A spectral sequence for tangent cohomology of algebraic operads
AA spectral sequence for tangent cohomology ofalgebras over algebraic operads J OSÉ
M. M
ORENO -F ERNÁNDEZ P EDRO T AMAROFF
Operadic tangent cohomology generalizes the existing theories of Harrison cohomology,Chevalley–Eilenberg cohomology and Hochschild cohomology to address the deformation the-ory of general types of algebras through the pivotal gadgets known as a deformation complexes.The cohomology of these is in general very non-trivial to compute, and in this paper we comple-ment the existing computational techniques by producing a spectral sequence that converges tothe operadic cohomology of a fixed algebra. Our main technical tool is that of filtrations arisingfrom towers of cofibrations of algebras, which play the same role cell attaching maps and skeletalfiltrations do for topological spaces.As an application, we consider the rational Adams–Hilton construction on topological spaces,where our spectral sequence gives rise to a seemingly new and completely algebraic descriptionof the Serre spectral sequence, which we also show is multiplicative and converges to the Chas–Sullivan loop product. We also consider relative Sullivan–de Rham models of a fibration p , whereour spectral sequence converges to the rational homotopy groups of the identity component of thespace of self-fiber-homotopy equivalences of p . MSC 2020:
Algebraic operads are an effective gadget to study different types of algebras through a commonlanguage. In particular, they provide us with tools to study the deformation theory of such algeb-ras. For example, the operads controlling Lie, associative and commutative algebras —collectivelyknown as the three graces of J.-L. Loday— along with the operadic formalism, recover for usswiftly the usual (co)homology theories of Chevalley–Eilenberg [15], Hochschild [32] and Har-rison [29], and shed light on the relation between these three. We point the reader to [28] for aninteresting example of this.More generally, for an operad P and a P -algebra A , there is a dg Lie algebra Def P ( id A ) , the de-formation complex of A (also known as the tangent Lie algebra of A ), that codifies all deformationproblems over A , in the spirit of [14, 18, 19, 26]. In particular, given a deformation problem, classesin the cohomology groups H ∗ ( A ) of Def P ( id A ) —which are now known as the André–Quillenor tangent cohomology groups of A — allow us, among other things, to determine obstructions tothe existence of solutions of such deformation problems. In this paper, we construct a spectral a r X i v : . [ m a t h . A T ] N ov A spectral sequence for tangent cohomology of algebraic operads sequence whose input is, in a precise sense, a “cellular decomposition of A ”, that converges to thecohomology groups H ∗ ( A ) .There is a rich interplay [30, 51] between the homological algebra that arises when studying suchdeformation complexes and the homotopical algebra of D. Quillen [44] and, in particular, withthe study of the homotopy category types of algebras. Some time after the work of Quillen, H.-J.Baues [4] developed the theory of cofibration categories, which plays a similar role in the devel-opment of homotopical algebra (and algebraic homotopy) as model categories do. The definitionof Baues, which is not self dual (as opposed to that of Quillen) allows us to focus our attentionon a class of cofibrations. This asymmetry, although perhaps slightly artificial in our case, appearsnaturally when doing, for example, ‘proper’ homotopy theory [6, 42, 43].Our main interest lies in towers of cofibrations of P -algebras for a fixed operad P . For simplicity,we assume P is non-dg, although most of what we do can be extended without too much effortfor dg operads. To take this point of view, we prove in Theorem 2.2 that the category of (dg) P -algebras is a cofibration category in the sense of Baues. This gives a very broad generalizationof the pioneering work done in [5], where the authors do this for Lie and associative algebras.With this theoretical framework at hand, we focus ourselves on the computation of the tangentcohomology of P -algebras.The relation between tangent cohomology and (co)derivations of types of algebras has appearedmany times in the literature, starting with the pioneering work of J. Stasheff and M. Schlessingerin [45, 48]. A definition for algebras over operads then naturally followed; we point the readerto [40] for an account of this development. A remark on terminology.
The term ‘André–Quillen cohomology’ seems to have been first usedin this generality by Hopkins–Goerss [27], while V. Hinich [30] prefers the term ‘(absolute) co-homology’. In order to avoid any confusion with the classical André–Quillen cohomology forcommutative algebras over fields of positive characteristic, whose definition also happens to coin-cide with that of Quillen (co)homology, we will use the more neutral term ‘tangent cohomology’,as in [45].
Definition.
Let f : B −→ B be a morphism of P -algebras, and M a B -module. The tangentcohomology of B relative to B with values in M is the cohomology of the complex Def P ( f , M ) : = Der Q ( Q , M ) where Q −→ Q is a cofibrant replacement of f : B −→ B , and w write it H ∗ B ( B , M ) .As we will explain, there is a functor from triples of algebras to short exact sequences —givenby the usual Jacobi–Zariski sequence in geometry [33, Section 2.4]— which yields a short ex-act sequence as long as the second map is a cofibration. This interplay between derivations andcofibrations is at the very heart of the construction of the spectral sequence appearing in our mainstatement, Theorem 3.7. There, we show that if one can exhibit Q −→ Q as a colimit of a towercofibrations T : Q (cid:26) Q (cid:26) · · · (cid:26) Q s (cid:26) A s + (cid:26) · · · (cid:26) lim −→ s Q s = Q , then one obtains a corresponding spectral sequence in terms of the tangent cohomology of theskeleta of T , where we allow coefficients to take values in a B -module, and not necessarily B itself. osé M. Moreno-Fernández and Pedro Tamaroff Theorem A.
Let Q be the colimit of the tower T of cofibrations of P -algebras. There is a functorialright half-plane spectral sequence with first pageE s , t = H s + t ( Der Q s ( Q s + , − )) s == ⇒ H s + t ( Der Q ( Q , − )) that is conditionally convergent in the sense of Boardman. Since Q −→ Q is a cofibration of P -algebras that is a cofibrant replacement of the morphism of P -algebras B −→ B , the target of this spectral sequence is H ∗ B ( B , − ) . We record this immediatereinterpretation of the theorem in Corollary 3.8; this is the situation we are interested in general.Having done this, we study how our construction specializes when considering the algebraic mod-els in rational homotopy of J. F. Adams and P. J. Hilton [1], and D. Sullivan [49]. One of the mainfeatures of these algebraic models of rational homotopy types is that they are built through towersof cofibrations in terms of a cellular decomposition of choice. Our first result in this direction isTheorem 4.1, which the reader is invited to compare with that of S. Shamir [46] and to the eponym-ous spectral sequence of J.-P. Serre. We expect this spectral sequence, in which all coefficients are rational , to be isomorphic to the one obtained in [17] by R. L. Cohen, J. D. S. Jones and J. Yan. Itgives us a completely algebraic description of a multiplicative spectral sequence converging to theChas–Sullivan loop product in string topology for simply connected oriented closed manifolds. Theorem B.
Let X be a CW complex with exactly one -cell, no -cells and all whose attachingmaps are based with respect to the only -cell. There is a first quadrant spectral sequence withE s , t = hom ( H s ( X ) , H t ( Ω X )) s == ⇒ H s + t ( LX ) , which is conditionally convergent in the sense of Boardman. Moreover, this is a spectral sequenceof algebras, its product in the E -page is given by the convolution product in hom ( H ∗ ( X ) , H ∗ ( Ω X )) .Whenever X is a simply connected oriented closed manifold, this product converges to the Chas–Sullivan loop product. We now turn our attention to Sullivan models. Let p : E −→ B be a fibration between simply-connected CW-complexes, with E having finitely many cells. The grouplike topological monoidAut ( p ) of self-fibre-homotopy equivalences of p is an important object in algebraic topology; wepoint out as an example the classification theorem of J. Stasheff [47], [36, Chapter 9], and the morerecent [10]). In [22], the authors show that the rational homotopy type of the connected componentAut ( p ) of the identity of Aut ( p ) is determined completely in terms of the Harrison cohomologyof the Sullivan model of p . We also point the reader to [9], where a general relation between theHarrison cohomology of Sullivan–de Rham algebras and homotopy types of function spaces isgiven, and to [13], where the authors show the rational homotopy groups of function spaces arealso determined completely in terms of the Harrison cohomology of a Sullivan model. Building ontop of the main result of [22], we obtain the following; see Theorem 4.7. A spectral sequence for tangent cohomology of algebraic operads
Theorem C.
