. The cells labeled bythe gauged binary trees which are immediate neighbours of the maximal gauged binary tree, areexactly the ones having a codimension 1 stratum in common with CT n ( t maxg ) . Choosing one suchtree t g , and gluing the cells CT n ( t g ) and CT n ( t maxg ) along their common boundary, one can readthe induced orientation on CT n ( t g ) . In the case when the immediate neighbour t g is obtained underthe covering relation (A), the cells CT n ( t g ) and CT n ( t maxg ) are in fact both oriented as subspacesof ] − ∞ , + ∞ [ × ]0 , + ∞ [ n − . In the case when the immediate neighbour t g is obtained under thecovering relations (B), we send the reader back to subsection 5.1.4 for explanations on why a − twist of the orientation has to be introduced. In each case, the induced orientation is exactly thecanonical orientation on CT n ( t g ) . This argument can now be repeated going down the poset, and theinduced orientation will always coincide with the canonical orientation on the cell, which concludesthe proof of proposition 3. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 55 Part 2 Geometry A ∞ and Ω BAs -algebra structures on the Morse cochains Let M be an oriented closed Riemannian manifold endowed with a Morse function f togetherwith a Morse-Smale metric. Following [Hut08], the Morse cochains C ∗ ( f ) form a homotopy retractof the singular cochains on M . The cup product naturally endows the singular cochains C ∗ sing ( M ) with a dg-algebra structure. The homotopy transfer theorem then ensures that it can be transferredto an A ∞ -algebra structure on the Morse cochains C ∗ ( f ) . The following question then naturallyarises. The differential on the Morse cochains is defined by a count of moduli spaces of gradienttrajectories connecting critical points of f . Is it possible to define higher multiplications m n on C ∗ ( f ) by a count of moduli spaces such that they fit in a structure of A ∞ -algebra ?We have seen in the previous part that the polytopes encoding the operad A ∞ are the associahedraand that they can be realized as the compactified moduli spaces of stable metric ribbon trees. Anatural candidate would thus be an interpretation of metric ribbon trees in Morse theory. A naiveapproach would be to define trees each edge of which corresponds to a Morse gradient trajectory asin figure 19. These moduli spaces are however not well defined, as two trajectories coming from twodistinct critical points cannot intersect. A second problem is that moduli spaces of trajectories issuedfrom the same critical point do not intersect transversely. In his article [Abo11], Abouzaid bypassesthis problem by perturbing the equation around each vertex, so that a transverse intersection canbe achieved. See also [Mes18]. This is illustrated in figure 19. x −∇ f −∇ f x −∇ f y −∇ f x −∇ f Perturbing the gradient vectorfield around each vertex of the tree x −∇ f −∇ f x −∇ f y −∇ f x −∇ f −∇ f + X −∇ f + X Figure 19 Trees obtained in this way will be called perturbed Morse gradient trees . Let t be a stable rib-bon tree type and y, x , . . . , x n a collection of critical points of the Morse function f . We prove inthis section that for a generic choice of perturbation data X t on the moduli space T n ( t ) , the mod-uli space of perturbed Morse gradient trees modeled on t and connecting x , . . . , x n to y , denoted T t ( y ; x , . . . , x n ) , is an orientable manifold (proposition 16). Under some additional generic assump-tions on the choices of perturbation data X t , these moduli spaces are compact in the 0-dimensionalcase, and can be compactified to compact manifolds with boundary in the 1-dimensional case (the-orems 7 and 8). We are finally able to define operations on the Morse cochains C ∗ ( f ) by counting the 0-dimensional moduli spaces of Morse gradient trees : these operations define an Ω BAs -algebrastructure on C ∗ ( f ) (theorem 9). Our constructions are carried out using the formalism introducedin [Abo11] and some terminology of [Mes18]. Technical details are moreover postponed to sec-tions 3 and 4.Note that in Floer theory, A ∞ -structures arise from the fact that moduli spaces of closed pointeddisks naturally yield the A ∞ -cell decompositions of the associahedra. This is not the case in oursituation, where it is the Ω BAs -cell decompositions that naturally arise.1.1. Conventions. We refer to section 4.2 for additional details on the moduli spaces introducedin this section. We will study Morse theory of the Morse function f : M → R using its negativegradient vector field −∇ f . Denote d the dimension of the manifold M and φ s the flow of −∇ f . Fora critical point x define its unstable and stable manifolds W U ( x ) := { z ∈ M, lim s →−∞ φ s ( z ) = x } W S ( x ) := { z ∈ M, lim s → + ∞ φ s ( z ) = x } . Their dimensions are such that dim( W U ( x )) + dim( W S ( x )) = d . We then define the degree of acritical point x to be | x | := dim( W S ( x )) . This degree is often referred to as the coindex of x in thelitterature.We will moreover work with Morse cochains. For two critical point x (cid:54) = y , define T ( y ; x ) := W S ( y ) ∩ W U ( x ) / R to be the moduli space of negative gradient trajectories connecting x to y . Denote moreover T ( x ; x ) = ∅ . Under the Morse-Smale assumption on f and the Riemannian metric on M , for x (cid:54) = y the moduli space T ( y ; x ) has dimension dim ( T ( y ; x )) = | y | − | x | − . The Morse differential ∂ Morse : C ∗ ( f ) → C ∗ ( f ) is then defined to count descending negative gradient trajectories ∂ Morse ( x ) := (cid:88) | y | = | x | +1 T ( y ; x ) · y . Perturbed Morse gradient trees.Definition 30 ([Abo11]) . Let T := ( t, { l e } e ∈ E ( t ) ) be a metric tree, where { l e } e ∈ E ( t ) are the lengthsof its internal edges. A perturbation data on T consists of the following data :(i) a vector field [0 , l e ] × M −→ X e T M , that vanishes on [1 , l e − , for every internal edge e of t ;(ii) a vector field [0 , + ∞ [ × M −→ X e T M , that vanishes away from [0 , , for the outgoing edge e of t ;(iii) a vector field ] − ∞ , × M −→ X ei T M , that vanishes away from [ − , , for every incoming edge e i (1 (cid:54) i (cid:54) n ) of t . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 57 Note that when l e (cid:54) , the vanishing condition on [1 , l e − is empty, that is we do not requireany specific vanishing property for X e . For brevity’s sake we will write D e for all segments [0 , l e ] aswell as for all semi-infinite segments ] − ∞ , and [0 , + ∞ [ in the rest of the paper. Definition 31 ([Abo11]) . A perturbed Morse gradient tree T Morse associated to ( T, X ) is the datafor each edge e of t of a smooth map γ e : D e → M such that γ e is a trajectory of the perturbednegative gradient −∇ f + X e , i.e. ˙ γ e ( s ) = −∇ f ( γ e ( s )) + X e ( s, γ e ( s )) , and such that the endpoints of these trajectories coincide as prescribed by the edges of the tree T . l l e e e e e e e e e e ff gg Figure 20. Choosing a perturbation datum X for this metric tree, we have that φ , X = φ l g, X ◦ φ l f, X ◦ φ e , X , φ , X = φ l g, X ◦ φ l f, X ◦ φ e , X , φ , X = φ l g, X ◦ φ e , X and φ , X = φ e , X A perturbed Morse gradient tree T Morse associated to ( T, X ) is determined by the data of thetime -1 points on its incoming edges plus the time 1 point on its outgoing edge. Indeed, for eachedge e of t , we write φ e, X for the flow of −∇ f + X e . We moreover define for every incoming edge e i (1 (cid:54) i (cid:54) n ) of T , the diffeomorphism φ i, X to be the composition of all flows obtained by followingthe time -1 point of the metric tree on e i along the only non-self crossing path connecting it to theroot. We also set φ , X for the flow of φ e , X at time -1, where e is the outgoing edge of t . This isdepicted on figure 20. Setting Φ T, X : M × · · · × M −→ φ , X ×···× φ n, X M × · · · × M , and ∆ for the thin diagonal of M × · · · × M , it is then clear that : Proposition 15 ([Abo11]) . There is a one-to-one correspondence (cid:26) perturbed Morse gradient treesassociated to ( T, X ) (cid:27) ←→ (Φ T, X ) − (∆) . The vector fields on the external edges are equal to −∇ f away from a length 1 segment, hencethe trajectories associated to these edges all converge to critical points of the function f . For criticalpoints y and x , . . . , x n , the map Φ T, X can be restricted to W S ( y ) × W U ( x ) × · · · × W U ( x n ) , such that the inverse image of the diagonal yields all perturbed Morse gradient trees associated to ( T, X ) connecting x , . . . , x n to y . Moduli spaces of perturbed Morse gradient trees. Recall that E ( t ) stands for the set ofinternal edges of t , and E ( t ) for the set of all its edges. We previously saw that a perturbation dataon a metric ribbon tree T := ( t, { l e } e ∈ E ( t ) ) is the data of maps X T,f : D f × M −→ T M , for everyedge f ∈ E ( t ) of t . Define the cone C f ⊂ T n ( t ) × R (cid:39) R e ( t )+1 to be(i) { (( l e ) e ∈ E ( t ) , s ) such that (cid:54) s (cid:54) l f } if f is an internal edge ;(ii) { (( l e ) e ∈ E ( t ) , s ) such that s (cid:54) } if f is an incoming edge ;(iii) { (( l e ) e ∈ E ( t ) , s ) such that s (cid:62) } if f is the outgoing edge.Then a choice of perturbation data for every metric ribbon tree in T n ( t ) yields a map X t,f : C f × M −→ T M , for every edge f of t . This choice of perturbation data is said to be smooth if all these maps aresmooth. Definition 32. Let X t be a smooth choice of perturbation data on T n ( t ) . For critical points y and x , . . . , x n , we define the moduli space T X t t ( y ; x , . . . , x n ) := (cid:26) perturbed Morse gradient trees associated to ( T, X T ) and connecting x , . . . , x n to y , for T ∈ T n ( t ) (cid:27) . Introduce now the map φ X t : T n ( t ) × W S ( y ) × W U ( x ) × · · · × W U ( x n ) −→ M × n +1 , whose restriction to every T ∈ T n ( t ) is as defined previously : Proposition 16. (i) The moduli space T X t t ( y ; x , . . . , x n ) can be rewritten as T X t t ( y ; x , . . . , x n ) = φ − X t (∆) , where ∆ is the thin diagonal of M × n +1 .(ii) Given a choice of perturbation data X t making φ X t transverse to the diagonal ∆ , the modulispace T X t t ( y ; x , . . . , x n ) is an orientable manifold of dimension dim ( T t ( y ; x , . . . , x n )) = e ( t ) + | y | − n (cid:88) i =1 | x i | . (iii) Choices of perturbation data X t such that φ X t is transverse to ∆ exist. Item (i) is straightforward and item (ii) stems from the fact that if φ X t transverse to ∆ , the modulispaces T X t t ( y ; x , . . . , x n ) are manifolds of codimension codim ( T t ( y ; x , . . . , x n )) = codim M × n +1 (∆) = nd , where d := dim( M ) . Note that we have chosen to grade the Morse cochains using the coindex inorder for this convenient dimension formula to hold. We refer to sections 3 for details on item (iii). IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 59 Compactifications. We now would like to compactify the moduli spaces T X t t ( y ; x , . . . , x n ) that have dimension 1 to 1-dimensional manifolds with boundary. They are defined as the inverseimage in T n ( t ) × W S ( y ) × W U ( x ) × · · · × W U ( x n ) of the diagonal ∆ under φ X t . The boundarycomponents in the compactification should hence come from those of T n ( t ) , of the W U ( x i ) , and of W S ( y ) : that is they will respectively come from internal edges of the perturbed Morse gradienttree collapsing, or breaking at a critical point (boundary of T n ( t ) ), its semi-infinite incoming edgesbreaking at a critical point (boundary of W U ( x i ) ) and its semi-infinite outgoing edge breaking at acritical point (boundary of W S ( y ) ). Some of these phenomena are represented on figure 21. x x yzx z x yx x Figure 21. Two examples of perturbed Morse gradient trees breaking at a criticalpointChoose smooth perturbation data X t for all t ∈ sRT i , (cid:54) i (cid:54) n . We denote X n := ( X t ) t ∈ sRT n and call it a choice of perturbation data on the moduli space T n . We construct the