MMorse Homotopy and Homological Conformal Field Theory
Viktor FrommAugust 31, 2018
Abstract
By studying spaces of flow graphs in a closed oriented manifold, we construct opera-tions on its cohomology, parametrized by the homology of the moduli spaces of compactRiemann surfaces with boundary marked points. We show that the operations satisfy thegluing axiom of an open homological conformal field theory. This complements previousconstructions due to R. Cohen et al., K. Costello and M. Kontsevich and is also the Morsetheoretic counterpart to a conjectural construction of operations on the Lagrangian Floerhomology of the zero section of a cotangent bundle, obtained by studying uncompactifiedmoduli spaces of higher genus pseudoholomorphic curves.
A flow graph in a manifold M consists of the following data: a graph G , a choice forevery edge e of G of a flow Ψ e on M , and a continuous map γ : G → M so that theimage of each edge e is a piece of a trajectory of the corresponding flow Ψ e . Flow graphscan be used to obtain invariants of manifolds. The simplest instance of this is the Morsecomplex, corresponding to the case when G consists of a single edge: by studying thespaces of trajectories of the gradient flow of a Morse function, one constructs a chaincomplex which computes the homology of a manifold. On the other hand, there are in-variants which are not visible in the classical Morse complex but can be recovered usingflow graphs ([BeCoh-94],[CohNor-12],[Fu-97]).In the work of R. Cohen and his collaborators ([BeCoh-94],[CohNor-12]), flow graphsin manifolds were used to construct invariants in the form of cohomology operations (weremark that our terminology is somewhat different in that we use the term ’flow graph’instead of ’graph flow’). The operations associated to certain special graphs can be iden-tified explicitly and turn out to correspond to invariants known from classical algebraictopology: the cup product, the Steenrod squares, the Stiefel-Whitney classes as well asthe Massey product (the latter using a somewhat different approach) can all be encodedin this way. Moreover, the operations satisfy a field-theoretic law: there is a compatibilitybetween gluing together graphs and composing the associated operations.In this paper we construct, building upon seminal ideas of K. Fukaya ([Fu-97]), theconformal version of this theory, obtained by studying flow graphs together with a ribbonstructure on the underlying graph. By taking the ribbon structure into account in asuitable way, we define operations that are parametrized by the homology of the modulispaces of Riemann surfaces with boundary. The construction draws upon ideas fromFloer homology and Gromov-Witten theory. a r X i v : . [ m a t h . G T ] M a y et Σ be a compact connected oriented surface with m > n + + n − ≥ g − m > M Σ the space of complex structures on Σ,together with labellings of the marked points by positive real numbers. We will associateto every closed oriented manifold M invariants in the form of linear maps( H ∗ ( M )) ⊗ n + → H ∗ ( M Σ ) ⊗ ( H ∗ ( M )) ⊗ n − . (1)Here H ∗ denotes singular cohomology with coefficients in a field of characteristic zero(more precisely, for M Σ cohomology with coefficients in a certain local system must beused). For example, if Σ is a disk with two incoming and one outgoing marked point,then M Σ is contractible and (1) is the cup product. The construction of the operations(1) in the general case proceeds as follows.Let g be a Riemannian metric on M and f a Morse function whose gradient flow withrespect to g is Morse-Smale. Denote by C ∗ ( f ) the associated Morse complex, with thegrading given by the Morse index and the codifferential given by counts of trajectories ofthe positive gradient flow. In order to define the operations (1), it suffices to constructcochain maps F M Σ : ( C ∗ ( f )) ⊗ n + → C ∗ ( M Σ ) ⊗ ( C ∗ ( f )) ⊗ n − , (2)where C ∗ ( M Σ ) is a complex computing the cohomology of M Σ . To this end, the ribbongraph decomposition of Riemann surfaces ([Har-88],[Pe-87]) will be used.A graph is a one-dimensional CW complex. We refer to the univalent vertices of agraph as the external vertices and to the vertices of valency greater than one as internal.An edge is called external if it is incident to a univalent vertex, and internal otherwise.Consider a graph G together with an embedding i : G (cid:44) → Σ, so that the univalent verticesof G are mapped to the boundary marked points, the remaining