Let F (cid:44) → E p −→ B be a fibration of -connected CW-complexes, with E finite. There isa spectral sequence withE s , t = hom ( π s ( F ) , H t ( E )) s == ⇒ π s − t ( Aut ( p )) . Again, all coefficients above are rational. In this case, it should be possible to show that the spectralsequence carries a multiplicative structure inherited from the convolution product on the E -pagemaking it a spectral sequences of Lie algebras by obtaining a result in the lines of Theorem 3in [13], by replacing the space of maps F ( X , Y ) by Aut ( p ) .We remark that A. Berglund and B. Saleh [7, Proposition 4.4] have independently considered aspectral sequence of the same shape as ours in order to determine a Quillen (that is, dg Lie) modelof the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix agiven subspace. The same spectral sequence appears in the work of A. Berglund and I. Madsen,see [8, Lemma 3.5]. This gives a third direction in which our methods can be pushed towards, andthe interested reader can consult that paper for further details.The paper is organized as follows. In Section 2 we recall the elements of operad theory andshow that the category of P -algebras carries the structure of a cofibration category in the sense ofBaues [4]. The main techical result we prove there is the fact the category of P -algebras satisfiesthe pushout axiom of Baues. In Section 3 we develop the main technical tool that we introducein this paper: the spectral sequence of Theorem 3.7. Having done this, we describe the differentpieces of the spectral sequence in that section. This includes the description of its differentials themaps involved in the exact couple that gives rise to it. We conclude this section by showing thatour spectral sequence degenerates in some natural situations. Finally, in Section 4, we give thementioned applications to rational homotopy theory.Throughout, we write P for a symmetric operad, which we assume is reduced, and identify withthe corresponding Schur endofunctor V (cid:55)−→ P ( V ) . All P -algebras are connected and dg unlessstated otherwise. We also make use of the following functors: Sing ∗ is the singular chains functor,Sing ∗ is the singular cochains functor, A is the Adams–Hilton construction, Cell is the cellularchain complex of a CW-complex, Ω is the based loop space, and L is the free loop space. The dualof a graded vector space V is V ∨ . For a topological space X , we write π ∗ ( X ) ∨ = ( π ∗ ( X ) ⊗ Q ) ∨ ,omitting the rational coefficients. All tensor products and hom-sets are understood to be taken overa fixed base ring k of characteristic zero, which in Section 3 will be Q . Similarly, in that section,all (co)homology groups are assumed to have rational coefficients. Acknowledgements
The first author thanks Aniceto Murillo for useful conversations, and acknowledges support fromthe Irish Research Council Postdoctoral Fellowship
GOIPD/2019/823 and partial support from theMINECO grant
MTM2016-78647-P . The second author thanks Andrea Solotar, Mariano Suárez-Álvarez and the participants of the homological algebra seminar in the University of Buenos Airesfor useful comments he received during his visit in July 2019 when he presented some very prelim- osé M. Moreno-Fernández and Pedro Tamaroff inary results that led to the results in this paper. We kindly thank Jim Stasheff for useful commentsand suggestions. P - Alg as a cofibration category
This section starts with a recollection of the necessary background on operad theory, mainly toset up notation. An excellent reference is [35]. We then recall the basics of cofibration categoriesas introduced by Baues [4] and prove that the category of P -algebras satisfies the pushout axiom.Therefore, the category of P -algebras admits the structure of a cofibration category where theweak equivalences are the quasi-isomorphisms and the cofibrations are maps obtained by freelyadjoining variables. Σ -modules and operads Write Σ dgMod for the category of dg Σ -modules. Recall there is a monoidal product, the composi-tion product − ◦ − : Σ dgMod × Σ dgMod −→ Σ dgMod with unit 1 the module concentrated in arity 1 where its value is k . It is useful to think of an elementin X ◦ Y as a corolla whose only vertex is labelled by x ∈ X of some arity k ∈ N and whose leavesare labelled in order by y , . . . , y k ∈ Y . This product restricts to the subcategory of non-graded Σ -modules, and a symmetric operad P is a monoid in this category, whose product we usually denoteby γ : P ◦ P −→ P . We can of course talk about graded or differential graded symmetric operads, but symmetric op-erads will suffice for our purposes. An operad is reduced if P ( ) =
0, and we will only considerthis kind of operads in what follows.We recall that there is a unital monad T in Σ dgMod and that symmetric operads are precisely the T -algebras. It follows in particular that if X is any dg Σ -module, T X is the free operad on X .Concretely, for n ∈ N , the space T X ( n ) consists of rooted trees t so that for each vertex of t isdecorated by an element in X ( d ) , where d is the number of inputs of it. The product µ : T ◦ T −→ T is obtained by substitution of trees into vertices, and the unit η : 1 −→ T sends an element of X to the corresponding corolla. We refer the reader to [35] for details. Fromthis we can present operads as quotients of free operads by ideals of relations. Classical examples A spectral sequence for tangent cohomology of algebraic operads include the operads As , Lie , Com , PreLie , Poiss , Ger and
Grav , whose presentations can be foundin the literature.
Recall we have fixed a reduced symmetric operad P and write P - Alg for the category of dg P -algebras. In particular, every P -algebra A includes the data of a square zero derivation d of A , thatis, is an endomorphism d : A −→ A so that d = µ of O of arity n ∈ N , d µ = µ d [ n ] , where d [ n ] is the induced differential on A ⊗ n . For example, if µ is binary (sothat n = d µ = µ ( d ⊗ + ⊗ d ) . Note that Koszul signs will appear in the previous formula when evaluating 1 ⊗ d on elements of A . Alternatively, we require that the structure map γ A : P −→ End A is one of complexes, where we view P as a dg operad concentrated in degree zero. Adjoint to thisis a map γ A : P ( A ) −→ A which we call the structure map of A . We will drop the prefix “dg” and speak simply of P -algebras.In all of what follows, with the exception of the “tilde” construction of Proposition 3.6, will only consider the category of P -algebras A that are connected and non-negatively homologically graded.Precisely, this means that A = k is the base ring and A vanishes in negative homological degrees. Fix a dg P -algebra A as before. An operadic A-module is a dg vector space M along with a unitalaction γ M : P ◦ ( A , M ) −→ M so that γ M ( ◦ ( γ A , γ M )) = γ M ( γ ◦ ( , )) . It is useful to note that if P = As and if A is an P -algebra or, what is the same, an associativealgebra, then an operadic A -module is the same as an A -bimodule and not a left (or right) A -module.Similarly, the operadic modules for commutative algebras are the symmetric bimodules, and theoperadic modules for Lie algebras coincide with the usual notion of Lie algebra representation. In this section we show that one may endow the category of P -algebras with the structure ofa cofibration category in the sense of Baues, see Theorem 2.2. The explicit description of the osé M. Moreno-Fernández and Pedro Tamaroff cofibrations given is useful to streamline the presentation of our results. One reason to take thispoint of view is that the cofibrations are simpler than those of [30]; they correspond to his ‘standardcofibrations’.Cofibration categories, introduced by Baues in [4], provide a framework for doing axiomatic ho-motopy theory in the spirit of Quillen [44] but under weaker assumptions; in particular, as its nameindicates, the cofibration categories of Baues are defined by choosing a class of cofibrations and aclass of weak-equivalences, but their definition does not require a class of fibrations. Definition 2.1 A cofibration category is a category endowed with two classes of morphisms, thecofibrations and weak equivalences, so that the following four axioms are satisfied:(C1) Composition axiom:
The isomorphisms are weak equivalences and cofibrations, andweak equivalences satisfy the two out of three property. Moreover, cofibrations are closedunder composition.(C2)
Pushout axiom: If i : B (cid:26) A is a cofibration and f : B −→ Y is any map, the pushout of A (cid:27) B −→ Y exists, and ¯ i is a cofibration. Moreover, (i) if f is a weak equivalence so is¯ f , and (ii) if i is a weak equivalence, so is ¯ i . B AY P if fi (C3)
Factorization axiom:
Every arrow can be factored into a cofibration followed by a weakequivalence.(C4)
Axiom on fibrant models:
For every object X there is an arrow X ∼ (cid:26) RX where RX is fibrant .An object R is fibrant if every trivial cofibration R ∼ (cid:26) Q splits. All P -algebras are fibrant (but wewill not need this) and the cofibrant objects are the P -algebras which are free as graded algebras.For A , B ∈ P - Alg , we write A (cid:63) B their coproduct as P -algebras. The work done in [30] showsthat all axioms except possibly the pushout axiom hold for this model structure. We check it inSection 2.5, and hence deduce the following result. We remark that this was done, without mentionto cofibration categories, by Baues–Lemaire [5] for associative and Lie algebras. Theorem 2.2
The category P - Alg of P -algebras carries a structure of cofibration category. Thecofibrations are those maps B −→ A for which there exists a submodule X of A such that theinduced map f (cid:63) B (cid:63) T X −→ A is an isomorphism, and the weak-equivalences are the quasi-isomorphisms.