boundary of thecompactification of the moduli space T X t t ( y ; x , . . . , x n ) by using the perturbation data ( X t ) t ∈ sRT i (cid:54) i (cid:54) n .It is given by the spaces(i) corresponding to an internal edge collapsing (int-collapse) : T X t (cid:48) t (cid:48) ( y ; x , . . . , x n ) where t (cid:48) ∈ sRT n are all the trees obtained by collapsing exactly one internal edge of t ;(ii) corresponding to an internal edge breaking (int-break) : T X t t ( y ; x , . . . , x i , z, x i + i +1 , . . . , x n ) × T X t t ( z ; x i +1 , . . . , x i + i ) , where t is seen to lie above the i + 1 -incoming edge of t ;(iii) corresponding to an external edge breaking (Morse) : T ( y ; z ) × T X t t ( z ; x , . . . , x n ) and T X t t ( y ; x , . . . , z, . . . , x n ) × T ( z ; x i ) . While the (Morse) boundary simply comes from the fact that external edges are Morse trajectoriesaway from a length 1 segment, the analysis for the (int-collapse) and (int-break) boundaries requiresto refine our definitions of perturbation data. It namely appears here why we had to choose moreperturbation data than X t , as they will appear in the boundary of the compactified moduli space.We begin by tackling the conditions coming with the (int-collapse) boundary. Let t be a stableribbon tree type and consider a choice of perturbation data on T n ( t ) : it is a choice of perturbationdata X T for every T ∈ T n ( t ) (cid:39) ]0 , + ∞ [ e ( t ) . Denote coll ( t ) ⊂ sRT n the set of all trees obtained bycollapsing internal edges of t . A choice of perturbation data ( X t (cid:48) ) t (cid:48) ∈ coll ( t ) then corresponds to a choice of perturbation data X T for every T ∈ [0 , + ∞ [ e ( t ) . Following section 1.3, such a choice ofperturbation data is equivalent to a map ˜ X t,f : ˜ C f × M −→ T M , for every edge f of t , where ˜ C f ⊂ [0 , + ∞ [ e ( t ) × R is defined in a similar fashion to C f . Definition 33. A choice of perturbation data ( X t (cid:48) ) t (cid:48) ∈ coll ( t ) is said to be smooth if all maps ˜ X t,f aresmooth. A choice of perturbation data X n is said to be smooth if for every t ∈ sRT n , the choice ofperturbation data ( X t (cid:48) ) t (cid:48) ∈ coll ( t ) is smooth.We now tackle the conditions coming with the (int-break) boundary. We work again with a fixedstable ribbon tree type t . Consider a choice of perturbation data X t = ( X t,e ) e ∈ E ( t ) on T n ( t ) . Wehave to specify what happens on the X t,e when the length of an internal edge f of t , denoted l f ,goes towards + ∞ . Write t and t for the trees obtained by breaking t at the edge f .(i) For e ∈ E ( t ) and (cid:54) = f , assuming for instance that e ∈ t , we require that lim l f → + ∞ X t,e = X t ,e . (ii) For f = e , X t,f yields two parts when l f → + ∞ : the part corresponding to the infiniteedge in t and the part corresponding to the infinite edge in t . We then require that theycoincide respectively with X t ,f and X t ,f .Two examples illustrating these two cases are detailed in the following paragraphs.Begin with an example of the first case, where e (cid:54) = f . This is represented on figure 22. We onlyrepresent the perturbation X t,f on this figure for clarity’s sake. The perturbation datum X ∞ t,f coulda priori depend on l f : the requirement X ∞ t,f = X t ,f says in particular that it is independent of l f . t f f f f f f X t,f l f −→ + ∞ t f f X ∞ t,f t f f Figure 22 Similarly, we illustrate the second case, where e = f , on figure 23. A priori, X + t,f and X − t,f candepend on both l f and l f : the requirement X + t,f = X t ,f says exactly that X + t,f is independent of l f , and similarly for X − t,f = X t ,f with respect to l f . Definition 34. A choice of perturbation data ( X i ) (cid:54) i (cid:54) n is said to be gluing-compatible if it satisfiesconditions (i) and (ii) for lengths of edges going toward + ∞ . A choice of perturbation data ( X n ) n (cid:62) being both smooth and gluing-compatible, and such that all maps φ X t are transverse to ∆ , is saidto be admissible . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 61 t f f f f f f X t,f l f −→ + ∞ t f f X − t,f t f f X + t,f Figure 23 Theorem 7. Admissible choices of perturbation data on the moduli spaces T n exist. Theorem 8. Let ( X n ) n (cid:62) be an admissible choice of perturbation data. The 0-dimensional modulispaces T X t t ( y ; x , . . . , x n ) are compact. The 1-dimensional moduli spaces T X t t ( y ; x , . . . , x n ) can becompactified to 1-dimensional manifolds with boundary T X t t ( y ; x , . . . , x n ) , whose boundary is de-scribed at the beginning of this section. We refer to section 3 for a proof of theorem 7. Theorem 8 is proven in chapter 6 of [Mes18].Using the results of [Weh12], we could in fact try to prove that all moduli spaces T X t t ( y ; x , . . . , x n ) can be compactified to compact manifolds with corners. The analysis involved therein goes howeverbeyond the scope of this paper.Consider now a stable ribbon tree t together with an internal edge f ∈ E ( t ) and write t and t forthe trees obtained by breaking t at the edge f , where t is seen to lie abpve t . Given critical points y, z, x , . . . , x n suppose moreover that the moduli spaces T t ( y ; x , . . . , x i , z, x i + i +1 , . . . , x n ) and T t ( z ; x i +1 , . . . , x i + i ) are 0-dimensional. Let T Morse and T Morse be two perturbed Morse gradienttrees which belong respectively to the former and the latter moduli spaces. Theorem 8 implies inparticular that there exists R > and an embedding T Morse ,T Morse : [ R, + ∞ ] −→ T t ( y ; x , . . . , x n ) parametrizing a neighborhood of the boundary { T Morse } × { T Morse } ⊂ ∂ T Morset , i.e. sending + ∞ to ( T Morse , T Morse ) ∈ ∂ T Morset . Such a map is called a gluing map for T Morse and T Morse . Explicitgluing maps are constructed in subsection 4.4.3.1.5. Ω BAs -algebra structure on the Morse cochains. We now have all the necessary materialto define an Ω BAs -algebra structure on the Morse cochains C ∗ ( f ) . Theorem 9. Let X := ( X n ) n (cid:62) be an admissible choice of perturbation data. Defining for every n and t ∈ sRT n the operations m t as m t : C ∗ ( f ) ⊗ · · · ⊗ C ∗ ( f ) −→ C ∗ ( f ) x ⊗ · · · ⊗ x n (cid:55)−→ (cid:88) | y | = (cid:80) ni =1 | x i |− e ( t ) T X t ( y ; x , · · · , x n ) · y , they endow the Morse cochains C ∗ ( f ) with an Ω BAs -algebra structure. The proof of this theorem is detailed in section 4.4. Putting it shortly, counting the boundarypoints of the 1-dimensional orientable compactified moduli spaces T X t ( y ; x , · · · , x n ) whose boundaryis described in the previous section yields the Ω BAs -equations [ ∂ Morse , m t ] = (cid:88) t (cid:48) ∈ coll ( t ) ± m t (cid:48) + (cid:88) t i t = t ± m t ◦ i m t . In fact, the collection of operations { m t } does not exactly define an Ω BAs -algebra structure : oneof the two differentials ∂ Morse appearing in the bracket [ ∂ Morse , · ] has to be twisted by a specificsign for the Ω BAs -equations to hold. We will speak about a twisted Ω BAs -algebra structure . In thecase when M is odd-dimensional, this twisted Ω BAs -algebra is exactly an Ω BAs -algebra.If we want to recover an A ∞ -algebra structure on the Morse cochains, whose existence is guaran-teed by the homotopy transfer theorem, it suffices to apply the morphism of operads A ∞ → Ω BAs described in section 3.1.5.2. A ∞ and Ω BAs -morphisms between the Morse cochains Let M be an oriented closed Riemannian manifold endowed with a Morse function f togetherwith a Morse-Smale metric. We have proven in the previous section that, upon choosing admissibleperturbation data on the moduli spaces of stable metric ribbon trees T n ( t ) , we can define modulispaces of perturbed Morse gradient trees, whose count will define the operations m t , t ∈ sRT , of an Ω BAs -algebra structure on the Morse cochains C ∗ ( f ) .Consider now another Morse function g on M . Apply again the homotopy transfer theorem to C ∗ ( f ) and C ∗ ( g ) , which are homotopy retracts of the singular cochains on M . Endowing them withtheir induced A ∞ -algebra structures, the theorem yields a diagram ( C ∗ ( f ) , m indn ) ˜ −→ ( C ∗ sing ( M ) , ∪ ) ˜ −→ ( C ∗ ( g ) , m indn ) , where each arrow is an A ∞ -quasi-isomorphism. Following the fundamental theorem of A ∞ -quasi-isomorphisms, there exists in particular an A ∞ -morphism ( C ∗ ( f ) , m indn ) −→ ( C ∗ ( g ) , m indn ) . Let X g be an admissible perturbation data for g . This motivates the following question : endowing C ∗ ( f ) and C ∗ ( g ) with their Ω BAs -algebra structures, can we construct an Ω BAs -morphism ( C ∗ ( f ) , m X f t ) −→ ( C ∗ ( g ) , m X g t ) ? While stable metric ribbon trees control Ω BAs -algebra structures, we have seen that two-coloredstable metric ribbon trees control Ω BAs -morphisms. The answer to the previous question is then ofcourse positive, and the morphism will be constructed using moduli spaces of two-colored perturbedMorse gradient trees . As in section 1, two-colored Morse gradient trees will be defined by perturbingMorse gradient equations around each vertex of the tree, where the Morse gradient is −∇ f abovethe gauge, and −∇ g below the gauge. This is illustrated in figure 24. The figure is incorrect, becausewe won’t choose the perturbation to be equal to X f above the gauge and to X g below, but gives thecorrect intuition on the construction we unfold in this section.The structure of this section follows the same lines as the previous section, and the only diffi-culty will consist in adapting properly our arguments to the combinatorics of two-colored ribbontrees. Under a generic choice of perturbation data on the moduli spaces CT n , the moduli spaces IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 63 −∇ f −∇ f −∇ f −∇ f −∇ f + X f −∇ f + X f −∇ f −∇ f −∇ f + Y −∇ g + Y −∇ g −∇ g −∇ g + X g −∇ g x x x x y Figure 24. An example of a perturbed two-colored Morse gradient tree, where the x i are critical points of f and y is a critical point of g of two-colored perturbed Morse gradient trees connecting x , . . . , x n ∈ Crit( g ) to y ∈ Crit( g ) ,that we denote CT t g ( y ; x , . . . , x n ) , are orientable manifolds. They are moreover compact when0-dimensional and can be compactified to compact manifolds with boundary when 1-dimensional(theorems 10 and 11). Counting 0-dimensional moduli spaces of two-colored Morse gradient treesthen defines an Ω BAs -morphism from C ∗ ( f ) to C ∗ ( g ) (theorem 12).2.1. Notation. A two-colored ribbon tree will be written t g using the gauge viewpoint, and t c usingthe colored vertices viewpoint. The tree t g then comes with an underlying stable ribbon tree t , whilethe tree t c is already a ribbon tree (though not necessarily stable because of its colored vertices).A two-colored stable metric ribbon tree T will be written ( t g , ( l e ) e ∈ E ( t ) , λ ) using the gauge view-point. The lengths associated to the underlying metric ribbon tree with colored vertices will thenbe written L f c (( l e ) e ∈ E ( t ) , λ ) where f c ∈ E ( t c ) . For instance, on figure 8, L = − λ L = l + λ L = − λ . For the sake of readability, we do not write the dependence on (( l e ) e ∈ E ( t ) , λ ) in the sequel.2.2. Perturbed two-colored Morse gradient trees.Definition 35. Let T g = ( t g , ( l e ) e ∈ E ( t ) , λ ) be a two-colored metric ribbon tree. A perturbationdatum Y on T is defined to be a perturbation datum on the associated metric ribbon tree ( t c , L f c ) in the sense of section 1.2. Definition 36. A two-colored perturbed Morse gradient tree T Morseg associated to a pair two-coloredmetric ribbon tree and perturbation datum ( T g , Y ) is the data(i) for each edge f c of t c which is above the gauge, of a smooth map D f c −→ γ fc M , such that γ f c is a trajectory of the perturbed negative gradient −∇ f + Y f c , (ii) for each edge f c of t c which is below the gauge, of a smooth map D f c −→ γ fc M , such that γ f c is a trajectory of the perturbed negative gradient −∇ g + Y f c ,and such that the endpoints of these