points of G are mappedto the interior of Σ and so that Σ deformation retracts to i ( G ). Two embeddings areidentified if one is obtained from the other by an isotopy which is constant on the univa-lent vertices. A ribbon structure on G is an equivalence class of embeddings. We write Γfor the ribbon graph and we say that Γ has Σ as its associated oriented surface. Figure 1: Two ribbon graphswhose associated surfaces area pair of pants and a toruswith a disk removed respec-tively. A half-edge of a graph G is a pair ( v, [ ϕ ]), where v is a vertex and [ ϕ ] is an isotopy classof embeddings ϕ : ([0 , , (cid:44) → ( G, v ). It is a classical observation that a ribbon structureon G can equivalently be defined as a cyclic ordering of the half-edges at every vertex.An orientation of an edge is defined as an ordering of the two-element set consisting ofthe corresponding half-edges. A metric structure on G is an assignment to every edge e ∈ E ( G ) of a non-negative real number l e . ssume that for every marked point o ∈ ∂ Σ, a critical point p o of f is fixed. In Section2, we associate to every ribbon graph Γ as above a space M Γ , x ( p + , p − ) of flow graphsin M . An element of M Γ , x ( p + , p − ) consists of a metric structure on Γ together with acontinuous map γ : Γ → M , so that the restrictions of the map to the edges are piecesof trajectories of given vector fields x on M and where incidence conditions correspond-ing to the points p + , p − are imposed at the external vertices. For generic choice of thevector field datum x , the space M Γ , x ( p + , p − ) is a smooth manifold. Moreover, the par-tial compactification M Γ , x ( p + , p − ) obtained by allowing internal edges of zero lengthas well as broken flow lines at the external edges is a manifold with corners. By theribbon graph decomposition of Riemann surfaces, the space of metric ribbon graphs Γ asabove is homeomorphic to M Σ (see equation (8) below) and thus there is a projection π Γ : M Γ , x ( p + , p − ) → M Σ obtained by forgetting γ . We show that π Γ is a proper map.These claims are established in Section 2.2.We view Z Γ , x ( p + , p − ) = ( M Γ , x ( p + , p − ) , π Γ ) as a geometric chain in M Σ . In Section 2.1we define a chain complex C BM ∗ ( M Σ ) generated by pairs ( P, f ), where P is a (not neces-sarily compact) oriented manifold with corners and f : P → M Σ is a proper continuousmap. The complex C BM ∗ ( M Σ ) computes the Borel-Moore homology of M Σ , the latterbeing isomorphic by rational Poincar´e duality to the cohomology. Theorem 1.1.
Let F M Σ : ( C ∗ ( f )) ⊗ n + → C BM ∗ ( M Σ ; det d ⊗ or ) ⊗ ( C ∗ ( f )) ⊗ n − (3) be the linear map defined by p + (cid:55)→ (cid:88) Γ , p − Z Γ , x ( p + , p − ) ⊗ p − , (4) where the summation is over all p − ∈ ( Crit ( f )) × n − and over all the ribbon graphs Γ , sothat every internal vertex of Γ is trivalent.1. F M Σ is a cochain map.2. The induced map HF M Σ : ( H ∗ ( M )) ⊗ n + → H ∗ ( M Σ ; det ⊗ d ) ⊗ ( H ∗ ( M )) ⊗ n − (5) is independent, up to a vector space isomorphism H ∗ ( M ) → H ∗ ( M ) , of all choices(i. e. independent of the vector field data, of the Riemannian metric and of theMorse function). Here d denotes the dimension of M and det and or are certain local systems on M Σ ,defined in Section 3.1. The appearance of local coefficients is due to the fact that thespace M Γ , x ( p + , p − ) is not canonically oriented, but rather its orientation is determinedby fixing linear orderings of the vertices and of the edges of Γ and orientations of theedges. The dependence of the orientation of M Γ , x ( p + , p − ) on these choices is describedin Section 3.1. Theorem 1.1 is proved in Section 3.2.We remark that evaluating the statement of Theorem 1.1 in the case when n + = n − = 0,i. e. there are no marked points, one obtaines invariants of closed oriented manifoldsin the form of cohomology classes in moduli spaces of Riemann surfaces with boundary.However, examining the degrees, one finds that these classes are trivial unless the dimen-sion of the manifold is at most three. e now outline the relationship of Theorem 1.1 to some previous results. As was men-tioned above, a construction of operations from spaces of flow graphs was first proposedin [BeCoh-94] (see [CohNor-12] for a more recent exposition). In this latter approach,one associates to a graph G a cohomology operation, parametrized by the homology ofthe classifying space of the automorphism group Aut ( G ). The