Let us introduce some useful definitions:• A cofibration B (cid:26) A is elementary of height d if A is obtained by adjoining finitely manygenerators in degree d + A spectral sequence for tangent cohomology of algebraic operads • An morphism f : B −→ A is is n-connected if it induces isomorphisms on homology in degree i < n and a surjection in degree n .• An algebra A is n-connected if the unique map ∗ −→ A is n -connected, and it is n-truncated if H s ( A ) = s > n .The elementary cofibrations correspond to the geometric situation where we add cells to a space.Note that we are not imposing any condition on B , which may very well have generators in arbitrarydegrees. Definition 2.3 If A −→ ∗ is an augmented P -algebra, we write A for the kernel of this augment-ation, and Ind P A for the cokernel of the structure map of the non-unital algebra A , which we callthe space of indecomposables of A . In this way we obtain a functor Ind P : P - Alg ∗ −→ Ch k fromaugmented P -algebras to chain complexes, which we call the functor of indecomposables .It is clear, for example, that if A = ( P ( V ) , d ) is quasi-free then Ind P A = ( V , d ( ) ) where d ( ) is thelinear part of the differential d of A , and we will use this later. In this way, the (derived) functorInd P A captures the “first order information” of a cofibrant resolution of the P -algebra A . Definition 2.4
We define the
Quillen homology of a P -algebra A , which we write H ∗ ( P , A ) , asthe left derived functor of Ind P . That is, if B ∼ −→ A is a weak equivalence and B is a cofibrant P -algebra, the homology of Ind P B is by definition H ∗ ( P , A ) .To illustrate, if A is an augmented associative algebra, this is just Tor A ∗ + ( k , k ) . Indeed, we can take Ω BA as a model of A , and then H ∗ ( P , A ) is the homology of the shifted reduced bar complex BA of A , which is precisely Tor A ∗ + ( k , k ) . This is a technical subsection which the uninterested reader may very well skip, since it is notgoing to play a role in obtaining our main result. We have decided to include it for completeness,to obtain Theorem 2.2. For details on the bar construction on operads we refer the reader to thebook [35].Write B ( P ) for the dg cooperad whose underlying cooperad is free on the suspension s P of theaugmentation ideal of P . Its differential collapses an edge of a tree and composes the elements of P accordingly. We call B ( P ) the bar construction on P . There is a twisting cochain τ : B ( P ) −→ P and the twisted complex E ( P ) = P ◦ τ B ( P ) is acyclic. These objects define functors E P and B P onthe category of P -algebras to complexes and B ( P ) -coalgebras, respectively. If, moreover, X is aquasi-free P -algebra, the quotient map B P ( X ) −→ Ind P ( X ) is a surjective quasi-isomorphism. We begin with a lemma. Compare with [5, Proposition 1.5]. osé M. Moreno-Fernández and Pedro Tamaroff Lemma 2.5
Let f : X −→ Y be a map between quasi-free P -algebras generated in non-negativedegrees. Then f is a quasi- isomorphism if, and only if, the induced map on indecomposables Ind P f is a quasi- isomorphism. One can think of this result as a “Whitehead Lemma” in the lines of [12, 4.2]. Indeed, in the caseof Lie algebras (in the category the reader may prefer), we have that Ind P ( L ) = L / [ L , L ] is theAbelianization functor in that paper. Proof.
The complexes Ind P ( X ) and Ind P ( Y ) compute the homology of the (non-unital) bar con-struction B P ( X ) and B P ( Y ) and the maps B P ( X ) −→ Ind P ( X ) are surjective quasi-isomorphisms.In this way, we can consider a commutative diagram of ‘fibration sequences’:0 X E P ( X ) B P ( X ) Y E P ( Y ) B P ( Y ) . The middle map is a quasi-isomorphism of contractible complexes, so the claim follows by Moore’scomparison theorem: the map B P ( X ) −→ B P ( Y ) is a quasi-isomorphism if and only if the map onindecomposables Ind P ( X ) −→ Ind P ( Y ) is one, if and only if X −→ Y is one. Proof of the Pushout Axiom. If f : B −→ Y is a quasi-isomorphism then, by the lemma, so isInd P f ⊕ P of indecomposables commutes with coproducts, we havethat Ind P f = Ind P f ⊕ QB ⊕ X −→ Ind P Y ⊕ X , and hence f is a quasi-isomorphism. Assume now that i is a quasi-isomorphism. Since the homo-topy cofibres of i and i are both isomorphic to P ( X ) with the induced differential, we see that i is aquasi-isomorphism if i is one. This section is the core of the paper. We pave the way for constructing the spectral sequence ofTheorem 3.7, the main result. We start by recalling the relevant facts on derivation complexes inSubsection 3.1, and recall the basic facts on tangent cohomology in Subsection 3.2. The mainresult and some corollaries are proven in Subsection 3.3. We take a closer look to the items ofthe spectral sequence in Subsection 3.4. We finish in Subsection 3.5 by studying some naturalsituations in which the spectral sequence degenerates.
In this section, we collect some general facts on complexes of derivations, which we then use toconstruct the spectral sequence of Theorem 3.7. Some applications of the spectral sequence will A spectral sequence for tangent cohomology of algebraic operads require, for convergence reasons, complexes concentrated in non-negative degrees. However, theresults of this section do not need this constraint.
Definition 3.1
Let B −→ A be a map of P -algebras, and let M be an operadic A -module. Wewrite Der B ( A , M ) for the cohomologically graded complex of derivations A −→ M that vanishon B . We write elements in Der B ( A , M ) by F : A | B −→ M . The differential of such an F is ∂ F = d M F − ( − ) | F | Fd A .In case u : A −→ U is a map of P -algebras, which in particular makes U into an A -bimodule, we willwrite Der B ( A , U ) without explicit mention to the map u , which will be understood from context.In case U = A and u is the identity, we write this complex simply by Der B ( A ) . Observe that in thiscase, the differential ∂ F is given by the bracket [ d A , F ] , and that Der B ( A ) is a dg Lie algebra underthe Lie bracket of derivations. Moreover, for each operadic A -module M , the complex Der B ( A , M ) is a left Lie module over Der B ( A ) .The following lemma says that this functor is well behaved when its arguments are cofibrations ofalgebras. The resulting exact sequence is called the Jacobi–Zariski sequence of the triple B −→ A −→ A (cid:48) . It is straightforward to see that it is functorial on maps of algebras A (cid:48) −→ U . Lemma 3.2
Let B −→ A −→ A (cid:48) be a sequence of morphisms of P -algebras, and let u : A (cid:48) −→ Ube a map of P -algebras. Then, there is a left exact sequence A ( A (cid:48) , U ) Der B ( A (cid:48) , U ) Der B ( A , U ) and it is exact if A −→ A (cid:48) is a cofibration.Proof. The map A −→ A (cid:48) induces a map Der B ( A (cid:48) , U ) −→ Der B ( A , U ) by precomposition F (cid:55)→ F ◦ i whose kernel is tautologically isomorphic to Der A ( ¯ A , U ) . Hence, it suffices to show the first mapis surjective in case A −→ A (cid:48) is a cofibration. To do this, we observe that we can always lift aderivation A −→ U that vanishes on B to one in A (cid:48) by declaring it to vanish on the module ofgenerators of A −→ A (cid:48) , and we will always assume, for consistency, that this is our choice of lift.This finishes the proof of the lemma. Let us now introduce the cohomology theory that will concern us and serves as the unifying conceptto present our results. We recommend the article [40] for an account on tangent cohomology foroperads.For a morphism of P -algebras, B −→ A and an operadic A -module M , we define the tangentcohomology of A relative to B with values in M as follows. Let P −→ Q be a cofibrant replacementof the map f : B −→ A : by this we mean that this map is a cofibration between cofibrant P -algebras osé M. Moreno-Fernández and Pedro Tamaroff that fits into a commutative diagram P QB A where the vertical maps are quasi-isomorphisms. The second vertical map makes M into an op-eradic Q -module, and hence we can consider the complex Der P ( Q , M ) of derivations Q −→ M thatvanish on P . Definition 3.3
The cohomology of this complex is, by definition, the tangent cohomology of Arelative to B with values in M , and we write it H ∗ B ( A , M ) . In case we take A = M , we will speakabout the tangent cohomology of the map f : B → A , and write it H ∗ ( f ) .Note that the Jacobi–Zariski sequence shows that for a triple B −→ A −→ A (cid:48) of algebras, we havea corresponding long exact sequence H ∗ B ( A (cid:48) , − ) −→ H ∗ A ( A (cid:48) , − ) −→ H ∗ A ( B , − ) −→ H ∗ B ( A (cid:48) , − )[ ] . Remark 3.4
Consider the situation when A is an associative algebra (without differential), whichwe think of as concentrated in degree zero, and take Q ∼ −→ A a cofibrant replacement of A in Alg ,the cofibration category of dga algebras. Then we have identifications for n ∈ Z , H n ( A , A ) = (cid:40) HH n + ( A ) for n (cid:62) ( A ) for n = ∗ ( A ) are the Hochschild cohomology groups of A .The way to remedy this difference between tangent cohomology of associative algebras and theirclassical Hochschild cohomology is as follows. Lemma 3.5
Let Ad Q : Q −→ Der ( Q ) be the adjoint map of Q. We have an isomorphism of (shifted)Lie algebras H ∗ ( cone ( Ad Q )) (cid:39) HH ∗ ( A ) . Observe that cone ( Ad Q ) is a (shifted) Lie algebra obtained as the semidirect product of Q and s − Der ( Q ) , where the bracket is given, for x + ϕ and y + ψ ∈ cone ( Ad Q ) , by the formula [ x + ϕ , y + ψ ] = [ x , y ] + ϕ ( y ) − ψ ( x ) − [ ϕ , ψ ] . Proof.