trajectories coincide as prescribed by the edges of the tree t c .Note that the above definitions still work for . A perturbation datum for is the data ofvector fields [0 , + ∞ [ × M −→ Y + T M , ] − ∞ , × M −→ Y − T M , which vanish away from a length 1 segment, and a two-colored perturbed Morse gradient treeassociated to ( , Y ) is then simply the data of two smooth maps ] − ∞ , −→ γ − M , [0 , + ∞ [ −→ γ + M , such that γ − is a trajectory of −∇ f + Y − and γ + is a trajectory of −∇ g + Y + .There is also an equivalent formulation for two-colored perturbed Morse gradient trees, by fol-lowing the flows of −∇ f + Y and −∇ g + Y along the the metric ribbon tree ( t c , L f c ) . That is, atwo-colored perturbed Morse gradient tree is determined by the data of the time -1 points on itsincoming edges plus the time 1 point on its outgoing edge. Introduce again the map Φ T g , Y : M × · · · × M −→ φ , Y ×···× φ n, Y M × · · · × M , defined as before, and set ∆ for the diagonal of M × n +1 Proposition 17. There is a one-to-one correspondence (cid:26) two-colored perturbed Morse gradient treesassociated to ( T g , Y ) (cid:27) ←→ (Φ T g , Y ) − (∆) . The vector fields on the incoming edges are equal to −∇ f away from a length 1 segment, hencethe trajectories associated to these edges all converge to critical points of the function f , while thevector field on the outgoing edge is equal to −∇ g away from a length 1 segment, hence the trajectoryassociated to these edge converges to a critical point of the function g . For critical points y of thefunction g and x , . . . , x n of the function f , the map Φ T, Y can be restricted to W Sg ( y ) × W Uf ( x ) × · · · × W Uf ( x n ) , such that the inverse image of the diagonal yields all two-colored perturbed Morse gradient treesassociated to ( T, Y ) connecting x , . . . , x n to y . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 65 Moduli spaces of two-colored perturbed Morse gradient trees. Choose a two-coloredstable ribbon tree t g ∈ sCRT n whose underlying stable ribbon tree is t and whose associated ribbontree with colored vertices is t c . We write ( ∗ ) t g for the set of inequalities and equalities on { l e } e ∈ E ( t ) and λ , which define the polyedral cone CT n ( t g ) ⊂ R e ( t )+1 . See part 1 section 5.2 for more details.Define for all f c ∈ E ( t c ) , the cone C f c ⊂ CT n ( t g ) × R ⊂ R e ( t )+1 × R to be(i) { (( l e ) e ∈ E ( T ) , λ, s ) such that ( ∗ ) t g , (cid:54) s (cid:54) L f c (( l e ) e ∈ E ( T ) , λ ) } if f c is an internal edge ;(ii) { (( l e ) e ∈ E ( T ) , λ, s ) such that ( ∗ ) t g , s (cid:54) } if f c is an incoming edge ;(iii) { (( l e ) e ∈ E ( T ) , λ, s ) such that ( ∗ ) t g , s (cid:62) } if f c is the outgoing edge.Then a choice of perturbation datum for every two-colored metric ribbon tree in CT n ( t g ) , yieldsmaps Y t g ,f c : C f c × M −→ T M for every edge f c of t c . These perturbation data are said to be smooth if all these maps are smooth. Definition 37. Let Y t g be a smooth choice of perturbation data on the stratum CT n ( t g ) . Given y ∈ Crit( g ) and x , . . . , x n ∈ Crit( f ) , we define the moduli spaces CT Y tg t g ( y ; x , . . . , x n ) := (cid:26) two-colored perturbed Morse gradient trees associated to ( T g , Y T g ) and connecting x , . . . , x n to y for T g ∈ CT n ( t g ) (cid:27) . Using the smooth map φ Y tg : CT n ( t g ) × W S ( y ) × W U ( x ) × · · · × W U ( x n ) −→ M × n +1 , this moduli space can be rewritten as CT Y tg t g ( y ; x , . . . , x n ) = φ − Y tg (∆) . Proposition 18. (i) Given a choice of perturbation data Y t g making φ Y tg transverse to thediagonal ∆ ⊂ M × n +1 , the moduli spaces CT Y tg t g ( y ; x , . . . , x n ) are orientable manifolds ofdimension dim (cid:0) CT t g ( y ; x , . . . , x n ) (cid:1) = + | y | − n (cid:88) i =1 | x i | − | t g | . (ii) Choices of perturbation data Y t g such that φ Y tg is transverse to the diagonal ∆ exist. The proof of this proposition is again postponed to section 3.2.4. Compactifications. We finally proceed to compactify the moduli spaces CT Y tg t g ( y ; x , . . . , x n ) that have dimension 1 to 1-dimensional manifolds with boundary. Their boundary components aregoing to be given by those coming from the compactification of CT n ( t g ) , and the compactificationsof the W U ( x i ) and of W S ( y ) .Choose admissible perturbation data X f and X g for the functions f and g . Choose moreoversmooth perturbation data Y t g for all t g ∈ sCRT i , (cid:54) i (cid:54) n . We will again denote Y n :=( Y t g ) t g ∈ sCRT n , and call it a choice of perturbation data on CT n . Fixing a two-colored stable ribbontree t g ∈ sCRT n we would like to compactify the 1-dimensional moduli space CT Y tg t g ( y ; x , . . . , x n ) using the perturbation data X f , X g and ( Y i ) (cid:54) i (cid:54) n . Its boundary will be given by the followingphenomena (i) an external edge breaks at a critical point (Morse) : T ( y ; z ) × CT Y tg t g ( z ; x , . . . , x n ) and CT Y tg t g ( y ; x , . . . , z, . . . , x n ) × T ( z ; x i ) ; (ii) an internal edge of the tree t collapses (int-collapse) : CT Y t (cid:48) g t (cid:48) g ( y ; x , . . . , x n ) where t (cid:48) g ∈ sCRT n are all the two-colored trees obtained by collapsing exactly one internaledge, which does not cross the gauge ;(iii) the gauge moves to cross exactly one additional vertex of the underlying stable ribbon tree(gauge-vertex) : CT Y t (cid:48) g t (cid:48) g ( y ; x , . . . , x n ) where t (cid:48) g ∈ sCRT n are all the two-colored trees obtained by moving the gauge to cross exactlyone additional vertex of t ;(iv) an internal edge located above the gauge or intersecting it breaks or, when the gauge is belowthe root, the outgoing edge breaks between the gauge and the root (above-break) : CT Y t g t g ( y ; x , . . . , x i , z, x i + i +1 , . . . , x n ) × T X ft t ( z ; x i +1 , . . . , x i + i ) ; (v) edges (internal or incoming) that are possibly intersecting the gauge, break below it, suchthat there is exactly one edge breaking in each non-self crossing path from an incoming edgeto the root (below-break) : T X gt t ( y ; y , . . . , y s ) × CT Y t g t g ( y ; x , . . . ) × · · · × CT Y tsg t sg ( y s ; . . . , x n ) . The (Morse) boundaries are again a simple consequence of the fact that external edges are Morsetrajectories away from a length 1 segment. Perturbation data that behave well with respect to the(int-collapse) and (gauge-vertex) boundaries are defined using simple adjustments of the discussionin section 1.4. Hence, it only remains to specify the required behaviours under the breaking of edges.We begin with the (above-break) boundary. Writing t c for the two-colored ribbon tree associatedto t g , it corresponds to the breaking of an internal edge f c of t c situated above the set of coloredvertices. Denote t c and t the trees obtained by breaking t c at the edge f c , where t is seen to lieabove t c . We have to specify, for each edge e c ∈ E ( t c ) , what happens to the perturbation Y t c ,e atthe limit.(i) For e c ∈ E ( t ) and (cid:54) = f c , we require that lim Y t c ,e c = X ft ,e c . (ii) For e c ∈ E ( t c ) and (cid:54) = f c , we require that lim Y t c ,e c = Y t c ,e c . (iii) For f c = e c , Y t c ,f c yields two parts at the limit : the part corresponding to the outgoingedge of t and the part corresponding to the incoming edge of t c . We then require that theycoincide respectively with the perturbation X ft ,e c and Y t c ,e c . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 67 t g Y t g ,e c lim Y t g ,e c = X ft ,e c t g t (above-break) case (i) t g Y t g ,e c lim Y t g ,e c = Y t g ,e c t g t (above-break) case (ii) t g Y t g ,e c lim t g Y t g ,e c = Y t g ,e c lim t Y t g ,e c = X ft ,e c t g t (above-break) case (iii) Figure 25 Leaving the notations aside, an example of each case is illustrated in figure 25.We conclude with the (below-break) boundary. Denote t g , . . . , t sg and t the trees obtained by thechosen breaking of t g below the gauge, where t g , . . . , t sg are seen to lie above t .(i) For e c ∈ E ( t ic ) and not among the breaking edges, we require that lim Y t c ,e c = Y t ic ,e c . (ii) For e c ∈ E ( t ) and not among the breaking edges, we require that lim Y t c ,e c = X gt ,e c . (iii) For f c among the breaking edges, Y t c ,f c yields two parts at the limit : the part correspondingto the outgoing edge of a t jc and the part corresponding to the incoming edge of t . We thenrequire that they coincide respectively with the perturbation Y t jc and X gt .This is again illustrated on figure 26. Definition 38. A choice of perturbation data Y on the moduli spaces CT n is said to be smooth if it iscompatible with the (int-collapse) and (gauge-vertex) boundaries. A smooth choice of perturbationdata is said to be gluing-compatible w.r.t. X f and X g if it satisfies the (above-break) and (below-break) conditions described in this section. Smooth and consistent choices of perturbation data t g Y t g ,e c lim Y t g ,e c = Y t g ,e c t t g t g (below-break) case (i) t g Y t g ,e c lim Y t g ,e c = X gt ,e c t t g t g (below-break) case (ii) t g Y t g ,e c lim t Y t g ,e c = X gt ,e c lim t g Y t g ,e c = Y t g ,e c t t g t g (below-break) case (iii) Figure 26 ( Y n ) n (cid:62) such that all maps φ Y tg are transverse to the diagonal ∆ are called admissible w.r.t. X f and X g or simply admissible . Theorem 10. Given admissible choices of perturbation data X f and X g on the moduli spaces T n ,choices of perturbation data on the moduli spaces CT n that are admissible w.r.t. X f and X g exist. Theorem 11. Let ( Y n ) n (cid:62) be an admissible choice of perturbation data on the moduli spaces CT n .The 0-dimensional moduli spaces CT Y tg t g ( y ; x , . . . , x n ) are compact. The 1-dimensional moduli spaces CT Y tg t g ( y ; x , . . . , x n ) can be compactified to 1-dimensional manifolds with boundary, whose boundaryis described at the beginning of this section.. Theorem 10 is proven in section 3. Theorem 11 is a consequence of the results in chapter 6of [Mes18]. We moreover point out that theorem 11 implies in particular the existence of gluingmaps above − breakT ,Morseg ,T ,Morse : [ R, + ∞ ] −→ CT t g ( y ; x , . . . , x n ) below − breakT ,Morse ,T ,Morseg ,...,T s,Morseg : [ R, + ∞ ] −→ CT t g ( y ; x , . . . , x n ) where notations are as in section 1.4. Such gluing maps are constructed in subsection 4.5.4. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 69 The Ω BAs -morphism between Morse cochains. Let X f and X g be admissible choices ofperturbation data for the Morse functions f and g . Denote ( C ∗ ( f ) , m X f t ) and ( C ∗ ( g ) , m X g t ) the Ω BAs -algebras constructed in section 1.5. Theorem 12. Let ( Y n ) n (cid:62) be a choice of perturbation on the moduli spaces CT n that is admissiblew.r.t. X f and X g . Defining for every n and t g ∈ sCRT n the operations µ t g as µ Y t g : C ∗ ( f ) ⊗ · · · ⊗ C ∗ ( f ) −→ C ∗ ( g ) x ⊗ · · · ⊗ x n (cid:55)−→ (cid:88) | y | = (cid:80) ni =1 | x i | + | t g | CT Y t g ( y ; x , · · · , x n ) · y . they fit into an Ω BAs -morphism µ Y : ( C ∗ ( f ) , m X f t ) → ( C ∗ ( g ) , m X g t ) . Again, the collection of operations { µ t g } does not exactly define an Ω BAs -morphism but rathera twisted Ω BAs -morphism . In the case when M is odd-dimensional, this twisted Ω BAs -morphismis exactly an Ω BAs -morphism between two Ω BAs -algebras. All sign computations are detailedin section 4. If we want to go back to the more classical algebraic framework of A ∞ -algebras, an A ∞ -morphism between the induced A ∞ -algebra structures on the Morse cochains is simply obtainedunder the morphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph .3. Transversality The goal of this section is to prove theorems 7 and 10. In this regard, we recall at first theparametric transversality lemma and then build an admissible choice of perturbation data ( X n ) n (cid:62) on the moduli spaces T n , proceeding by induction on the number of internal edges e ( t ) of a stableribbon tree t . It moreover appears in our construction that all arguments adapt nicely to theframework of two-colored trees and admissible choices of perturbation data ( Y n ) n (cid:62) on the modulispaces CT n .3.1. Parametric transversality lemma. We begin by recalling Smale’s generalization of the clas-sical Sard theorem. See [Sma65] or [MS12] for a detailed proof : Theorem 13 (Sard-Smale theorem) . Let X and Y be separable Banach manifolds. Suppose that f : X → Y is a Fredholm map of class C l with l (cid:62) max(1 , ind( f ) + 1) . Then the set Y reg ( f ) ofregular values of f is residual in Y in the sense of Baire. This theorem implies in particular the following corollary in transversality theory, that will constitutethe cornerstone of our proof of theorem 7 : Corollary 1 (Parametric transversality lemma) . Let X be a Banach space, M and N two finite-dimensional manifolds and S ⊂ N a submanifold of N . Suppose that f : X × M → N is a map ofclass C l with l (cid:62) max(1 , dim( M ) + dim( S ) − dim( N ) + 1) and that it is transverse to S . Then theset X (cid:116) S := { X ∈ X such that f X (cid:116) S } is residual in X in the sense of Baire. Proof. The map f being transverse to S , the inverse image f − ( S ) is a Banach submanifold of X × M . Consider the standard projection p X : X × M → X and denote π := p X | f − ( S ) . Followinglemma 19.2 in [AR67], this map is Fredholm and has index dim( M ) + dim( S ) − dim( N ) . Moreover,drawing from an argument in section 3.2. of [MS12], there is an equality X reg ( π ) = X (cid:116) S . One canthen conclude by applying the Sard-Smale theorem to the map π . (cid:3) Proof of theorem 7. The case e ( t ) = 0 . If e ( t ) = 0 , the tree t is a corolla. Fix an integer l such that l (cid:62) max (cid:32) , e ( t ) + | y | − n (cid:88) i =1 | x i | + 1 (cid:33) . We define C l -choices of perturbation data in a similar fashion to smooth choices of perturbation data.A C l -choice of perturbation data X t on T n ( t ) then simply corresponds to a C l -choice of perturbationdatum on each external edge of t . Define the parametrization space X lt := { C l -perturbation data X t on the moduli space T n ( t ) } . This parametrization space is a Banach space. The linear combination of choices of perturbationdata is simply defined as the linear combination of each perturbation datum X t,e with e an externaledge of t . The vector space X lt is moreover Banach as each perturbation datum X t,e vanishes awayfrom a length 1 segment in D e .Given critical points y and x , . . . , x n , introduce the C l -map φ t : X lt × T n ( t ) × W S ( y ) × W U ( x ) × · · · × W U ( x n ) −→ M × n +1 , such that for every X t ∈ X lt , φ t ( X t , · ) = φ X t . Note that we should in fact write φ y,x ,...,x n t asthe domain of φ t depends on y, x , . . . , x n . The map φ t is then a submersion. This is proven inLemma 7.3. of [Abo11] and Abouzaid explains it informally in the following terms : "[this lemma] isthe infinitesimal version of the fact that perturbing the gradient flow equation on a bounded subsetof an edge integrates to an essentially arbitrary diffeomorphism".In particular the map φ t is transverse to the diagonal ∆ ⊂ M × n +1 . Applying the parametrictransversality theorem of subsection 3.1, there exists a residual set Y l ; y,x ,...x n t ⊂ X lt such thatfor every choice of perturbation data X t ∈ Y l ; y,x ,...x n t the map φ X t is transverse to the diagonal ∆ ⊂ M × n +1 . Considering the intersection Y lt := (cid:92) y,x ,...,x n Y y,x ,...x n t ⊂ X t which is again residual, any X t ∈ Y lt yields a C l -choice of perturbation data on T n ( t ) such that allthe maps φ X t are transverse to the diagonal ∆ ⊂ M × n +1 . It remains to prove this statement in thesmooth case.3.2.2. Achieving smoothness à la Taubes. Using an argument drawn from section 3.2. of [MS12] andattributed to Taubes, we now prove that the set Y t := (cid:26) smooth choices of perturbation data X t on T n ( t ) such thatall the maps φ X t are transverse to the diagonal ∆ ⊂ M × n +1 (cid:27) IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 71 is residual in the Fréchet space X t := { smooth choices of perturbation data X t on T n ( t ) } . Choose an exhaustion by compact sets L ⊂ L ⊂ L ⊂ · · · of the space T n ( t ) × W S ( y ) × W U ( x ) ×· · · × W U ( x n ) . Define Y t,L m := (cid:26) smooth choices of perturbation data X t on T n ( t ) such thatall maps φ X t are transverse on L m to the diagonal of M × n +1 (cid:27) and note that Y t = + ∞ (cid:92) m =0 Y t,L m . We will prove that each Y t,L m ⊂ Y t is open and dense in X t to conclude that Y t is indeed residual.Fix m (cid:62) . To prove that the set Y t,L m is open in X t it suffices to prove that for every l , theset Y lt,L m is open in X lt , where Y lt,L m is defined by replacing "smooth" by " C l " in the definitionof Y t,L m . This last result is a simple consequence of the fact that "being transverse on a compactsubset" is an open property : if the map φ X t is transverse on L m to the diagonal ∆ ⊂ M × n +1 thenfor X t ∈ X lt sufficiently close to X t the map φ X t is again transverse on L m to the diagonal on L m .Let now X t ∈ X t . As X t ∈ X lt and the set Y lt is dense in X lt , there exists a sequence X lt ∈ Y lt suchthat for all l || X t − X lt || C l (cid:54) − l . Note that X lt ∈ Y lt,L m . Now since the set Y lt,L m is open in X lt for the C l -topology, there exists ε l > such that for all X (cid:48) lt ∈ X lt if || X lt − X (cid:48) lt || C l (cid:54) min(2 − l , ε l ) , then X (cid:48) lt ∈ Y lt,L m . Choosing X (cid:48) lt to be smooth, this yields a sequence of smooth choices of perturbationdata lying in Y t,L m and converging to X t , which concludes the proof.3.2.3. Induction step and conclusion. Let k (cid:62) and suppose that we have constructed an admissiblechoice of perturbation data ( X t ) e ( t ) (cid:54) k . This notation should not be confused with the notation ( X i ) i (cid:54) k : the former corresponds to a choice of perturbation data on the strata T ( t ) of dimension (cid:54) k while the latter corresponds to a choice of perturbation data on the moduli spaces T i with i (cid:54) k .Let t be a stable ribbon tree with e ( t ) = k + 1 . We want to construct a choice of perturbation data X t on T n ( t ) which is smooth, gluing-compatible and such that each map φ X t is transverse to thediagonal ∆ ⊂ M × n +1 .Under a choice of identification T n ( t ) (cid:39) [0 , + ∞ ] e ( t ) , define T n ( t ) ⊂ T n ( t ) as the inverse image of [0 , + ∞ [ e ( t ) . Introduce the parametrization space X lt := C l -perturbation data X t on T n ( t ) such that X t | T ( t (cid:48) ) = X t (cid:48) for all t (cid:48) ∈ coll ( t ) and such that lim l e → + ∞ X t = X t e X t for all e ∈ E ( t ) , where t e t = t , and lim l e → + ∞ X t = X t e X t denotes the gluing-compatibility condition describedin section 1.4. Following [Mes18] this parametrization space is an affine space which is Banach. Onecan indeed show that the l e → + ∞ conditions imply that each X t ∈ X lt is bounded in the C l -norm,and that the C l -norm is thus well defined on X lt although T n ( t ) is not compact. Consider the C l -map φ t : X lt × T n ( t ) × W S ( y ) × W U ( x ) × · · · × W U ( x n ) −→ M × n +1 . Using the same argument as in subsection 3.2.1, the map φ t is again transverse to the diagonal ∆ ⊂ M × n +1 . Applying the parametric transversality theorem and proceeding as in the case e ( t ) = 0 ,there exists a residual set Y lt ⊂ X lt such that for every choice of perturbation data X t ∈ Y lt the map φ X t is transverse to the diagonal ∆ ⊂ M × n +1 . Using the previous argument à la Taubes, we canmoreover prove the same statement in the smooth context. By definition of the parametrizationspaces X t this construction yields indeed an admissible choice of perturbation data ( X t ) e ( t ) (cid:54) k +1 ,which concludes the proof of theorem 7 by induction.4. Signs, orientations and gluing We now complete and conclude the proofs of theorems 9 and 12, by expliciting all orientationsconventions on the moduli spaces of Morse gradient trees and computing the signs involved therein.We use to this extent the ad hoc formalism of signed short exact sequences of vector bundles.Particular attention will be paid to the behaviour of orientations under gluing in our proof.4.1. More on signs and orientations. Additional tools for orientations. Consider a short exact sequence of vector spaces −→ V −→ W −→ V −→ . It induces a direct sum decomposition W = V ⊕ V . Suppose that the vector spaces W , V and V are oriented. We denote ( − ε the sign obtained by comparing the orientation on W to the oneinduced by the direct sum V ⊕ V . We will then say that the short exact sequence has sign ( − ε .In particular, when ( − ε = 1 , we will say that the short exact sequence is positive .Now, consider two short exact sequences −→ V −→ W −→ V −→ and −→ V (cid:48) −→ W (cid:48) −→ V (cid:48) −→ , of respective signs ( − ε and ( − ε (cid:48) . Then the short exact sequence obtained by summing them −→ V ⊕ V (cid:48) −→ W ⊕ W (cid:48) −→ V ⊕ V (cid:48) −→ , has sign ( − ε + ε (cid:48) +dim( V (cid:48) )dim( V ) . Indeed, the direct sum decomposition writes as W ⊕ W (cid:48) = ( − ε ( V ⊕ V ) ⊕ ( − ε (cid:48) ( V (cid:48) ⊕ V (cid:48) ) (cid:39) ( − ε + ε (cid:48) +dim( V (cid:48) )dim( V ) V ⊕ V (cid:48) ⊕ V ⊕ V (cid:48) . Orientation and transversality. Given two manifolds M, N , a codimension k submanifold S ⊂ N and a smooth map φ : M −→ N which is tranverse to S , the inverse image φ − ( S ) is a codimension k submanifold of M . Moreover,choosing a complementary ν S to T S , the transversality assumption yields the following short exactsequence of vector bundles −→ T φ − ( S ) −→ T M | φ − ( S ) −→ dφ ν S −→ . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 73 Suppose now that M , N and S are oriented. The orientations on N and S induce an orientation on ν S . The submanifold φ − ( S ) is then oriented by requiring that the previous short exact sequencebe positive. We will refer to this choice of orientation as the natural orientation on φ − ( S ) .In the particular case of two submanifolds S and R of M which intersect transversely, we will usethe inclusion map S (cid:44) → M , which is transverse to R ⊂ M , to define the intersection S ∩ R . Theorientation will then be defined using the positive short exact sequence −→ T ( S ∩ R ) −→ T S | S ∩ R −→ ν R −→ , or equivalently with the direct sum decomposition T S = ν R ⊕ T ( S ∩ R ) . The intersection R ∩ S (in contrast to S ∩ R ) is oriented by interchanging S and R in the abovediscussion. The two orientations on the intersection differ then by a ( − codim( S )codim( R ) sign.4.2. Basic moduli spaces in Morse theory and their orientations. Orienting the unstable and stable manifolds. Recall that for a critical point x of a Morsefunction f , its unstable and stable manifolds are respectively defined as W U ( x ) := { z ∈ M, lim s →−∞ φ s ( z ) = x } W S ( x ) := { z ∈ M, lim s → + ∞ φ s ( z ) = x } , where we denote φ s the flow of −∇ f , and its degree is defined as | x | := dim( W S ( x )) .The unstable and stable manifolds are respectively diffeomorphic to a ( d − | x | ) -dimensional balland a | x | -dimensional ball. They are hence orientable. They intersect moreover transversely in aunique point, which is x . Assume now that the manifold M is orientable and oriented. We choosefor the rest of this section an arbitrary orientation on W U ( x ) , and endow W S ( x ) with the uniqueorientation such that the concatenation of orientations or W U ( x ) ∧ or W S ( x ) at x coincides with theorientation or M .4.2.2. Orienting the moduli spaces T ( y ; x ) . For two critical points x (cid:54) = y , the moduli spaces ofnegative gradient trajectories T ( y ; x ) can be defined in two ways. The first point of view hinges onthe fact that R acts on W S ( y ) ∩ W U ( x ) , by defining s · p = φ s ( p ) for s ∈ R and p ∈ W S ( y ) ∩ W U ( x ) .The moduli space T ( y ; x ) is then defined by considering the quotient associated to this action, i.e.by defining T ( y ; x ) := W S ( y ) ∩ W U ( x ) / R . The second point of view is to consider the transverseintersection with the level set of a regular value a , T ( y ; x ) := W S ( y ) ∩ W U ( x ) ∩ f − ( a ) . Using this description, and coorienting the level set f − ( a ) with −∇ f , the spaces T ( y ; x ) caneasily be oriented with the formalism of section 4.1.2 on transverse intersections : T W S ( y ) (cid:39) T W S ( x ) ⊕ T (cid:0) W S ( y ) ∩ W U ( x ) (cid:1) (cid:39) T W S ( x ) ⊕ −∇ f ⊕ T T ( y ; x ) . Note that the space W S ( y ) ∩ W U ( x ) consists in a union of negative gradient trajectories γ : R → M .We will therefore use the notation ˙ γ for −∇ f , which will become handy in the next section. We point out that the moduli spaces T ( y ; x ) are constructed in a different way than the modulispaces T t ( y ; x , . . . , x n ) : they cannot naturally be viewed as an arity 1 case of the moduli spaces ofgradient trees. This observation will be of importance in our upcoming discussion on signs for the Ω BAs -algebra structure on the Morse cochains.Finally, the moduli spaces T ( y ; x ) are manifolds of dimension dim( T ( y ; x )) = | y | − | x | − , which can be compactified to manifolds with corners T ( y ; x ) , by allowing convergence towards brokennegative gradient trajectories. See for instance [Weh12]. In the case where they are 1-dimensional,their boundary is given by the signed union ∂ T ( y ; x ) = (cid:91) z ∈ Crit( f ) −T ( y ; z ) × T ( z ; x ) . We moreover recall from section 1.1 that we work under the convention T ( x ; x ) = ∅ .4.2.3. Compactifications of the unstable and stable manifolds. Using the moduli spaces T ( y ; x ) , wecan now compactify the manifolds W S ( y ) and W U ( x ) to compact manifolds with corners W S ( y ) and W U ( x ) . See [Hut08] for instance. With our choices of orientations detailed in the previoussection, the top dimensional strata in their boundary are given by ∂W S ( y ) = (cid:91) z ∈ Crit( f ) ( − | z | +1 W S ( z ) × T ( y ; z ) ,∂W U ( x ) = (cid:91) z ∈ Crit( f ) ( − ( d −| z | )( | x | +1) W U ( z ) × T ( z ; x ) , where d is the dimension of the ambient manifold M .The pictures in the neighborhood of the critical point z are represented in figure 27. For instance,in the case of ∂W S ( y ) , an element of W S ( y ) is seen as lying on a negative semi-infinite trajectoryconverging to y , and an outward-pointing vector to the boundary is given by − ˙ γ . We hence havethat − ˙ γ ⊕ T W S ( z ) ⊕ T T ( y ; z ) = ( − | z | T W S ( z ) ⊕ − ˙ γ ⊕ T T ( y ; z ) = ( − | z | +1 T W S ( y ) .γW S ( z ) − ˙ γ W S ( y ) z y W U ( x ) γ W U ( z ) ˙ γ xz . Figure 27 IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 75 Euclidean neighborhood of a critical point. Following [Weh12], we will assume in the rest ofthis part that the pair (Morse function,metric) on the manifold M is Euclidean. Denote B kδ := { x ∈ R k , | x | < δ } . Such a pair is said to Euclidean if it is Morse-Smale and is such that for each criticalpoint z ∈ Crit( f ) there exists a local chart φ : B d −| z | δ × B | z | δ ˜ −→ U z ⊂ M , such that φ (0) = z andsuch that the function f and the metric g read as f ( x , . . . , x n −| z | , y , . . . , y | z | ) = f ( p ) − 12 ( x + · · · + x n −| z | ) + 12 ( y + · · · + y | z | ) g = n −| z | (cid:88) i =1 dx i ⊗ dx i + | z | (cid:88) i =1 dy i ⊗ dy i in the chart φ . In this chart, we then have that W U ( z ) := { y = · · · = y | z | = 0 } W S ( z ) := { x = · · · = x n −| z | = 0 } , and M = W U ( z ) × W S ( z ) . Hence any point of U z can be uniquely written as a sum x + y where x ∈ W U ( z ) and y ∈ W S ( z ) . Choosing now s ∈ R such that the the image of x + y under the Morseflow map φ s still lies in U z , we have that φ s ( x + y ) = e s x + e − s y . These observations will reveal crucial in the proof of subsection 4.4.3.4.3. Preliminaries for section 4.4. Counting the points on the boundary of an oriented 1-dimensional manifold. Consider anoriented 1-dimensional manifold with boundary. Then its boundary ∂M is oriented. Assume it canbe written set-theoretically as a disjoint union ∂M = (cid:71) i N i . Suppose now that each N i comes with its own orientation, and write ( − † i for the sign obtained bycomparing this orientation to the boundary orientation. As orientable manifolds, the union writesas ∂M = (cid:71) i ( − † i N i . The N i being 0-dimensional, they can be seen as collections of points each coming with a + or − sign. Noticing that an orientable 1-dimensional manifold with boundary is either a segment or acircle, and writing N i for the signed count of points of N i , the previous equality finally impliesthat (cid:88) ( − † i N i = 0 . This basic observation is key to constructing most algebraic structures arising in symplectic topology(and in particular Morse theory). For instance, for a critical point x , counting the boundary points of the 1-dimensional manifolds T ( y ; x ) proves that ∂ Morse ◦ ∂ Morse ( x ) = (cid:88) y ∈ Crit( f ) | y | = | x | +2 (cid:88) z ∈ Crit( f ) | z | = | x | +1 T ( y ; z ) T ( z ; x ) · y = 0 . The equations for Ω BAs -algebras and Ω BAs -morphisms will be proven using this method in thefollowing two subsections.4.3.2. Reformulating the Ω BAs -equations. We fix for each t ∈ sRT n an orientation ω t . Given a t ∈ sRT n the orientation ω t defines an orientation of the moduli space T n ( t ) , and we write moreover m t for the operations ( t, ω ) . The Ω BAs -equations for an Ω BAs -algebra then read as [ ∂, m t ] = (cid:88) t (cid:48) ∈ coll ( t ) ( − † Ω BAs m t (cid:48) + (cid:88) t i t = t ( − † Ω BAs m t ◦ i m t , where the notations for trees are as defined previously. The signs ( − † Ω BAs are obtained as insection 5.1, by computing the signs of T n ( t (cid:48) ) and T i +1+ i ( t ) × i T i ( t ) in the boundary of T n ( t ) .We will not need to compute their explicit value, and will hence keep this useful notation ( − Ω BAs to refer to them.4.3.3. Twisted A ∞ -algebras and twisted Ω BAs -algebras. It is clear using this counting method, thatthe operations m t of section 1.5 will endow the Morse cochains C ∗ ( f ) with a structure of Ω BAs -algebra over Z / . Working over integers will prove more difficult, and we will prove a weaker result inthis case. We introduce to this extent the notion of twisted A ∞ -algebras and twisted Ω BAs -algebras. Definition 39. A twisted A ∞ -algebra is a dg- Z -module A endowed with two different differentials ∂ and ∂ , and a sequence of degree − n operations m n : A ⊗ n → A such that [ ∂, m n ] = − (cid:88) i + i + i = n (cid:54) i (cid:54) n − ( − i + i i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) , where [ ∂, · ] denotes the bracket for the maps ( A ⊗ n , ∂ ) → ( A, ∂ ) . A twisted Ω BAs -algebra is definedsimilarly.We make explicit the formulae obtained by evaluating the Ω BAs -equations on A ⊗ n , as we willneed them in our next proof : − ∂ m t ( a , . . . , a n ) + ( − | t | + (cid:80) i − j =1 | a j | m t ( a , . . . , a i − , ∂ a i , a i +1 , . . . , a n )+ (cid:88) t t = t ( − † Ω BAs + | t | (cid:80) i j =1 | a j | m t ( a , . . . , a i , m t ( a i +1 , . . . , a i + i ) , a i + i +1 , . . . , a n )+ (cid:88) t (cid:48) ∈ coll ( t ) ( − † Ω BAs m t (cid:48) ( a , . . . , a n )= 0 . We refer to them as "twisted", as these algebras will occur in the upcoming lines by setting ∂ :=( − σ ∂ , that is by simply twisting the differential ∂ by a specific sign. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 77 Note that these two definitions cannot be phrased in terms of operads, as Hom(( A, ∂ ) , ( A, ∂ )) is an (Hom( A, ∂ ) , Hom( A, ∂ )) -operadic bimodule but is NOT an operad : the composition mapson Hom(( A, ∂ ) , ( A, ∂ )) are associative, but they fail to be compatible with the differential [ ∂, · ] .As a result, a twisted A ∞ -algebra cannot be described as a morphism of operads from A ∞ to Hom(( A, ∂ ) , ( A, ∂ )) . However, a twisted Ω BAs -algebra structure always transfers to a twisted A ∞ -algebra structure. Indeed, while the functorial proof of 3.1.5 does not work anymore, we point outthat the morphism of operads A ∞ → Ω BAs still contains the proof that a sequence of operations m t defining a twisted Ω BAs -algebra structure on A can always be arranged in a sequence of operations m n defining a twisted A ∞ -algebra structure on A .4.3.4. The maps ψ e i , X t . Consider again a stable ribbon tree t and order its external edges clockwise,starting with e at the outgoing edge. Given a choice of perturbation data X t , we illustrate infigure 28 a mean to visualize the map φ X t : T n ( t ) × W S ( y ) × W U ( x ) × · · · × W U ( x n ) −→ M × n +1 defined in section 1.3. We introduce a family of maps defined in a similar fashion. Consider e i anincoming edge of t . Define the map ψ e i , X t : T n ( t ) × W S ( y ) × W U ( x ) × · · · × (cid:92) W U ( x i ) × · · · × · · · × W U ( x n ) −→ M × n to be the map which for a fixed metric tree T takes a point of a W U ( x j ) for j (cid:54) = i to the point in M obtained by following the only non-self crossing path from the time − point on e j to the time − point on e i in T through the perturbed gradient flow maps associated to X T , and which takes apoint of W s ( y ) to the point in M obtained by following the only non-self crossing path from the time point on e to the time − point on e i in T through the perturbed gradient flow maps associatedto X T . The map ψ e , X t is defined similarly for the outgoing edge e . These two definitions are twobe understood as depicted on two examples in figure 28. W U ( x ) W S ( y ) W U ( x ) W U ( x ) φ X t W U ( x ) W S ( y ) W U ( x ) ψ e , X t W U ( x ) W U ( x ) W U ( x ) ψ e , X t Figure 28. Representations of a map φ X t , a map ψ e , X t and a map ψ e , X t The twisted Ω BAs -algebra structure on the Morse cochains. Summary of the proof of theorem 9. Definition 40. (i) We define (cid:101) T X t ( y ; x , . . . , x n ) to be the oriented manifold T X t ( y ; x , . . . , x n ) whose natural orientation has been twisted by a sign of parity σ ( t ; y ; x , . . . , x n ) := dn (1 + | y | + | t | ) + | t || y | + d n (cid:88) i =1 | x i | ( n − i ) . (ii) Similarly, we define (cid:101) T ( y ; x ) to be the oriented manifold T ( y ; x ) whose natural orientationhas been twisted by a sign of parity σ ( y ; x ) := 1 . The operations m t and the differential on C ∗ ( f ) are then defined as m t ( x , . . . , x n ) = (cid:88) | y | = (cid:80) ni =1 | x i | + | t | (cid:101) T X t ( y ; x , . . . , x n ) · y ,∂ Morse ( x ) = (cid:88) | y | = | x | +1 (cid:101) T ( y ; x ) · y . Proposition 19. If (cid:101) T t ( y ; x , . . . , x n ) is 1-dimensional, its boundary decomposes as the disjoint unionof the following components(i) ( − | y | + † Ω BAs + | t | (cid:80) i i =1 | x i | (cid:101) T t ( y ; x , . . . , x i , z, x i + i +1 , . . . , x n ) × (cid:101) T t ( z ; x i +1 , . . . , x i + i ) ;(ii) ( − | y | + † Ω BAs (cid:101) T t (cid:48) ( y ; x , . . . , x n ) for t (cid:48) ∈ coll ( t ) ;(iii) ( − | y | + † Koszul +( d +1) | x i | (cid:101) T t ( y ; x , . . . , z, . . . , x n ) × (cid:101) T ( z ; x i ) where † Koszul = | t | + (cid:80) i − j =1 | x j | ;(iv) ( − | y | +1 (cid:101) T ( y ; z ) × (cid:101) T t ( z ; x , . . . , x n ) . Applying the method of subsection 4.3.1 finally proves that : Theorem 9. The operations m t define a twisted Ω BAs -algebra structure on ( C ∗ ( f ) , ∂ T wMorse , ∂ Morse ) ,where ( ∂ T wMorse ) k = ( − ( d +1) k ∂ kMorse . Signs for the (int-break) boundary. We resort to the formalism of short exact sequences ofvector bundles to handle orientations in this section. For the sake of readability, we will write N rather than T N for the tangent bundle of a manifold N in the upcoming computations.The moduli space T t ( y ; x , . . . , x n ) is defined as the inverse image of the diagonal ∆ ⊂ M × n +1 under the map φ X t : T n ( t ) × W S ( y ) × W U ( x ) × · · · × W U ( x n ) −→ M × n +1 , where the factors of M × n +1 are labeled in the order M y × M x × · · · × M x n . Orienting the domainand codomain of φ X t by taking the product orientations, and orienting ∆ as M , defines the nat-ural orientation on T t ( y ; x , . . . , x n ) as in subsection 4.1.2. Choose M × n labeled by x , . . . , x n ascomplementary to ∆ . Then the orientation induced on M × n by the orientations on M × n +1 and on ∆ , differs by a ( − d n sign from the product orientation of M × n . In the language of short exactsequences, T t ( y ; x , . . . , x n ) is oriented by the short exact sequence −→ T t ( y ; x , . . . , x n ) −→ T n ( t ) × W S ( y ) × n (cid:89) i =1 W U ( x i ) −→ M × n −→ , which has a sign of parity dn . (A)In the case of T Morset := T t ( y ; x , . . . , x i , z, x i + i +1 , . . . , x n ) , we choose M × i +1+ i labeled by y, x , . . . , x i , x i + i +1 , . . . , x n as complementary to ∆ . The orientation induced on M × i +1+ i , by IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 79 the orientations on M × i +2+ i and on ∆ , differs by a ( − d i sign from the product orientation of M × i +1+ i . Hence the short exact sequence −→ T Morset −→ T i +1+ i ( t ) × W S ( y ) × i (cid:89) i =1 W U ( x i ) × W U ( z ) × n (cid:89) i = i + i +1 W U ( x i ) −→ M × i +1+ i → , has a sign of parity di . (B)In the case of T Morset := T t ( z ; x i +1 , . . . , x i + i ) , we choose M × i labeled by x i +1 , . . . , x i + i ascomplementary to ∆ . The orientation induced on M × i differs this time by a ( − d i sign from theproduct orientation. The short exact sequence −→ T Morset −→ T i ( t ) × W S ( z ) × i + i (cid:89) i = i +1 W U ( x i ) −→ M × i → , has now a sign given by the parity of di . (C)Following the convention of subsection 4.1.1, taking the product −→ T Morset × T Morset −→ T i +1+ i ( t ) × W S ( y ) × i (cid:89) i =1 W U ( x i ) × W U ( z ) × n (cid:89) i = i + i +1 W U ( x i ) × T i ( t ) × W S ( z ) × i + i (cid:89) i = i +1 W U ( x i ) −→ M × i +1+ i × M × i → doesn’t introduce a sign, as T Morset and T Morset are 0-dimensional.In the previous short exact sequence, M × i +1+ i × M × i is labeled by y, x , . . . , x i , x i + i +1 , . . . , x n , x i +1 , . . . , x i + i . We rearrange this labeling into y, x , . . . , x n , which induces a sign given by the parity of di i . (D)We also rearrange the expression T i +1+ i ( t ) × W S ( y ) × i (cid:89) i =1 W U ( x i ) × W U ( z ) × n (cid:89) i = i + i +1 W U ( x i ) × T i ( t ) × W S ( z ) × i + i (cid:89) i = i +1 W U ( x i ) , into W U ( z ) × W S ( z ) × T i +1+ i ( t ) × T i ( t ) × W S ( y ) × n (cid:89) i =1 W U ( x i ) . The parity of the produced sign is that of | z | (cid:32) | t | + n (cid:88) i = i + i +1 ( d − | x i | ) (cid:33) + m (cid:32) | t | + | y | + i (cid:88) i =1 ( d − | x i | ) (cid:33) (E) + | t | (cid:32) | y | + i (cid:88) i =1 ( d − | x i | ) + n (cid:88) i = i + i +1 ( d − | x i | ) (cid:33) + (cid:32) i + i (cid:88) i = i +1 ( d − | x i | ) (cid:33) (cid:32) n (cid:88) i = i + i +1 ( d − | x i | ) (cid:33) . Introduce now the factor [ L, + ∞ [ , corresponding to the length l e increasing towards + ∞ , where e is the edge of t whose breaking produces t and t . Following convention 4.1.2, the short exactsequence −→ [ L, + ∞ [ ×T Morset × T Morset −→ [ L, + ∞ [ × W U ( z ) × W S ( z ) × T ( t ) × T ( t ) × × W S ( y ) × n (cid:89) i =1 W U ( x i ) −→ M × n +1 −→ , induces a sign change whose parity is given by d ( n + 1) . (F)Define the map ψ : M × T n ( t ) × W S ( y ) × n (cid:89) i =1 W U ( x i ) −→ M × M × n +1 , which is defined on the factors T n ( t ) × W S ( y ) × (cid:81) ni =1 W U ( x i ) as φ and is defined on M × T n ( t ) byseeing M as the point lying in the middle of the edge e in t . This map is depicted on figure 29. Theinverse image of the diagonal of M × M × n +1 is exactly T t ( y ; x , . . . , x n ) . Fix now a sufficiently great L > . We prove in subsection 4.4.3 that orienting [ L, + ∞ [ ×T Morset × T Morset with the previousshort exact sequence, the orientation induced on T Morset by gluing is the exactly the one given bythe short exact sequence T Morset [ L, + ∞ [ × M × T ( t ) × T ( t ) × W S ( y ) × (cid:81) ni =1 W U ( x i ) M × n +1 dψ , where our convention on orientations for the unstable and stable manifolds of z implies that W U ( z ) × W S ( z ) yields indeed the orientation of M , and M × n +1 is labeled by y, x , . . . , x n . M W U ( x ) W S ( y ) W U ( x ) W U ( x ) ψ Figure 29. Representation of the map ψ Transform the coorientation labeled by y, x , . . . , x n into the coorientation labeled by M, x , . . . , x n and rearrange the factors [ L, + ∞ [ × M × T ( t ) × T ( t ) × · · · into M × [ L, + ∞ [ × T ( t ) × T ( t ) × · · · IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 81 This produces a sign change of parity d + d ≡ . (G)We can moreover now delete the two M factors associated to the label M to obtain the short exactsequence −→ T t ( y ; x , . . . , x n ) −→ [ L, + ∞ [ ×T ( t ) × T ( t ) × W S ( y ) × n (cid:89) i =1 W U ( x i ) −→ M × n −→ , where M × n = M x × · · · × M x n .Transforming finally [ L, + ∞ [ ×T ( t ) × T ( t ) into T n ( t ) gives a sign of parity † Ω BAs . (H)In closing, the short exact sequence −→ T t ( y ; x , . . . , x n ) −→ T n ( t ) × W S ( y ) × n (cid:89) i =1 W U ( x i ) −→ M × n −→ , has sign given by the parity of A when T Morset is endowed with its natural orientation. It has signgiven by the parity of B + C + D + E + F + G + H when T Morset is endowed with the orientationinduced by [ L, + ∞ [ ×T Morset × T Morset , where the first factor is the length l e and determines theoutward-pointing direction ν e to the boundary component T Morset × T Morset .We thus obtain that with our choice of orientation on the moduli spaces T Morset , the sign of T t ( y ; x , . . . , x i , z, x i + i +1 , . . . , x n ) × T t ( z ; x i +1 , . . . , x i + i ) in the boundary of the 1-dimensionalmoduli space T t ( y ; x , . . . , x n ) is given by the parity of ( ∗ ) A + B + C + D + E + F + G + H = | z || t | + d | y | + d | t | + ( n + 1) d + i (cid:88) i =1 d | x i | + | t || y | + di | t | + di n (cid:88) i = i + i +1 | x i | + † Ω BAs + | t | i (cid:88) i =1 | x i | . Hence the sign of (cid:101) T t ( y ; x , . . . , x i , z, x i + i +1 , . . . , x n ) × (cid:101) T t ( z ; x i +1 , . . . , x i + i ) in the boundary ofthe 1-dimensional moduli space (cid:101) T t ( y ; x , . . . , x n ) is given by the parity of σ ( t ; y ; x , . . . , x n ) + σ ( t ; y ; x , . . . , x i , z, x i + i +1 , . . . , x n ) + σ ( t ; z ; x i +1 , . . . , x i + i ) + ( ∗ )= | y | + † Ω BAs + | t | i (cid:88) i =1 | x i | . Gluing and orientations. We prove in this subsection that after orienting [ L, + ∞ [ ×T Morset ×T Morset with the short exact sequence L, + ∞ [ × T Morset × T Morset [ L, + ∞ [ × W U ( z ) × W S ( z ) × T ( t ) × T ( t ) × W S ( y ) × (cid:81) ni =1 W U ( x i ) M × n +1 , the orientation induced on T Morset by gluing is the one given by the short exact sequence T Morset [ L, + ∞ [ × M × T ( t ) × T ( t ) × W S ( y ) × (cid:81) ni =1 W U ( x i ) M × n +1 dψ . The proof boils down to the following lemma. Lemma 1. Let M and N be manifolds and S ⊂ N a submanifold of N . Suppose that M , N and S areorientable and oriented. Let f : [0 , × M → N be a smooth map such that f := f (1 , · ) : M → N is transverse to S . Let x ∈ f − ( S ) . Then there exist an open subset V of M containing x and (cid:54) t < such that(i) The map f | [ t , × V : [ t , × V → N is transverse to S . In particular the inverse image f | − t , × V ( S ) is then a submanifold of [ t , × V .(ii) There exists an orientation-preserving embedding f | − t , × V ( S ) −→ [ t , × f − ( S ) equal to the identity on f | − V ( S ) and preserving the t coordinate, where we orient [ t , × f − ( S ) with the short exact sequence −→ [ t , × f − ( S ) −→ [0 , × M −→ ν S −→ and we orient f | − t , × V ( S ) with the short exact sequence −→ f | − t , × V ( S ) −→ [0 , × M −→ ν S −→ . Proof. Choose an adapted chart for S around f ( x ) , i.e. a chart φ : U (cid:48) ⊂ N → R n such that φ ( U (cid:48) ∩ S ) = { ( y , . . . , y n − s , x , . . . , x s ) ∈ R n , y = · · · = y n − s = 0 } , where n and s respectively denote the dimensions of N and S . Using the local normal form theoremfor submersions, there exists a local chart ψ : U ⊂ M → R m around x such that the map f readsas ( y , . . . , y n − s , x , . . . , x m + s − n ) (cid:55)−→ ( y , . . . , y n − s , F ( (cid:126)y, (cid:126)x ) , . . . , F s ( (cid:126)y, (cid:126)x )) in the local charts ψ and φ , where the F i are smooth maps and (cid:126)y := y , . . . , y n − s , (cid:126)x := x , . . . , x m + s − n and m := dim( M ) . In these local charts, U ∩ f − ( U (cid:48) ∩ S ) = { ( y , . . . , y n − s , x , . . . , x m + s − n ) ∈ R m , y = · · · = y n − s = 0 } . The property "being transverse to S " being open, there exists a neighborhood W of x in M and t ∈ [0 , such that the map f | [ t , × W : [ t , × W → N is transverse to S . Suppose W ⊂ U andconsider now the projection π : R m → R m + s − n given by ( y , . . . , y n − s , x , . . . , x m + s − n ) (cid:55)−→ ( x , . . . , x m + s − n ) and define the smooth map ι := id t × π : f | − t , × W ( S ) −→ [0 , × f − ( S ) in the local charts φ and ψ . The differential of this map is invertible at (1 , x ) . The inverse functiontheorem then ensures that there exits t ∈ [ t , and a neighborhood V ⊂ W of x such that themap ι : f | − t , × V ( S ) −→ [0 , × f − ( S ) is a diffeomorphism on its image.Orient now [0 , × f − ( S ) and f | − t , × V ( S ) with the previous short exact sequences. It remainsto show that the map ι is orientation-preserving. The proof of this result can be reduced to a proof IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 83 in linear algebra, i.e. by considering a smooth family of linear maps f : [0 , × R m → R n such that f reads as ( y , . . . , y n − s , x , . . . , x m + s − n ) (cid:55)−→ ( y , . . . , y n − s , F ( (cid:126)y, (cid:126)x ) , . . . , F s ( (cid:126)y, (cid:126)x )) , and the linear subspace S = { } × R s ⊂ R n . Then there exists t ∈ [0 , such that f | [ t , × R m istransverse to S , and we can consider the smooth map ι := id t × π : f | − t , × R m ( S ) −→ [0 , × f − ( S ) which is a diffeomorphism on its image. Basic computations finally show that the map ι is indeedorientation-preserving. (cid:3) We now go back to our initial problem. Let T Morse ∈ T Morse and T Morse ∈ T Morse , where werefer to subsection 4.4.2 for notations. Consider a local Euclidean chart φ z : U z → R d for the criticalpoint z as in subsection 4.2.4. Introduce the map ev : [0 , + ∞ ] × U z → U z × U z reading as ( δ, x + y ) (cid:55)−→ ( e − δ x + y, x + e − δ y ) in the chart φ z . The pair ev ( δ, x + y ) corresponds to the two endpoints of the unique finite Morsetrajectory parametrized by [ − δ, δ ] and meeting e − δ x + e − δ y at time .Consider the trajectory γ e, : ] − ∞ , → M and the trajectory γ e, : [0 , + ∞ [ → M , respectivelyassociated to the incoming edge of T Morse and to the outgoing edge of T Morse which result from thebreaking of the edge e in t . Choose L large enough such that γ e, ( − L ) and γ e, ( L ) belong to U z .Introduce the map f := ev × ( φ − ( L − ) × i +1+ i ◦ ψ e, X t × ( φ L − ) × i ◦ ψ e, X t acting as [0 , + ∞ ] × U z × T i +1+ i ( t ) × W S ( y ) × (cid:81) i i =1 W U ( x i ) × (cid:81) ni = i + i +1 W U ( x i ) × T i ( t ) × (cid:81) i + i i = i +1 W U ( x i ) −→ M × × M × i +1+ i × M × i , where φ L − stands for the time L − Morse flow and the maps ψ e, X t and ψ e, X t have been introducedin subsection 4.2.4. This map is depicted in figure 30. W U ( x ) W U ( x ) W U ( x ) W S ( y ) zU z Mψ e, X t φ L − ψ e, X t φ − ( L − ev δ Figure 30. Representation of the map f . The label M corresponds to the point e − δ x + e − δ y and not to the point x + y . Define the d -dimensional submanifold Λ ⊂ M × × M × i +1+ i × M × i to be Λ := ( m z , m z , m y , m , . . . , m i , m i +1+ i , . . . , m