main difference of the con-struction of Theorem 1.1 to these previous ideas is the consideration of ribbon structures.As a consequence, the operations constructed here are parametrized by the homology ofthe moduli spaces of Riemann surfaces instead.While the operations (5) bear some formal resemblence to the structure of Gromov-Witteninvariants, there are two pronounced differences: firstly, Σ has non-empty boundary andsecondly, the operations are parametrized by the homology of M Σ instead of the homol-ogy of the Deligne-Mumford compactification. The operations (5) fit into the frameworkof an open homological conformal field theory ([Se-04],[Ge-94]) - the corresponding gluingaxiom is proved in Section 3.3.There are two previously known constructions of open HCFTs (in fact, more generally,of open TCFTs. i. e. topological conformal field theories). The first is an algebraic-combinatorial approach based on an idea of M. Kontsevich in [Ko-94] (see also [Cos-07a],[HamLaz-08]) and produces an open HCFT starting with a (finite-dimensional minimal)cyclic A ∞ -algebra - the homotopy associative analogue of a differential graded algebra,equipped with a compatible inner product. The second construction, due to K. Costello,is more analytic and uses heat kernels ([Cos-07b]). The starting point here is a so-calledCalabi-Yau elliptic space - one of the simplest examples of such an object is the de Rhamalgebra of a closed oriented manifold. The relationship between these two approachesis well-understood: a cyclic A ∞ -algebra can be associated to each Calabi-Yau ellipticspace and the open HCFT obtained via the analytic construction on the elliptic space isequivalent to the result of applying the algebraic approach to the associated A ∞ -algebra([Cos-07b], Section 5).Applying these ideas to the de Rham algebra, one obtains operations which have thesame form as those constructed in Theorem 1.1 and which we denote by HF dR Σ . In aforthcoming paper, we will show that the constructions are equivalent, i. e. the oper-ations HF M Σ and HF dR Σ coincide up to a vector space isomorphism H ∗ ( M ) → H ∗ ( M ).This explains how to compute the maps HF M Σ from the de Rham algebra of the man-ifold with its product and inner product using the projection and homotopy which areprovided (after choosing a Riemannian metric) by classical Hodge theory.It is known that for every open TCFT there is an associated universal open-closed, andthus in particular a closed TCFT ([Cos-07a]). A direct geometric construction of an open-closed TCFT on a manifold, following ideas different from what is presented here, wasoutlined in [BlCohTe-09]. The resulting closed TCFT is expected to be closely related tothe BV algebra of loop homology ([Ge-94]).We finish the introduction by sketching the relationship of the construction of Theorem1.1 to Floer homology and the study of pseudoholomorphic curves in cotangent bundles.For a smooth function f on a manifold M , the graph L df of the differential df is anexact Lagrangian submanifold of the total space of the cotangent bundle T ∗ M . Twographs L df and L df intersect transversally if and only if the difference f − f is a Morsefunction. In this case the Lagrangian Floer cohomology of the pair L df , L df is defined [Fl-89]). The differential of Lagrangian Floer cohomology is constructed by counting theelements of the zero-dimensional components of the moduli spaces of pseudoholomorphicstrips in T ∗ M with L df and L df as boundary conditions.Extending this idea, we can consider for any oriented surface Σ as in Theorem 1.1 thespace of all pseudoholomorphic maps from Σ to T ∗ M with Lagrangian boundary con-ditions of the form L df , f ∈ C ∞ ( M, R ) imposed on the components of the complementin ∂ Σ of the set of marked points. For example, in the case when there are no markedpoints, we consider the space M T ∗ M Σ of all pseudoholomorphic curves in T ∗ M which mapevery boundary component of Σ to a submanifold L df for some fixed f ∈ C ∞ ( M, R ). Thespace M T ∗ M Σ is usually non-compact, but if it carries a fundamental class in Borel-Moorehomology, then using the fact that the projection π Σ : M T ∗ M Σ → M Σ is proper (this isa consequnce of Gromov compactness) and rational Poincar´e duality, one would obtain acohomology class in M Σ . More generally, by considering surfaces with marked points onthe boundary, we would be led to operations on Lagrangian Floer cohomology, analogousto the ones constructed in Theorem 