Since H ∗ ( Q ) = A , the long exact sequence of the cone complex gives us isomorphisms H n ( cone Ad Q ) = H n − ( Der Q ) for n (cid:62)
2, giving the claim of the theorem in that range. The remainder of the long exact sequenceis a four term exact sequence0 −→ H ( cone Ad Q ) −→ H ( Q ) −→ H ( Der ( Q )) −→ H ( cone Ad Q ) −→ . A spectral sequence for tangent cohomology of algebraic operads
Now observe that H ( Ad Q ) is just the adjoint map A −→ Der ( A ) of A . It follows the above exactsequence identifies H ( cone ( Ad Q )) with the kernel of the map A −→ Der ( A ) and H ( cone Ad Q ) with its cokernel, which is what we wanted.It will be useful for us to have an alternative description of this cone. It is shown in [23] that tocompute HH ∗ ( A ) —even in the case we allow A to be dg— one may instead consider a complexof derivations obtained from the algebra Q as follows. If Q = ( T ( V ) , d ) , consider the algebra (cid:101) Q = ( P ( V ⊕ ε ) , ¯ d ) where | ε | = − , ¯ d ε = ε , ¯ dv = dv + [ ε , v ] Note that ε is a Maurer–Cartan element and that F = ( T ( ε ) , ¯ d ) is acyclic —that is, H ∗ ( F ) = k —since for each n ∈ N we have that d ( ε n − ) = ε n . Proposition 3.6
For any quasi-free associative algebra Q = ( T ( V ) , d ) the Lie algebra Der ( (cid:101) Q ) isquasi-isomorphic to Lie algebra given by the cone of the adjoint map Ad Q .Proof. There is a quotient map π : (cid:101) Q −→ Q with fibre F and a cofibration F −→ (cid:101) Q , which givesus a short exact sequence0 Der F ( (cid:101) Q ) Der ( (cid:101) Q ) Der ( F , (cid:101) Q ) . On the other hand, we have an identification and a quasi-isomorphismDer ( F , (cid:101) Q ) = (cid:101) Q and Der F ( (cid:101) Q ) −→ Der ( Q ) . coming from the fact that F is quasi-free and contractible. The takeaway is a commutative diagramwhose rows are exact and whose columns consist of quasi-isomorphisms:0 Der F ( (cid:101) Q ) Der ( (cid:101) Q ) Der ( F , (cid:101) Q )
00 Der ( Q ) Der ( (cid:101) Q ) (cid:101) Q
00 Der ( Q ) cone ( Ad Q ) Q ev ε π This shows that the Jacobi–Zariski short exact sequence above is quasi-isomorphic to the shortexact sequence for the cone of Q , and hence that Der ( (cid:101) Q ) is quasi-isomorphic to cone ( Ad Q ) . In this section, we present our main technical tool, a spectral sequence converging to the cohomo-logy of the derivations of a P -algebra that is the colimit of a tower of cofibrations. The broad osé M. Moreno-Fernández and Pedro Tamaroff generality in which this spectral sequence appears means we can only guarantee it to be condi-tionally convergent in the sense of Boardman [11]. We will study the differentials in this spectralsequence, and give natural conditions that ensure its strong convergence in Section 3.5. Theorem 3.7
Let A be the colimit of a tower of cofibrations of P -algebrasA (cid:26) A (cid:26) · · · (cid:26) A s (cid:26) A s + (cid:26) · · · (cid:26) lim −→ s A s = A . There is a functorial right half-plane spectral sequence with first pageE s , t = H s + t ( Der A s ( A s + , − )) s == ⇒ H s + t ( Der A ( A , − )) that is conditionally convergent in the sense of Boardman. A consequence of this result is the following corollary which involves tangent cohomology. In fact,we will be mostly concerned in the situation described in the statement of this corollary:
Corollary 3.8
Let f : A −→ A be a cofibration of P -algebras that is a cofibrant replacement ofa map of P -algebras B −→ B. If f is the colimit of a tower of cofibrations as above, there is afunctorial right half-plane spectral sequence with first pageE s , t = H s + t ( Der A s ( A s + , − )) s == ⇒ H s + tB ( B , − ) that is conditionally convergent in the sense of Boardman. We now proceed to prove Theorem 3.7.
Proof.
Let M be an operadic A -module, and for each s ≥
0, consider the exact sequence of deriva-tions induced by the triple A (cid:26) A s (cid:26) A , as in Lemma 3.2:0 −→ Der A s ( A s + , M ) −→ Der A ( A s + , U ) −→ Der A ( A s , M ) −→ . Form the exact couple associated to the long exact sequence in cohomology, explicitly given forevery s , t ∈ N by D s , t = H s + t ( Der A ( A s + , M )) , and E s , t = H s + t ( Der A s ( A s + , M )) . The maps i , j , k forming the exact couple ( D , E , i , j , k ) have bidegrees ( − , ) , ( , ) and ( , ) ,respectively and are described after the proof. In the bidegree pair ( s , t ) , we are denoting by s thefiltration degree and by t the complementary degree. The differential d = jk produced on E = E of bidegree ( , ) . The general procedure of forming iterated derived couples produces the spectralsequence, finding that the r th differential d r has bidegree ( r , − r ) . Thus, differentials entering intoa fixed module E s , tr originate at points outside the right-half-plane for all r > t , so eventually vanish.This guarantees there is conditional convergence in the sense of Boardman; see [11, Theorem 7.3]. A spectral sequence for tangent cohomology of algebraic operads
The spectral sequence is functorial, for if M −→ N is map of A -modules, then the morphisminduced between the corresponding short exact sequences of derivations induces a morphism ofthe corresponding exact couples, producing the desired morphism of spectral sequences.Another corollary of our main result is a spectral sequence that in many situations degeneratesand exhibits tangent cohomology as a twisted algebra defined in terms of Quillen homology andordinary homology of P -algebras, in the same spirit as the usual way of computing Hochschildcohomology of an associative algebra A through a twisted complex hom τ ( C , A ) where C is an A ∞ -coalgebra model of BA with associative twisting cochain τ : C −→ A .To state it, let us take A to be a cofibrant P -algebra of the form P ( V ) that is a cofibrant replacementof the P -algebra B , and write A as the colimit of the tower of cofibrations of the form k (cid:26) P ( V (cid:54) ) (cid:26) P ( V (cid:54) ) (cid:26) · · · (cid:26) P ( V (cid:54) s ) (cid:26) · · · , where for each s ∈ N , we write A s + = P ( V (cid:54) s ) for the subalgebra of A generated by elements ofdegree at most s . Corollary 3.9
Assume that B is -connected. There is a functorial right half-plane spectral se-quence with second pageE s , t = hom ( H s ( P , B ) , H t ( − )) s == ⇒ H s + t ( B , − ) that is conditionally convergent in the sense of Boardman. We remark that if both B and M have zero differential, the spectral sequence collapses and yieldsan isomorphism H ∗ ( hom ( H ∗ ( P , B ) , M )) −→ H ∗ ( B , M ) that exhibits tangent cohomology as the cohomology of a twisted complex. This follows immedi-ately from Lemma 3.12 which shows that the E -page is concentrated in one row. In this sense,the tower of cofibrations above is rather crude, and it is perhaps not unreasonable to consider othermore refined towers to expect a more meaningful spectral sequence. Proof of Corollary 3.9.
Let us consider the cofibration k = A (cid:26) P ( V ) = A and compute the ho-mology of the complex of derivations of A relative to A . For each s ∈ N the cofibration A s (cid:26) A s + is obtained by adding generators in degree s , and it is elementary.We can easily identify the E -page: a closed derivation of degree s + t inDer A s ( A s + , M ) is determined on the generators of A s + living in degree s , whose image are in homological degree t . From this and the fact f ( dv ) = v of degree s , we see that f must have image inthe cycles of M . Therefore, we have an identification E s , t = hom ( V s , H t ( M )) . osé M. Moreno-Fernández and Pedro Tamaroff Since B is 0-connected, we can assume that V =
0. Using this, we now check that the differentialon the first page is hom ( d ( ) , ) where d ( ) : V −→ V is the linear part of the differential of P ( V ) .Indeed, let us take a cocycle representative f : A s + | A s −→ M , let F be extension by zero to A s + ,so that d (cid:74) f (cid:75) = (cid:74) [ d , F ] (cid:75) . We now observe that if w is a generator of A s + (cid:26) A s + , then F vanisheson w . The remaining term is ( − ) | F | + f ( dw ) , and now we note that any term in dw that is notlinear must be a product of lower degree terms, which f vanishes on. Since d ( ) computes theQuillen homology of B , the description of the E -page is what we have claimed.The spectral sequence of Theorem 3.7 can also be obtained from a filtration. Indeed, to filterDer A ( A , M ) , consider the exact sequence of derivations induced by the triple A (cid:26) A s (cid:26) A . Itidentifies the kernel of the restrictionDer A ( A , M ) −→ Der A ( A s , M ) with Der A s ( A , M ) . If we let F s = Der A s ( A , M ) we obtain a complete decreasing filtration · · · ⊆ F s ⊆ F s − ⊆ · · · ⊆ F = Der A ( A , M ) . Then the exact sequence for the triple A s (cid:26) A s + (cid:26) A identifies the quotient F s / F s + with thespace E s , ∗ = Der A s ( A s + , M ) , giving rise to a spectral sequence with the same first page.We have chosen to construct the spectral sequence from an exact couple that does not arise fromthe filtration above. The relationship between these two approaches is that we instead consider thetower of surjections T : · · · −→ F / F s −→ · · · −→ F / F −→ F / F −→ F = Der A ( A , M ) . From the discussion above, see that F / F s is naturallyisomorphic to Der A ( A s , M ) so that we have exact sequences W s : 0 −→ Der A s ( A s + , M ) −→ Der A ( A s + , M ) −→ Der A ( A s , M ) −→ . The following proposition follows from the standard way in which an exact couples is built from afamily of exact sequences ( W s ) s ∈ N as above, and we omit its proof. Proposition 3.10
These short exact sequences give rise to the cohomological exact couple (cid:104) D , E ; i , j , k (cid:105) where, for each s , t ∈ N we haveD s , t = H s + t Der A ( A s + , M ) , E s , t = H s + t Der A s ( A s + , M ) , and the maps i , j and k are of the formi : D s , t −→ D s − , t + , j : D s , t −→ E s + , t , k : E s , t −→ D s , t . Thus, | i | = ( − , ) , | j | = ( , ) and | k | = ( , ) . A spectral sequence for tangent cohomology of algebraic operads
Let us now take a closer look at the exact couple that gives rise to our spectral sequence.