n , m i +1 , . . . , m i + i ) such that m z = m i +1 = · · · = m i + i and m z = m y = m = · · · = m i = m i +1+ i = · · · = m n . The pair ( T Morse , T Morse ) then belongs to the inverse image f − ∞ (Λ) . By assumption on the choiceof perturbation data ( X n ) n (cid:62) , the map f + ∞ is moreover transverse to Λ . Applying lemma 1 to themap f at the point ( T Morse , T Morse ) , there exists R > and an embedding T Morse ,T Morse : [ R, + ∞ ] −→ T t ( y ; x , . . . , x n ) . Note that the parameter δ corresponds to an edge of length L + 2 δ in the resulting glued tree. Uponreordering the factors of the domain of f , it is finally easy to check that this lemma also implies theresult on orientations stated at the beginning of this subsection.4.4.4. Signs for the (int-collapse) and (Morse) boundary. Repeating the beginning of the previoussection, for the moduli spaces T t (cid:48) ( y ; x , . . . , x n ) , where t (cid:48) ∈ coll ( t ) , and T t ( y ; x , . . . , x n ) , we choose M × n labeled by x , . . . , x n as complementary to the diagonal ∆ ⊂ M × n +1 . The parity of the totalsign change coming from these coorientation choices is dn + dn = 0 . (A)Introduce the factor ]0 , L ] , corresponding to the length l e going towards , where e is the edge of t whose collapsing produces t (cid:48) . Applying again lemma 1 and following convention 4.1.1, the shortexact sequence −→ T t ( y ; x , . . . , x n ) =]0 , L ] × T t (cid:48) ( y ; x , . . . , x n ) −→ ]0 , L ] × T n ( t (cid:48) ) × W S ( y ) × n (cid:89) i =1 W U ( x i ) −→ M × n −→ , introduces a sign change whose parity is given by dn . (B)Transforming finally ]0 , L ] × T n ( t (cid:48) ) into T n ( t ) gives a sign of parity † Ω BAs . (C)Adding these contributions, we obtain that the sign of T t (cid:48) ( y ; x , . . . , x n ) in the boundary of the1-dimensional moduli space T t ( y ; x , . . . , x n ) is given by the parity of A + B + C = dn + † Ω BAs . (*)The sign of (cid:101) T t (cid:48) ( y ; x , . . . , x n ) in the boundary of the 1-dimensional moduli space (cid:101) T t ( y ; x , . . . , x n ) ishence given by the parity of σ ( t ; y ; x , . . . , x n ) + σ ( t (cid:48) ; y ; x , . . . , x n ) + ( ∗ ) = | y | + † Ω BAs . Finally, the signs for the (Morse) boundary can be computed following the exact same lines of thetwo previous proofs.4.5. The twisted Ω BAs -morphism between the Morse cochains. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 85 Reformulating the Ω BAs -equations. We set again for the rest of this section an orientation ω for each t g ∈ sCRT n , which endows each moduli space CT n ( t g ) with an orientation, and writemoreover µ t g for the operations ( t g , ω ) of Ω BAs − Morph . The Ω BAs -equations for an Ω BAs -morphism then read as [ ∂, µ t g ] = (cid:88) t (cid:48) g ∈ coll ( t g ) ( − † Ω BAs µ t (cid:48) g + (cid:88) t (cid:48) g ∈ g − vert ( t g ) ( − † Ω BAs µ t (cid:48) g + (cid:88) t g i t = t g ( − † Ω BAs µ t g ◦ i m t + (cid:88) t t g ,...,t sg )= t g ( − † Ω BAs m t ◦ ( µ t g ⊗ · · · ⊗ µ t sg ) , where the notations for trees are transparent. The signs ( − † Ω BAs are obtained as in subsection 4.3.2.4.5.2. Twisted A ∞ -morphisms and twisted Ω BAs -morphisms. Again, it is clear using the countingmethod of 4.3.1 that if we work over Z / , the operations µ t g of 2.5 define an Ω BAs -morphism. Wewill prove a weaker result in the case of integers, introducing for this matter the notion of twisted A ∞ -morphisms and twisted Ω BAs -morphisms. Definition 41. Let ( A, ∂ , ∂ , m n ) and ( B, ∂ , ∂ , m n ) be two twisted A ∞ -algebras. A twisted A ∞ -morphism from A to B is defined to be a sequence of degree − n operations f n : A ⊗ n → B suchthat [ ∂, f n ] = (cid:88) i + i + i = ni (cid:62) ( − i + i i f i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) − (cid:88) i + ··· + i s = ns (cid:62) ( − (cid:15) B m s ( f i ⊗ · · · ⊗ f i s ) , where [ ∂, · ] denotes the bracket for the maps ( A ⊗ n , ∂ ) → ( B, ∂ ) . A twisted Ω BAs -morphism between twisted Ω BAs -algebras is defined similarly.The formulae obtained by evaluating the Ω BAs -equations on A ⊗ n then become − ∂ µ t g ( a , . . . , a n ) + ( − | t g | + (cid:80) i − j =1 | a j | µ t g ( a , . . . , a i − , ∂ a i , a i +1 , . . . , a n )+ (cid:88) t g t = t ( − † Ω BAs + | t | (cid:80) i j =1 | a j | µ t g ( a , . . . , a i , m t ( a i +1 , . . . , a i + i ) , a i + i +1 , . . . , a n )+ (cid:88) t t g ,...,t sg )= t g ( − † Ω BAs + † Koszul m t ( µ t g ( a , . . . , a i ) , . . . , µ t sg ( a i + ··· + i s − +1 , . . . , a n ))+ (cid:88) t (cid:48) g ∈ coll ( t g ) ( − † Ω BAs µ t (cid:48) g ( a , . . . , a n ) + (cid:88) t (cid:48) g ∈ g − vert ( t g ) ( − † Ω BAs µ t (cid:48) g ( a , . . . , a n )= 0 , where † Koszul = s (cid:88) r =1 | t rg | r − (cid:88) t =1 i t (cid:88) j =1 | a i + ··· + a it − + j | . Again these two definitions cannot be phrased using an operadic viewpoint. However, a twisted Ω BAs -morphism between twisted Ω BAs -algebras always descends to a twisted A ∞ -morphism be-tween twisted A ∞ -algebras, for the same reason as in subsection 4.3.3. Summary of the proof of theorem 12. Let X f and X g be admissible choices of perturbationdata on the moduli spaces T n for the Morse functions f and g , and Y be a choice of perturbationdata on the moduli spaces CT n that is admissible w.r.t. X f and X g . Definition 42. We define (cid:102) CT Y t g ( y ; x , . . . , x n ) to be the oriented manifold CT Y t g ( y ; x , . . . , x n ) whosenatural orientation has been twisted by a sign of parity σ ( t g ; y ; x , . . . , x n ) := dn (1 + | y | + | t g | ) + | t g || y | + d n (cid:88) i =1 | x i | ( n − i ) . The moduli spaces (cid:101) T ( y ; x ) and (cid:101) T t ( y ; x , . . . , x n ) are moreover defined as in section 4.4. We definethe operations µ t g : C ∗ ( f ) ⊗ n → C ∗ ( g ) as µ t g ( x , . . . , x n ) = (cid:88) | y | = (cid:80) ni =1 | x i | + | t g | (cid:102) CT Y t g ( y ; x , . . . , x n ) · y . Proposition 20. If (cid:102) CT t g ( y ; x , . . . , x n ) is 1-dimensional, its boundary decomposes as the disjointunion of the following components(i) ( − | y | + † Ω BAs + | t | (cid:80) i i =1 | x i | (cid:102) CT t g ( y ; x , . . . , x i , z, x i + i +1 , . . . , x n ) × (cid:101) T t ( z ; x i +1 , . . . , x i + i ) ;(ii) ( − | y | + † Ω BAs + † Koszul (cid:101) T t ( y ; y , . . . , y s ) × (cid:102) CT t g ( y ; x , . . . ) × · · · × (cid:102) CT t sg ( y s ; . . . , x n ) ;(iii) ( − | y | + † Ω BAs (cid:102) CT t (cid:48) g ( y ; x , . . . , x n ) for t (cid:48) ∈ coll ( t ) ;(iv) ( − | y | + † Ω BAs (cid:102) CT t (cid:48) g ( y ; x , . . . , x n ) for t (cid:48) ∈ g − vert ( t ) ;(v) ( − | y | + † Koszul +( m +1) | x i | (cid:102) CT t g ( y ; x , . . . , z, . . . , x n ) × (cid:101) T ( z ; x i ) where † Koszul = | t g | + i − (cid:88) j =1 | x j | ;(vi) ( − | y | +1 (cid:101) T ( y ; z ) × (cid:102) CT t g ( z ; x , . . . , x n ) . Applying the method of subsection 4.3.1 again finally proves that : Theorem 12. The operations µ t g define a twisted Ω BAs -morphism between the Morse cochains ( C ∗ ( f ) , ∂ T wMorse , ∂ Morse ) and ( C ∗ ( g ) , ∂ T wMorse , ∂ Morse ) . Gluing. We construct explicit gluing maps in the two-colored framework using lemma 1.Gluing maps for the (above-break) boundary components are built as in subsection 4.4.3. In the(below-break) case, consider critical points y, y , . . . , y s ∈ Crit( g ) and x , . . . , x n ∈ Crit( f ) such thatthe moduli spaces T t ( y ; y , . . . , y s ) and CT t rg ( y r ; x i + ··· + i r − +1 , . . . , x i + ··· + i r ) are 0-dimensional. Let T ,Morse ∈ T Morset and T r,Morseg ∈ CT Morset rg . Fix moreover an Euclidean neighborhood U z r ofeach critical point z r and choose L large enough such that for r = 1 , . . . , s , γ e r ,T ,Morse ( − L ) and γ e ,T r,Morseg ( L ) belong to U z r . Define finally the map σ e , X t : M → M × s in a similar fashion tothe maps ψ e i , X t , as depicted for instance in figure 31. Gluing maps for the perturbed Morse trees T ,Morse and T r,Morseg can then be defined by applying lemma 1 to the map [0 , + ∞ ] × s (cid:89) r =1 U z r × T s ( t ) × W S ( y ) × s (cid:89) r =1 CT i r ( t rg ) × i + ··· + i r (cid:89) i = i + ··· + i r − +1 W U ( x i ) −→ M × s × M × s × s (cid:89) r =1 M × i r . defined as follows : IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 87 (i) the factor T s ( t ) × W S ( y ) is sent to M × s under the map ( φ − ( L − ) × s ◦ σ e ,t ;(ii) the factor CT i r ( t rg ) × (cid:81) W U ( x i ) is sent to M × i r under the map ( φ ( L − ) × i r ◦ σ e ,t rg ;(iii) the factor [0 , + ∞ ] × (cid:81) sr =1 U z r is sent to M × s under the map ev U z l δ × · · · × ev U zs l sδ where δ denotes the parameter in [0 , + ∞ ] and the lengths l rδ are defined as in subsection 5.2.7 ofpart 1 in order for them to define a two-colored metric ribbon tree. In particular, we haveexplicit formulae depending on δ for the resulting edges in the glued tree. W S ( y ) σ e , X t Figure 31. Representation of the map σ e , X t .4.6. On these twisted structures. Note first that if we work with coefficients in Z / , the opera-tions m t define of course an Ω BAs -algebra structure on the Morse cochains. The operations µ t g thendefine an Ω BAs -morphism between two Ω BAs -algebras. We will say that the structure we definedare untwisted . We hence work now over the integers Z . It appears from the definition of ∂ T wMorse thatwhen M is odd-dimensional, the structures we define are untwisted. In the even-dimensional case,the structures are twisted, and it remains to be proven that all the operations m t could be twistedin order to get an untwisted structure.We also point out that the twisted structures arise from the two uncompatible orientation con-ventions on an intersection R ∩ S and S ∩ R detailed in 4.1.2. Indeed, we decided to orient T ( y ; x ) inside the intersection W S ( y ) ∩ W U ( x ) . The signs then compute nicely for the boundary component (cid:101) T ( y ; z ) × (cid:102) CT t g ( z ; x , . . . , x n ) , and the twist in ∂ T wMorse arises in (cid:102) CT t g ( y ; x , . . . , z, . . . , x n ) × (cid:101) T ( z ; x i ) .Orienting T ( y ; x ) inside the intersection W U ( x ) ∩ W S ( y ) makes these two boundary componentsswitch roles. In that case, redefining the twist on the orientation of the moduli space T ( y ; x ) asgiven by the parity of σ ( y ; x ) := 1 + | x | , we check that the operations m t define a twisted Ω BAs -algebra structure on ( C ∗ ( f ) , ∂ Morse , ∂ T wMorse ) .The operations µ t g on their side define a twisted Ω BAs -morphism between ( C ∗ ( f ) , ∂ Morse , ∂ T wMorse ) and ( C ∗ ( g ) , ∂ Morse , ∂ T wMorse ) . Part 3 Further developments The map µ Y is a quasi-isomorphism The goal of this section is to prove the following proposition : Proposition 21. The twisted Ω BAs -morphism µ Y : ( C ∗ ( f ) , m X f t ) −→ ( C ∗ ( g ) , m X g t ) constructed intheorem 12 is a quasi-isomorphism. In other words we want to prove that the arity 1 component µ Y : C ∗ ( f ) → C ∗ ( g ) is a map whichinduces an isomorphism in cohomology. The map µ Y is a dg-map ( C ∗ ( f ) , ∂ T wMorse ) → ( C ∗ ( g ) , ∂ Morse ) ,but the cohomologies defined by the differentials ∂ T wMorse and ∂ Morse are equal.In this regard, we will prove that given three perturbation data on CT := { } , Y fg , Y gf and Y ff , defining dg-maps µ Y ij : ( C ∗ ( i ) , ∂ T wMorse ) −→ ( C ∗ ( j ) , ∂ Morse ) , we can construct a homotopy h : C ∗ ( f ) → C ∗ ( f ) such that ( − d µ Y gf ◦ µ Y fg − µ Y ff = ∂ Morse h + h∂ T wMorse . Specializing to the case where Y ff is null, µ Y ff = id and this yields the desired result. For thesake of readability, we will write Y ij := Y ij in the rest of this section. Note also that the choice ofperturbation data X f and X g are not necessary for this construction.1.1. The moduli space H ( y ; x ) . Begin by considering the moduli space of metric trees H , repre-sented in two equivalent ways in figure 32. Adapting the discussions of section 1.2, we infer withoutdifficulty