The first differential.
The differential d = jk on E is of bidegree ( , ) . Explicitly, if a class (cid:74) f (cid:75) ∈ E s , t is represented by a cocycle f : A s + | A s −→ M , then d (cid:74) f (cid:75) is represented by ∂ f ( s + ) : A s + | A s + → M ; that is, we extend f by zero to A s + and then take its usual differential. In otherwords, d (cid:74) f (cid:75) = (cid:74) [ d , f ( s + ) ] (cid:75) . The first derived couple.
Let us write E (cid:48) for the homology of E , D (cid:48) = im ( i ) and explain how toobtain the first derived couple ( D (cid:48) , E (cid:48) , i (cid:48) , j (cid:48) , k (cid:48) ) :(1) First, i (cid:48) is just induced by i , since i ( D (cid:48) ) ⊆ D (cid:48) . Note that i (cid:48) has the same bidegree ( − , ) asthat of i .(2) To define j (cid:48) , write an element of D (cid:48) as ix , and consider the class of jx in E (cid:48) ; note that this isa cycle since k j = k (cid:48) (cid:74) x (cid:75) = kx . Note that since jkx = kx is indeed in D (cid:48) .One can see that j (cid:48) has bidegree ( , − ) and k (cid:48) has bidegree ( , ) , so that d = j (cid:48) k (cid:48) has bidegree ( , − ) .More generally, when forming the ( r − ) th derived couple of E , which gives rise to the r th page E r of our spectral sequence, obtained by simply iterating the prescription we just gave, the bidegreeof the differential d r : E r −→ E r is ( r , − r ) , as expected. r − r d r Figure 1: A half-plane spectral sequence.
The rth page of the spectral sequence.
Recall, see for instance [37, Proposition 2.9], that we candescribe the r th page of our spectral sequence as a subquotient of E by using the iterated image osé M. Moreno-Fernández and Pedro Tamaroff spaces D r = im ( i ◦ · · · ◦ i ) = im ( i r − ) where i appears r − k and the map j , as follows. If we put Z s , tr = k − im ( i r − : D s + r , t − r −→ D s , t ) , B s , tr = j ( ker i r − : D s − , t −→ D s − r − , t + r ) then we have the following identifications: E s , tr = Z s , tr / B s , tr , and E s , t ∞ = (cid:92) r Z s , tr / (cid:91) r B s , tr . Diagrammatically, the r th page is the homology of the diagonal of the commutative diagram withexact rows and exact column of Figure 2. We now observe that for each r ∈ N , the term D s , tr ker i r − E s , t D s − , t D s , t D s − r − , t + r D s + r , t − r coker i r − jk ι i r − i r − π Figure 2: computing the r th page.consists of classes (cid:74) f (cid:75) represented by a cocycle f : A s + | A −→ M of degree s + t which is therestriction of a cocycle F : A s + r + | A −→ M of the same degree, and the map i r − : D s + r , t − rr −→ D s , tr is just the map induced by restriction on homology. The higher differentials.
From our construction we see that the kernel of the map jk = d : E −→ E consists of classes of those ( s + t ) -cocycles f : A s + | A s −→ M that are restrictions of cocycles A s + | A s −→ M . Indeed, if jk (cid:74) f (cid:75) = k (cid:74) f (cid:75) = i (cid:74) G (cid:75) for somecocycle G : A s + | A −→ M , and this with some rearranging of terms implies f is the restriction of some cocycle g : A s + | A −→ M in D s + , t − . In fact, we can make it so g = f ( s + ) + h for some derivation h which is notnecessarily a cocycle. The map k (cid:48) : ( E (cid:48) ) s , t −→ ( D (cid:48) ) s , t then sends the class of g to g and hence thedifferential d s , t : E s , t −→ E s + , t − A spectral sequence for tangent cohomology of algebraic operads is obtained by applying j (cid:48) to the class of g or, what is the same, the connecting morphism j to g itself. This can be iterated to describe the successive differentials.Let us now consider the second derived couple and obtain a description of d before stating ageneral result describing the successive differentials in the spectral sequence. To do this, let uscontinue with the notation at the beginning of the section, and assume that we are given a cocycle f : A s + | A s −→ M for which d (cid:74) f (cid:75) = d (cid:74) f (cid:75) =
0. As explained there, the first conditionsimplies there exists an extension of f to a cocycle h : A s + | A s −→ M , and the definition says now that d (cid:74) f (cid:75) is the class in E (cid:48) of the connecting morphism j applied to (cid:74) h (cid:75) . If this vanishes, then we have an equality j (cid:74) h (cid:75) = j (cid:48) k (cid:48) ( b (cid:48) ) in E for some b (cid:48) = (cid:74) f (cid:75) . Unwinding the definitions, it follows that there are• a cocycle f : A s + | A s + −→ M ,• a derivation h : A s + | A s + −→ M ,so that f ( s + ) + h ( s + ) + f ( s + ) + h : A s + | A s −→ M is a cocycle. The end result is that we have extended f to a cocycle up to A s + . Since d r is inducedby k , j , and ( r − ) -preimages of i , we obtain the following: Proposition 3.11
For each r (cid:62) , the kernel of d r is generated by classes η of cocyclesf : A s + | A s −→ Mof degree s + t that admit a lift to a cocycle g : A s + r + | A s −→ M of the same degree. In thissituation, the differential d r + ( η ) is the class of the connecting morphism on g. In particular, if the class of some f : A s + | A s −→ M of degree s + t survives to Z s , t ∞ , what we obtainis an extension to a cocycle f (cid:48) : A | A s −→ M of degree s + t in F s = Der A s ( A , M ) . In Theorem 3.7, we have given the most general form of the spectral sequence of derivations.When giving applications, it is convenient that it be concentrated in a single quadrant in orderto have strong, rather than conditional, convergence. To this end, we collect in this section somenatural conditions that guarantee certain vanishing patterns on the E -page of the spectral sequence.Recall that a P -algebra U is b -truncated if H p ( U ) = p ≥ b +
1, and that a cofibration B (cid:26) A is elementary of height s if A = B (cid:63) P ( V ) with V = V s + . In this case, one has that d ( V ) ⊆ B . osé M. Moreno-Fernández and Pedro Tamaroff Lemma 3.12
Let B (cid:26)
A be an elementary cofibration of height s, and u : A −→ U a morphism of P -algebras. The following holds:(1) If U = U ≥ k for some k, then Der pB ( A , U ) = for all p ≥ s + − k . (2) If U is b-truncated for some b, then H p ( Der B ( A , U )) = for all p ≤ s − b . Proof.
The fact that B (cid:26) A is an elementary cofibration implies that any F ∈ Der B ( A , U ) is com-pletely determined by its image on V , where A = B (cid:63) TV and V = V s + .To prove the first statement, let us take F ∈ Der pB ( A , U ) . Since F has degree − p the image of F lies in U s + − p . It follows that F vanishes if p ≥ s + − k . For the second statement, let us take acocycle F ∈ Der pB ( A , U ) of degree p ≤ s − t , and show that F = ∂ G for some G ∈ Der p − B ( A , U ) .Recall that F is determined by its image on V , and that for any generator v ∈ V we have F ( v ) ∈ U s + − p . Given that F is a cocyle, for any such generator v we have that ∂ F ( v ) = dF ( v ) − ( − ) p Fd ( v ) = . But d ( v ) ∈ B , because B (cid:26) A is elementary, so Fd ( v ) = F ( v ) is a cycle in U s + − p . Since p ≤ s − b , it turns out that F ( v ) ∈ U (cid:62) b + is a boundary. Let G ( v ) ∈ U be such that dG ( v ) = F ( v ) .This defines G ∈ Der p − B ( A , U ) such that ∂ G = F , which is what we wanted. Indeed, ∂ G and F arederivations that coincide on every v ∈ V , and hence coincide on all of A .A nice filtration and a truncated target. Definition 3.13
A tower of cofibrations A (cid:26) A (cid:26) · · · (cid:26) A s (cid:26) · · · (cid:26) lim −→ s A s = A is cellular if each A s (cid:26) A s + is elementary of height s −
1. That is, if for every s ∈ N the algebra A s + is obtained from A s by freely adjoining a space of generators in homological degree s . A spectral sequence for tangent cohomology of algebraic operads
Algebras that are quasi-free and triangulated are good examples of colimits of cellular towers ofcofibrations, and in particular the three algebraic models of Adams–Hilton, Quillen and Sullivan,respectively, are good examples of colimits of towers of cofibrations relevant to topology. In Sec-tion 4 we will exploit this observation to apply our methods to rational homotopy theory.
Corollary 3.14
Let A be the colimit of a cellular tower of cofibrations of P -algebrasA (cid:26) A (cid:26) · · · (cid:26) A s (cid:26) · · · (cid:26) lim −→ s A s = A , and let u : A −→ U be a morphism of P -algebras. The following holds:(1) If U = U ≥ k for some k, then E s , t = for all t ≥ − k.(2) If U is b-truncated for some b, then E s , t = for all t ≤ − ( b + ) .In particular, if U is -truncated and U = U (cid:62) , the spectral sequence degenerates at the secondpage.Proof. The description of the E -page of our spectral sequence and Lemma 3.12 imply the firsttwo statements, if we recall that in a cellular tower each A s (cid:26) A s + is elementary of height s − b = k =
0, we obtain that E s , t = t =
0, so that the first page is concentrated in onerow, and the claim follows.
In this section, we apply the spectral sequence of Theorem 3.7 to classical rational homotopytheory using the models for spaces due to Adams–Hilton and Sullivan. We assume the reader hascertain familiarity with rationa homotopy theory and recommend the book [20] as a reference tothis subject. Since these models are constructed as colimits of towers of cofibrations of associativeand associative commutative dg algebras, respectively, this context is a natural starting point toconsider what the specialization of our spectral sequence look like.We remark that, unlike what we do here, one can use this spectral sequence over positive charac-teristic provided the corresponding operad is well-behaved in that setting, such as the associativeone. This may provide with further applications to p -local homotopy theory by exploiting, forinstance, the Adams-Hilton construction carried over the ring of integers localized at a prime; seefor example [2, 3, 41]. Conventions for the section.
In this section, we change the bidegree in the spectral sequence ofTheorem 3.7 by ( s , t ) (cid:55)→ ( s , − t ) . This gives more natural formulas, avoiding negative signs inthe applications that follow. The spectral sequence is still right-half-plane, but the differential d r changes its bidegree to ( r , r − ) , and under convergence assumptions, the E ∞ -page recovers thecohomology of the target by the formula ( gr H ) p = (cid:77) s = p + t E s , t ∞ . osé M. Moreno-Fernández and Pedro Tamaroff We also revert to using the qualifier “dg” in front of algebras to avoid any confusion when dealingwith (co)chain algebras, for example.
Let us now draw up some simple connections between the existing geometrical models of spacesfrom the work of F. Adams and P. Hilton, D. Quillen, and D. Sullivan, and the various homologyfunctors we have at hand, namely:(1) The functor H ∗ ( − ) : P - Alg −→ P - Alg that assigns to a (dg) P -algebra A its homology groups H ∗ ( A ) . As our notation suggests, these are also (non dg) P -algebras.(2) The functor H ∗ ( P , − ) : P - Alg −→ P ∞ - Cog that assigns to a P -algebra A its Quillen homologygroups H ∗ ( P , A ) . As our notation suggests, these are homotopy P -coalgebras. We will usethis only for P the associative operad.(3) The assignment (but not functor) H ∗ : P - Alg −→ Lie - Alg that assigns to a P -algebra A itstangent cohomology groups with values in itself H ∗ ( A , A ) .On the other hand, we have three classical constructions on topological spaces:(1) The Quillen functor λ : Top ∗ , −→ Lie - Alg that assigns to a pointed 1-connected X a dg Liealgebra λ ( X ) that models the Lie algebra of homotopy groups π ∗ ( Ω X ) .(2) The Sullivan functor A PL : Top −→ Com - Alg that assigns to a space X a commutative dgaalgebra A ∗ PL ( X ) that is a model of the cochain algebra Sing ∗ ( X ) .(3) The Adams–Hilton construction A : Top ∗ , −→ Ass - Alg that assigns to a pointed 1-connected X a dga algebra A ∗ ( X ) that is a model of the Pontryagin algebra of chains Sing ∗ ( Ω X ) .In the following table we record some of the relations between these algebraic models of spaces andthe three flavours of homology above. We point the reader to [7, 9, 45] for useful references wherethese relations are studied in detail. We now proceed to recall the last three functors above andapply our operadic formalism to obtain spectral sequences to compute their tangent cohomologygroups, which we will identify with invariants of known geometrical objects.Homology theoryModels Ordinary Quillen TangentQuillen π ∗ ( Ω X ) H ∗ + ( X ) π ∗ ( LX ) Sullivan–de Rham H ∗ ( X ) π ∗ + ( X ) ∨ π ∗ ( F ( X , X )) Adams–Hilton H ∗ ( Ω X ) H ∗ + ( X ) H ∗ ( LX ) In [23] the authors exploit the identification of Hochschild cohomology and the homology of de-rivations of Remark 3.4 along with the identification of the loop homology of a closed orientable A spectral sequence for tangent cohomology of algebraic operads manifold with the Hochschild cohomology of Sing ∗ ( Ω X ) to compute the loop bracket of X . Thisis done under the hypothesis the identification of Jones and Cohen [16] between loop homologyand Hochschild cohomology of Sing ∗ ( X ) commutes with the bracket.Explicitly, one can compute Hochschild cohomology of Sing ∗ ( Ω X ) through the Adams–Hiltonmodel A ∗ ( X ) of X , and then the loop space bracket as the Lie bracket of derivations of its associatedLie algebra of derivations Der ( A ∗ ( X )) . Since the model A ∗ ( X ) has generators that are in bijectionwith the cells of a CW decomposition of the manifold X , this method lends itself quite nicely tocomputations. We now recall the construction of Adams and Hilton.Let X be a CW complex with exactly one 0-cell, no 1-cells, and such that all the attaching maps ofhigher dimensional cells are based with respect to this only 0-cell. In [1], the authors construct acofibrant model A ∗ ( X ) of the dg algebra Sing ∗ ( Ω X ) , where Ω X is the Moore loop-space of X . Inthe following, for each n ∈ N write X n for the n -skeleton of X . Theorem (Adams–Hilton [1])
There is a cofibrant modelf X : A ∗ ( X ) = ( TV , d ) −→ ( Sing ∗ ( Ω X ) , d ) of the Pontryagin dga algebra of X such that for each n ∈ N , (L1) the space V n has basis the ( n + ) -cells of X , so that V = V (cid:62) , (L2) the map f X restricts to quasi-isomorphisms A ∗ ( X n ) −→ ( Sing ∗ ( Ω X n ) , d ) , (L3) if g : S n −→ X n is the attaching map of a cell e in X , then ( f X ) ∗ (cid:74) dv e (cid:75) = K (cid:74) g (cid:75) , where K is the isomorphism π n X n → π n − Ω X n followed by the Hurewicz map. These conditions determine the dg algebra A ∗ ( X ) uniquely up to isomorphism, and we call it the Adams–Hilton model of X . Of course, this model depends on the CW structure of X , whichwe take as part of the input data for its construction. As we noted, homology of the shift ofindecomposables s Ind P A ∗ ( X ) of the Adams–Hilton model is the (reduced) homology H ∗ ( X ) . Inother words, s Ind P A ∗ ( X ) is the reduced cellular chain complex Cell ∗ ( X ) of X .We remark that in [1] the authors actually produce a model of the fibration sequence of a CWcomplex X . That is, there is a commutative diagram of complexes of k -modules A ∗ ( X ) Cell ∗ ( X ) ⊗ A ∗ ( X ) Cell ∗ ( X ) Sing ∗ ( Ω X ) Sing ∗ ( LX ) Sing ∗ ( X ) where all the vertical maps are quasi-isomorphisms, the top row is the classical “algebraic fibration”coming from a cobar construction, and the bottom row is obtained by applying the singular chainsfunctor to the based path-space fibration. Moreover, the first vertical map is a map of dga algebras osé M. Moreno-Fernández and Pedro Tamaroff and the second vertical map is a map of right modules with respect to the obvious action of A ∗ ( X ) on Cell ∗ ( X ) ⊗ A ∗ ( X ) and the action of Ω X on LX . The Adams-Hilton model and, more generally,non-commutative dg algebra models have proven successful in the study of the p -local homotopytheory of spaces, for example; see the article [2] and the book [3].We now show how to run our spectral sequence to compute loop space homology of a space X byusing the filtration by degree in the Adams–Hilton model. In doing so, we obtain the followingresult, which the reader can compare with the spectral sequence of S. Shamir [46] and to theeponymous spectral sequence of J.-P. Serre. We remark that our spectral sequence has differentialsthat we can control provided we can control the Adams–Hilton model. This gives us a spectralsequence of the same shape as that in [17], produced by R. L. Cohen, J. D. S. Jones and J. Yan. Theorem 4.1