the notion of smooth choice of perturbation data on H . Given such a choice of perturbationdata W , we then say that it is consistent with the Y ij if it is such that, when l → , lim( W ) = Y ff ,and when l → + ∞ , the limit lim( W ) on the above part of the broken tree is Y fg and the limit lim( W ) on the bottom part of the broken tree is Y gf . l l Figure 32 For x and y critical points of the function f , introduce now the moduli space H W ( y ; x ) consistingof perturbed Morse gradient trees modeled on , and such that the two external edges correspond toperturbed Morse equations for f , and the internal edge corresponds to a perturbed Morse equation IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 89 for g . We then check that a generic choice of perturbation data W makes them into orientablemanifolds of dimension dim( H W ( y ; x )) = | y | − | x | + 1 . The 1-dimensional moduli spaces H ( y ; x ) can be compactified into compact manifolds with boundary H ( y ; x ) , whose boundary is given by the three following phenomena :(i) an external edge breaks at a critical point of f (Morse) ;(ii) the length of the internal edge tends towards 0 : this yields the moduli spaces CT Y ff ( y ; x ) ; (iii) the internal edge breaks at a critical point of g : this yields the moduli spaces (cid:91) z ∈ Crit( g ) CT Y gf ( y ; z ) × CT Y fg ( z ; x ) . Defining the map h : C ∗ ( f ) → C ∗ ( f ) as h ( x ) := (cid:80) | y | = | x |− H W ( y ; x ) · y , a signed count of theboundary points of the 1-dimensional compactified moduli spaces H W ( y ; x ) then proves that : Proposition 22. The map h defines an homotopy between ( − d µ Y gf ◦ µ Y fg and µ Y ff i.e. is suchthat ( − d µ Y gf ◦ µ Y fg − µ Y ff = ∂ Morse h + h∂ T wMorse . Proposition 21 is then a simple corollary to this proposition.1.2. Proof of propositions 21 and 22. We define the moduli space H ( y ; x ) as before, by intro-ducing the map φ W : H × W S ( y ) × W U ( x ) −→ M × M , and setting H ( y ; x ) := φ − (∆) where ∆ is the diagonal of M × M . We recall moreover that σ ( ; y ; x ) = d (1 + | y | ) , σ ( y ; x ) = 1 and that µ Y ij ( x ) = (cid:88) | y | = | x | (cid:102) CT Y ij ( y ; x ) · y ∂ Morse ( x ) = (cid:88) | y | = | x | +1 (cid:101) T ( y ; x ) · y . We then set σ ( ; y ; x ) = ( d + 1) | y | , and write (cid:101) H ( y ; x ) for the moduli space H ( y ; x ) endowed with the orientation obtained by twistingits natural orientation by a sign of parity σ ( ; y ; x ) . We can now define the map h : C ∗ ( f ) → C ∗ ( f ) by h ( x ) := (cid:88) | y | = | x |− (cid:101) H ( y ; x ) · y . If (cid:101) H ( y ; x ) is 1-dimensional, its boundary decomposes as the disjoint union of the following fourtypes of components ( − | y | + d (cid:102) CT Y gf ( y ; z ) × (cid:102) CT Y fg ( z ; x ) ( − | y | +1 (cid:102) CT Y ff ( y ; x )( − | y | +1 (cid:101) T ( y ; z ) × (cid:101) H ( z ; x ) ( − | y | +1+( d +1) | x | (cid:101) H ( y ; z ) × (cid:101) T ( z ; x ) . Counting the boundary points of these 1-dimensional moduli spaces implies that ( − d µ Y gf ◦ µ Y fg − µ Y ff = ∂ Morse h + h∂ T wMorse . To prove proposition 21, it remains to note that this relation descends in cohomology to the relation ( − d [ µ Y gf ] ◦ [ µ Y fg ] = [ µ Y ff ] . More on the Ω BAs viewpoint We stated in section 1.6 that because the two-colored operad A ∞ is a fibrant-cofibrant replace-ment of As in the model category of two-colored operads, the category of A ∞ -algebras with A ∞ -morphisms between them yields a nice homotopic framework to study the notion of "dg-algebraswhich are associative up to homotopy". In fact, most classical theorems for A ∞ -algebras can beproven using the machinery of model categories, on the model category of two-colored operads indg- Z -modules. We can thus similarly introduce the two-colored operad Ω BAs , which is again afibrant-cofibrant replacement of As in the model category of two-colored operads. The category of Ω BAs -algebras with Ω BAs -morphisms between them yields another satisfactory homotopic frame-work to study "dg-algebras which are associative up to homotopy", in which most classical theoremsfor A ∞ -algebras still hold.We also point out that while there exists a morphism of operads A ∞ → Ω BAs which is canoni-cally given by refining the cell decompositions on the associahedra, Markl and Shnider constructedin [MS06] an explicit non-canonical morphism of operads Ω BAs → A ∞ . The operads Ω BAs and A ∞ being fibrant-cofibrant replacements of As , model category theory tells us that there necessarilyexist two morphisms A ∞ → Ω BAs and Ω BAs → A ∞ . Hence the noteworthy property of these twomorphisms is not that they exist, but that they are explicit and computable .Switching to the two-colored operadic viewpoint, model category theory tells us again that therenecessarily exist two morphisms A ∞ → Ω BAs and Ω BAs → A ∞ . We have already introduced thenecessary material to define an explicit and computable morphism of two-colored operads A ∞ → Ω BAs . To render explicit a morphism Ω BAs → A ∞ it would be enough to construct a morphismof operadic bimodules Ω BAs − Morph → A ∞ − Morph . To our knowledge, this has not yet been done,but we conjecture that the construction of Markl-Shnider should adapt nicely to the multiplihedrato define such a morphism. 3. Quilted disks We explained in this article how the multiplihedra can be realized as compactified moduli spacesof stable two-colored metric ribbon trees. In fact, Mau-Woodward also prove in [MW10] that themultiplihedra J n can be realized as the compactified moduli spaces of stable quilted disks QD n, .The objects of QD n, are disks with n + 1 points z , z , · · · , z n marked on the boundary, withan additional interior disk passing through the point z . An instance is depicted in figure 33.These moduli spaces however only contain the A ∞ -cell decompositions of the multiplihedra, anddo not contain their Ω BAs -cell decompositions. They can moreover be realized in Floer theory asin [MWW18], to construct A ∞ -morphisms between Fukaya categories. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 91 Figure 33. An example of a quilted disk in QD , Towards higher algebra In closing, two questions naturally arise from this construction. They will respectively representthe starting points to the parts II and III to this article. Problem 1. Given two Morse functions f, g , choices of perturbation data X f and X g , and choicesof perturbation data Y and Y (cid:48) , is µ Y always A ∞ -homotopic (resp. Ω BAs -homotopic) to µ Y (cid:48) ? I.e.,when can the following diagram be filled in the A ∞ (resp. Ω BAs ) world C ∗ ( f ) C ∗ ( g ) µ Y µ Y (cid:48) ? In which sense, with which notion of homotopy can it be filled ? And in general, which notion ofhigher operadic algebra naturally encodes this type of problem ? Problem 2. Given three Morse functions f , f , f , choices of perturbation data X i , and choices ofperturbation data Y ij defining morphisms µ Y : ( C ∗ ( f ) , m X t ) −→ ( C ∗ ( f ) , m X t ) ,µ Y : ( C ∗ ( f ) , m X t ) −→ ( C ∗ ( f ) , m X t ) ,µ Y : ( C ∗ ( f ) , m X t ) −→ ( C ∗ ( f ) , m X t ) , can we construct an A ∞ -homotopy (or an Ω BAs -homotopy), such that µ Y ◦ µ Y (cid:39) µ Y throughthis homotopy ? That is, can the following cone be filled in the A ∞ (resp. Ω BAs ) world C ∗ ( f ) C ∗ ( f ) C ∗ ( f ) µ Y µ Y µ Y ? Which higher operadic algebra naturally arises from this basic question ? Note that the constructionof section 1 solves the arity 1 step of this problem.Problem 1 is solved in [Maz21] by introducing the notions of n − A ∞ -morphisms and n − Ω BAs -morphisms. Problem 2 will be adressed in an upcoming paper, in which it will appear that thehigher algebra of n − A ∞ -morphisms provides a natural framework to solve this problem. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY I 93 References [Abo11] Mohammed Abouzaid. A topological model for the Fukaya categories of plumbings. J. Differential Geom. ,87(1):1–80, 2011.[AR67] Ralph Abraham and Joel Robbin. Transversal mappings and flows . An appendix by Al Kelley. W. A.Benjamin, Inc., New York-Amsterdam, 1967.[Aur14] Denis Auroux. A beginner’s introduction to Fukaya categories. In Contact and symplectic topology , vol-ume 26 of Bolyai Soc. Math. Stud. , pages 85–136. János Bolyai Math. Soc., Budapest, 2014.[Bot19a] Nathaniel Bottman. 2-associahedra. Algebr. Geom. Topol. , 19(2):743–806, 2019.[Bot19b] Nathaniel Bottman. Moduli spaces of witch curves topologically realize the 2-associahedra. J. SymplecticGeom. , 17(6):1649–1682, 2019.[BV73] J. M. Boardman and R. M. Vogt. Homotopy invariant algebraic structures on topological spaces . LectureNotes in Mathematics, Vol. 347. Springer-Verlag, Berlin-New York, 1973.[For08] Stefan Forcey. Convex hull realizations of the multiplihedra. Topology Appl. , 156(2):326–347, 2008.[Fuk97] Kenji Fukaya. Morse homotopy and its quantization. In Geometric topology (Athens, GA, 1993) , volume 2of AMS/IP Stud. Adv. Math. , pages 409–440. Amer. Math. Soc., Providence, RI, 1997.[Gro85] M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math. , 82(2):307–347, 1985.[Hut08] Michael Hutchings. Floer homology of families. I. Algebr. Geom. Topol. , 8(1):435–492, 2008.[LH02] Kenji Lefevre-Hasegawa. Sur les A ∞ -catégories . PhD thesis, Ph. D. thesis, Université Paris 7, UFR deMathématiques, 2003, math. CT/0310337, 2002.[Lod04] Jean-Louis Loday. Realization of the Stasheff polytope. Arch. Math. (Basel) , 83(3):267–278, 2004.[LV12] Jean-Louis Loday and Bruno Vallette. Algebraic operads , volume 346 of Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences] . Springer, Heidelberg, 2012.[Mar06] Martin Markl. Transferring A ∞ (strongly homotopy associative) structures. Rend. Circ. Mat. Palermo (2)Suppl. , (79):139–151, 2006.[Maz21] Thibaut Mazuir. Higher algebra of A ∞ and Ω BAs -algebras in Morse theory II. arXiv preprint, 2021.[Mes18] Stephan Mescher. Perturbed gradient flow trees and A ∞ -algebra structures in Morse cohomology , volume 6of Atlantis Studies in Dynamical Systems . Atlantis Press, [Paris]; Springer, Cham, 2018.[MS06] Martin Markl and Steve Shnider. Associahedra, cellular W -construction and products of A ∞ -algebras. Trans. Amer. Math. Soc. , 358(6):2353–2372, 2006.[MS12] Dusa McDuff and Dietmar Salamon. J -holomorphic curves and symplectic topology , volume 52 of AmericanMathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, secondedition, 2012.[MTTV19] Naruki Masuda, Hugh Thomas, Andy Tonks, and Bruno Vallette. The diagonal of the associahedra.arXiv:1902.08059, 2019.[MV] Naruki Masuda and Bruno Vallette. The diagonal of the multiplihedra and the product of A ∞ -categories.In preparation.[MW10] S. Ma’u and C. Woodward. Geometric realizations of the multiplihedra. Compos. Math. , 146(4):1002–1028,2010.[MWW18] S. Ma’u, K. Wehrheim, and C. Woodward. A ∞ functors for Lagrangian correspondences. Selecta Math.(N.S.) , 24(3):1913–2002, 2018.[Sei08] Paul Seidel. Fukaya categories and Picard-Lefschetz theory . Zurich Lectures in Advanced Mathematics.European Mathematical Society (EMS), Zürich, 2008.[Sma65] S. Smale. An infinite dimensional version of Sard’s theorem. Amer. J. Math. , 87:861–866, 1965.[Smi15] Ivan Smith. A symplectic prolegomenon. Bull. Amer. Math. Soc. (N.S.) , 52(3):415–464, 2015.[Sta63] James Dillon Stasheff. Homotopy associativity of H -spaces. I, II. Trans. Amer. Math. Soc. 108 (1963),275-292; ibid. , 108:293–312, 1963.[Tam54] Dov Tamari. Monoïdes préordonnés et chaînes de Malcev. Bulletin de la Société mathématique de France ,82:53–96, 1954.[Val14] Bruno Vallette. Algebra + homotopy = operad. In Symplectic, Poisson, and noncommutative geometry ,volume 62 of Math. Sci. Res. Inst. Publ. , pages 229–290. Cambridge Univ. Press, New York, 2014.[Val20] Bruno Vallette. Homotopy theory of homotopy algebras. Ann. Inst. Fourier (Grenoble) , 70(2):683–738,2020. [Weh12] Katrin Wehrheim. Smooth structures on Morse trajectory spaces, featuring finite ends and associativegluing. In Proceedings of the Freedman Fest , volume 18 of