Let X be a CW complex with exactly one -cell, no -cells and all whose attachingmaps are based with respect to the only -cell. There is a first quadrant spectral sequence withE s , t = hom ( H s ( X ) , H t ( Ω X )) s == ⇒ H s + t ( LX ) , which is conditionally convergent in the sense of Boardman.Proof. Since X is simply connected, the Adams–Hilton model of X is a 0-connected dga algebra,and then Corollary 3.9 applies. We already noted that the Quillen homology of A ∗ ( X ) is H ∗ + ( X ) ,while its homology is H ∗ ( Ω X ) .To conclude, we need only address the identification of the target of our spectral sequence and thefact we have replaced the positive homology groups of X with with the homology groups of X .To do the second, we recall from Proposition 3.6 that we may pass from tangent cohomology toHochschild cohomology of the dga algebra A ∗ ( X ) by taking the cone of the adjoint mapAd : A ∗ ( X ) −→ Der A ∗ ( X ) . We then filter the cone by only filtering the summand corresponding to derivations. This has littleeffect in our spectral sequence, except that it adds the summand corresponding to the homology of A ∗ ( X ) , namely H ∗ ( Ω X ) = hom ( H ( X ) , H ∗ ( Ω X )) , in the second page and throughout the computation, and produces the desired shift (there is a shiftin the cone).Having addressed this, we now recall from [34, Chapter 4] and [16] that the Hochschild cohomo-logy groups of the Pontryagin algebra Sing ∗ ( Ω X ) of X are functorially isomorphic to the homologygroups of the free loop space LX = Map ( S , X ) of X . Remark 4.2
Observe that if A ∗ ( X ) is minimal or, what is the same, if the linear part of its dif-ferential is zero, then the description of the E -page is greatly simplified. If there are only cells ineven degree or only in odd degree, then A ∗ ( X ) will be minimal, for example. A spectral sequence for tangent cohomology of algebraic operads
For X a simply connected closed oriented manifold of dimension m , write H ∗ ( LX ) = H ∗ + m ( LX ) . We recall from [25] that there is a loop product − • − : H ∗ ( LX ) ⊗ H ∗ ( LX ) −→ H ∗ ( LX ) . One ofthe main results of that paper is as follows: Theorem
For every simply connected closed oriented manifold X , there is a natural isomorphismof graded commutative associative algebras ( H ∗ ( LX ) , − • − ) −→ ( HH ∗ ( Sing ∗ ( X )) , − (cid:94) − ) . Following the notation of Section 4.2, let A ∗ ( X ) = ( TV , d ) be an Adams–Hilton model for X , sothat the cone of the adjoint map of A ∗ ( X ) computes H ∗ ( LX ) . In our spectral sequence, the E -pagehas the form E s , t = hom ( H s ( X ) , H t ( Ω X )) and, since H ∗ ( X ) is a associative coalgebra and H ∗ ( Ω X ) an associative algebra, this page inheritsan associative convolution product given by the following composite: H ∗ ( X ) ∆ −→ H ∗ ( X ) ⊗ H ∗ ( X ) f ⊗ g −→ H ∗ ( Ω X ) ⊗ H ∗ ( Ω X ) µ −→ H ∗ ( Ω X ) . Observe that in case we consider elements with domain in H ( X ) = Q , this identifies with thePontryagin product of H ∗ ( Ω X ) . We remark that this is in line with Theorem 1 in [17], and alsopoint the reader to [38]. Theorem 4.3
The spectral sequence of Theorem 4.1 is multiplicative. The convolution product onthe E -page converges to the cup product on the Hochschild cohomology of A ∗ ( X ) , which equalsthe Chas–Sullivan product in the loop homology groups of X .Proof. The cup product on HH ∗ ( A ∗ ( X )) is induced from the differential d of Der ( A ∗ ( X )) as fol-lows. Let f and g be linear maps V −→ TV which correspond to derivations F and G . Then F (cid:94) G is determined by the derivation induced from the brace operation { d ; f , g } [31, 39]. Ex-plicitly, this is obtained by all possible ways of inserting f and g (in this order) into the operators d = d + d + d + · · · . The first few terms are as follows: { d ; f , g } = d ( f , g ) + d ( f , g , ) + d ( f , , g ) + d ( , f , g ) + · · · . Using the filtration of that theorem, on the E -page we are left only with the term induced inhomology by the derivation associated to d ( f , g ) , which on V restricts precisely to µ ◦ ( f ⊗ g ) ◦ ∆ ,which is what we wanted.We point out a similar description for a Lie bracket in the cohomology groups of derivations ofSullivan models of spaces through brace operations is given in [13, Theorem 3]. osé M. Moreno-Fernández and Pedro Tamaroff The Sullivan model of a topological space has proved to be quite successful and versatile in study-ing rational homotopy theory and its connection to other fields. We refer the reader to [20, 24]for a survey on the various results obtained through this formalism. In this section, we study thespectral sequence of Theorem 3.7 in this context, that is, when considering cofibrant commutativedga algebras that model the rational homotopy type of topological spaces and fibrations betweenthese.
The cohomological conventions.
As a chain algebra, a Sullivan algebra is concentrated in non-positive degrees, and the complexes of derivations of [22], which we will be using, are the same asours. Since we intend to apply Corollary 3.14 to Sullivan algebras, which are naturally cohomo-logically graded and concentrated in non-negative degrees, we substitute the “truncated” conditionfollowing Theorem 2.2 by the following.
Definition 4.4
Let b ∈ N . A dga algebra A is b-truncated if H p ( A ) vanishes for all p ≥ b + n -manifold has a n -truncated Sullivan model. Using these cohomological conven-tions, the items of Lemma 3.12 read as follows: Lemma 4.5
Let U be a cohomologically graded P -algebra.(1) If U = U ≥ k for some k, then Der pB ( A , U ) = for all p ≤ k − s − , (2) If U is b-truncated for some b, then H p ( Der B ( A , U )) = for all p ≥ b − s. Similarly, the items of Corollary 3.14 read as follows, where again t is replaced by − t : Lemma 4.6
Let U be a cohomologically graded P -algebra.(1) If U = U ≥ k for some k, then E s , t = for all s ≤ k + t − .(2) If U is b-truncated for some b, then E s , t = for all s ≥ t + b + . Since Sullivan algebras are concentrated in non-negative degrees, we will have that E s , t = t ≥ s +
1. This implies strong convergence in the spectral sequence of Theorem 4.7. If moreover X is an n -manifold, or any space whose rational cohomology is concentrated in degrees ≤ n , thenwe have the sharper vanishing of E s , t whenever t + n + ≤ s .Let us fix a fibration p : E −→ B , where E , B are 1-connected CW-complexes and E is finite, andlet us take ( Λ W , d ) (cid:26) ( Λ W ⊗ Λ V , D ) a relative Sullivan model of this fibration. In other words,the following diagram of cdga algebras commutes and the vertical maps are quasi-isomorphismsof cdga algebras: A PL ( B ) A PL ( E )( Λ W , d ) ( Λ W ⊗ Λ V , D ) . A PL ( p ) (cid:39) (cid:39) A spectral sequence for tangent cohomology of algebraic operads
The main result in [22] show that this cofibration codifies the homotopy type of Aut ( p ) , thecomponent of the identity in the topological monoid Aut ( p ) of fibre-homotopy self equivalences f : E −→ E . We recall this means that f is a homotopy equivalence such that p f = p . Theorem (Theorem 1 in [22])
There is an isomorphism of graded Lie algebrasH ∗ ( Der Λ W ( Λ W ⊗ Λ V )) −→ π ∗ ( Aut ( p )) . In the language of this paper, this result states the following:
Theorem (Operadic version)
The tangent cohomology groups of the mapA PL ( p ) : A PL ( B ) −→ A PL ( E ) are isomorphic, as a graded Lie algebra, to the rational Samelson Lie algebra π ∗ ( Aut ( p )) : thereis an isomorphism of graded Lie algebras H ∗ ( A PL ( p )) −→ π ∗ ( Aut ( p )) . It is useful to remark that, since each connected component of Aut ( p ) has the same rational ho-motopy type, the construction above determines the rational homotopy type of the space Aut ( p ) interms of the rational homotopy type of Aut ( p ) and the group structure in π ( Aut ( p )) . See [22]for complete details.We now observe that we can exhibit the cofibration ( Λ W , d ) −→ ( Λ W ⊗ Λ V , D ) as the tower of cofibrations obtained by adding the cells of Λ V “one by one”. With this in mind,let us put the technical tools developed in Section 3.1 into the appropriate context to apply them toSullivan algebras.The construction of the relative Sullivan model [20, Proposition 15.6], does not require any finitetype hypotheses on the spaces involved in the fibration, see [21, Theorem 3.1]. On the otherhand, for any simply connected space X with Sullivan model ( Λ V , d ) , we do need π ∗ ( X ) finitedimensional in each degree in order to have an identification V ∗ = π ∗ ( X ) . Otherwise, we can onlyconclude that there is an isomorphism V ∗ −→ hom ( π ∗ ( X ) , Q ) . Since in most applications we will take X to be a finite type CW-complex, a classical result ofSerre, see for instance [50, Theorem 20.6.3], let us identify V ∗ = π ∗ ( X ) , and we will write it likethis in the statements. eferences Theorem 4.7
Let F (cid:44) → E p −→ B be a fibration of -connected CW-complexes, with E finite. Thereis a spectral sequence withE s , t = hom ( π s ( F ) , H t ( E )) s == ⇒ π s − t ( Aut ( p )) . Proof.
Let ( Λ W , d ) (cid:26) ( Λ W ⊗ Λ V , D ) be a relative Sullivan model of the fibration. The Sullivancondition provides a filtration V ( k ) of V such that DV ( k ) ⊆ Λ W ⊗ Λ V ( k − ) for all k ≥ . Since W = W ≥ , we can assume that its Sullivan filtration is given by degree, W ( k ) = W ≤ k .Equally, we can assume the same degree filtration on V . It thus makes sense to consider thecellular tower of cofibrations of cdga’s given by A s = Λ W ⊗ Λ V < s , whose colimit is the relativeSullivan algebra ( Λ W , d ) (cid:26) ( Λ V ⊗ Λ W , D ) . We are in the cohomological situation of Corollary 3.9, and this gives us the second page if wenote that the cohomology groups of Λ W ⊗ Λ V are those of E . On the other hand, the quotientalgebra ( Λ V , d (cid:48) ) is a model for the fibre, so its Quillen homology gives the homotopy groupsof F . To conclude, recall that the target of the spectral sequence are the cohomology groups ofDer Λ W ( Λ W ⊗ Λ V ) , so an application of [22, Theorem 1] finishes the proof.Applying this result to the trivial fibration X −→ ∗ yields the following result. Corollary 4.8
Let X be a finite, -connected CW-complex. There is a conditionally convergentfirst quadrant spectral sequence withE s , t = hom ( π s ( X ) , H t ( X )) s == ⇒ π s − t ( Aut ( X )) To obtain the corollary above, we just fed a specific fibration to the spectral sequence of The-orem 4.7. The following table collects some other fibrations and the target of the correspondingspectral sequence. Input Shape Targetloop-space fibration PX −→ X π ∗ ( Ω X ) trivial fibration F × X −→ X π ∗ ( F ( X , Aut ( F )) principal G -bundle E −→ B π ∗ ( G ◦ ) We remark that the spectral sequence of Theorem 4.7 is multiplicative for a graded Lie bracket that,on the second page, identifies with the convolution bracket obtained from the Lie coalgebra π ∗ ( F ) with the Whitehead cobracket and the commutative algebra H ∗ ( E ) with the cup product. A resultanalogous to Theorem 3 in [13] for F ( X , Y ) replaced with Aut ( p ) should then give the conclusionanalogous to that of Theorem 4.3 that this product converges to the Whitehead product in the targethomotopy groups π ∗ ( Aut ( p )) . References
References [1] J. F. Adams and P. J. Hilton,
On the chain algebraof a loop space , Comment. Math. Helv. (1956),305–330. MR77929[2] D. J. Anick, The Adams-Hilton model for a fibra-tion over a sphere , J. Pure Appl. Algebra (1991),no. 1, 1–35. MR1137159[3] , Differential algebras in topology , ResearchNotes in Mathematics, vol. 3, A K Peters, Ltd.,Wellesley, MA, 1993. MR1213682[4] H.-J. Baues,
Algebraic homotopy , Cambridge Stud-ies in Advanced Mathematics, vol. 15, CambridgeUniversity Press, Cambridge, 1989. MR985099[5] H.-J. Baues and J.-M. Lemaire,
Minimal models inhomotopy theory , Math. Ann. (1977), no. 3,219–242. MR431172[6] H.-J. Baues and A. Quintero,
Infinite homotopytheory , K -Monographs in Mathematics, vol. 6,Kluwer Academic Publishers, Dordrecht, 2001.MR1848146[7] A. Berglund and B. Saleh, A dg lie model for relativehomotopy automorphisms , 2019.[8] Alexander Berglund and Ib Madsen,
Rational ho-motopy theory of automorphisms of manifolds , ActaMathematica (2020), no. 1, 67–185.[9] J. Block and A. Lazarev,
André-Quillen cohomo-logy and rational homotopy of function spaces , Adv.Math. (2005), no. 1, 18–39. MR2132759[10] M. Blomgren and W. Chachólski,
On the classi-fication of fibrations , Trans. Amer. Math. Soc. (2015), no. 1, 519–557. MR3271269[11] J. M. Boardman,
Conditionally convergent spec- tral sequences , Homotopy invariant algebraic struc-tures (Baltimore, MD, 1998), 1999, pp. 49–84.MR1718076[12] A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G.Quillen, D. L. Rector, and J. W. Schlesinger,
The mod − p lower central series and the Adams spectralsequence , Topology (1966), 331–342. MR199862[13] U. Buijs and A. Murillo, The rational homotopy Liealgebra of function spaces , Comment. Math. Helv. (2008), no. 4, 723–739. MR2442961[14] D. Calaque, R. Campos, and J. Nuiten, Moduli prob-lems for operadic algebras , 2019. [15] C. Chevalley and S. Eilenberg,
Cohomology theoryof lie groups and lie algebras , Transactions of theAmerican Mathematical Society (January 1948),no. 1, 85–85.[16] R. L. Cohen and J. D. S. Jones, A homotopy theor-etic realization of string topology , Math. Ann. (2002), no. 4, 773–798. MR1942249[17] R. L. Cohen, J. D. S. Jones, and Jun Yan,
The loophomology algebra of spheres and projective spaces ,Categorical decomposition techniques in algebraictopology (Isle of Skye, 2001), 2004, pp. 77–92.MR2039760[18] V. Dotsenko, S. Shadrin, and B. Vallette,
The twist-ing procedure , 2018.[19] M. Doubek, M. Markl, and P. Zima,
Deforma- tion theory (lecture notes) , Arch. Math. (Brno) (2007), no. 5, 333–371. MR2381782[20] Y. Félix, S. Halperin, and J.-C. Thomas, Rationalhomotopy theory , Graduate Texts in Mathemat-ics, vol. 205, Springer-Verlag, New York, 2001.MR1802847[21] ,
Rational homotopy theory. II , World Sci-entific Publishing Co. Pte. Ltd., Hackensack, NJ,2015. MR3379890[22] Y. Félix, G. Lupton, and S. B. Smith,
The rationalhomotopy type of the space of self-equivalences ofa fibration , Homology Homotopy Appl. (2010),no. 2, 371–400. MR2771595[23] Y. Félix, L. Menichi, and J.-C. Thomas, Ger-stenhaber duality in Hochschild cohomology , J.Pure Appl. Algebra (2005), no. 1-3, 43–59.MR2134291[24] Y. Félix, J. Oprea, and D. Tanré,
Algebraic modelsin geometry , Oxford Graduate Texts in Mathemat-ics, vol. 17, Oxford University Press, Oxford, 2008.MR2403898[25] Y. Félix, J.-C. Thomas, and M. Vigué-Poirrier,
Ra-tional string topology , J. Eur. Math. Soc. (JEMS) (2007), no. 1, 123–156. MR2283106[26] T. F. Fox, An introduction to algebraic deforma-tion theory , Journal of Pure and Applied Algebra (1993), no. 1, 17 –41.[27] Paul G. Goerss and Michael J. Hopkins, André- quillen (co)-homology for simplicial algebras oversimplicial operads , Une degustation topologique:Homotopy theory in the swiss alps, 2000, pp. 41–85. eferences [28] J. Griffin, Operadic comodules and (co)homologytheories , 2014.[29] D. K. Harrison,
Commutative algebras and cohomo-logy , Trans. Amer. Math. Soc. (1962), 191–204.MR142607[30] V. Hinich,
Homological algebra of homotopy algeb-ras , Comm. Algebra (1997), no. 10, 3291–3323.MR1465117[31] , Tamarkin’s proof of Kontsevich formalitytheorem , Forum Math. (2003), no. 4, 591–614.MR1978336[32] G. Hochschild, Cohomology and representations ofassociative algebras , Duke Math. J. (1947), 921–948. MR22842[33] S. Iyengar, André-Quillen homology of commutative algebras , Interactions between homotopy theory andalgebra, 2007, pp. 203–234. MR2355775[34] J.-L. Loday,
Free loop space and homology , 2011.[35] J.-L. Loday and B. Vallette,
Algebraic operads ,Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences],vol. 346, Springer, Heidelberg, 2012. MR2954392[36] J. P. May,
Classifying spaces and fibrations , Mem.Amer. Math. Soc. (1975), no. 1, no, 1, no. 155,xiii+98. MR370579[37] J. McCleary, A user’s guide to spectral sequences ,Second, Cambridge Studies in Advanced Mathem-atics, vol. 58, Cambridge University Press, Cam-bridge, 2001. MR1793722[38] L. Meier,
Spectral sequences in string topology , Al-gebraic & Geometric Topology (October 2011),no. 5, 2829–2860.[39] S. Mescher, A primer on a-infinity-algebras andtheir hochschild homology , 2016.[40] J. Millès,
André-quillen cohomology of algebrasover an operad , Adv. Math. (2011), no. 6, 5120–5164. [41] J. A. Neisendorfer,
What is loop multiplication any-how? , J. Homotopy Relat. Struct. (2017), no. 3,659–690. MR3691300[42] T. Porter, Proper homotopy theory , Handbook of al-gebraic topology, 1995, pp. 127–167. MR1361888[43] ,
Variations on a theme of homotopy , Ann.Fac. Sci. Toulouse Math. (6) (2013), no. 5, 1045–1089. MR3154586[44] D. G. Quillen, Homotopical algebra , Lecture Notesin Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR0223432[45] M. Schlessinger and J. Stasheff,
The Lie algebrastructure of tangent cohomology and deformationtheory , J. Pure Appl. Algebra (1985), no. 2-3,313–322. MR814187 [46] S. Shamir, A spectral sequence for the Hochschildcohomology of a coconnective DGA , Math. Scand. (2013), no. 2, 182–215. MR3073454[47] J. Stasheff,
A classification theorem for fibre spaces ,Topology (1963), 239–246. MR154286[48] , The intrinsic bracket on the deformationcomplex of an associative algebra , J. Pure Appl. Al-gebra (1993), no. 1-2, 231–235. MR1239562[49] D. Sullivan, Infinitesimal computations in topology ,Inst. Hautes Études Sci. Publ. Math. (1977), 269–331 (1978). MR646078[50] T. tom Dieck, Algebraic topology , EMS Textbooksin Mathematics, European Mathematical Society(EMS), Zürich, 2008. MR2456045[51] B. Vallette,
Homotopy theory of homotopy algebras ,arXiv preprint arXiv:1411.5533 (2014).S
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