1.1. If Σ is a disk, this could be made rigorous by themethods developed in [FOOO-09]. The general case is, to the knowledge of the author,conjectural.A. Floer showed that for a suitable choice of almost complex structures on T ∗ M andof a Riemannian metric on M , the chain complex of Lagrangian Floer homology of apair L df , L df is isomorphic to the Morse complex of f − f ([Fl-89]). The main ideais sometimes referred to as an ’adiabatic limit’ argument: Floer demonstrated that aftermultiplying f and f by a sufficiently small number, there is a one-to-one correspondencebetween isolated pseudoholomorphic strips in T ∗ M and isolated gradient flow trajecto-ries in M . In [FuOh-98], this idea was generalized to obtain an identification betweenspaces of pseudoholomorphic disks with arbitrary number of boundary marked points andspaces of flows of ribbon trees was obtained. These results suggest that the operationson the Lagrangian Floer homology of the zero section of the cotangent bundle, defined asoutlined above from the study of psudoholomorphic curves in T ∗ M , should correspondto operations on the cohomology of M , obtained from spaces of flows of general ribbongraphs. Theorem 1.1 provides the construction of these latter operations.Throughout this paper, homology and cohomology with real coefficients is used and thecoefficient ring is omitted from the notation. Theorem 1.1 continues to hold for coeffi-cients in an arbitrary field of characteristic zero.The author would like to thank Klaus Mohnke for many fruitful conversations. In this Section we discuss the complex C BM ∗ ( M Σ ) of locally finite geometric chains in M Σ and use flow graphs in a manifold to construct elements of this complex. The Borel-Moore homology, or homology with closed support, of a topological space X is defined as the homology of the chain complex of locally finite singular chains in X , i. e. of linear combinations (cid:80) n σ σ of singular simplices σ : ∆ → X , so that forevery compact subset K ⊂ X there are only finitely many non-vanishing coefficients n σ with σ (∆) ∩ K (cid:54) = ∅ . Summing the top-dimensional simplices of a triangulation of a not necessarily compact) oriented manifold N d , one defines its fundamental class [ N ] ∈ H BMd ( N ). The map x (cid:55)→ [ N ] ∩ x yields a Poincar´e duality isomorphism H k ( N ) ∼ −→ H BMd − k ( N ) . (6)While Borel-Moore homology is not functorial in general, there is a pushforward f ∗ : H BM ∗ ( X ) → H BM ∗ ( Y ) provided that f : X → Y is proper. If f : X → Y is continu-ous and on both X and Y there are Poincar´e duality isomorphisms as in (6), then f ∗ : H ∗ ( Y ) → H ∗ ( X ) yields a transfer homomorphism f ! BM ∗ : H BM ∗ ( Y ) → H BM ∗ + d X − d Y ( X ) inBorel-Moore homology.Similarly to the case of singular homology discussed in [Ja-00], Borel-Moore homologymay be computed as the homology of a complex of geometric chains. To define this com-plex, we recall that a d -dimensional manifold with corners P is a paracompact Hausdorffspace locally modelled on the products R k ≥ × R d − k , 0 ≤ k ≤ d . A local boundary com-ponent β at a point x of P consists of a choice, for a coordinate neighbourhood U of x ,of a connected component of the points of U which lie on the codimension one stratum.With the convention ∂P = { ( x, β ) : x ∈ P, β a local boundary component of P at x } , (7)the boundary of a manifold with corners is again a manifold with corners and there is anatural map i ∂P : ∂P → P .We denote by C BM ∗ ( X ) the vector space of all sums of the form (cid:80) P n P ( P, f P ), where P isan oriented manifold with corners and f P : P → X is a proper continuous map. As before,we require that the geometric chain (cid:80) P n P ( P, f P ) be locally finite: for every K ⊂⊂ X ,there are only finitely many summands so that n P is non-zero and f P ( P ) ∩ K (cid:54) = ∅ . Wemake the following two identifications. Firstly, if P (cid:48) is obtained from P by reversingthe orientation, then we identify ( P (cid:48) , f P (cid:48) ) with − ( P, f P ). Secondly, if P is the union P = Q ∪ X R of two codimension zero submanifolds with corners Q and R along a (pos-sibly empty) submanifold with corners X ⊂ ∂P, ∂Q , then we identify ( P, f P ) with thesum of ( Q, f P | Q ) and ( R, f P | R ). The degree of ( P, f P ) is the dimension of P and theboundary is defined by ∂ ( P, f P ) = ( ∂P, f P ◦ i ∂P ). Proposition 2.1.
For any topological space X , the homology of the complex C BM ∗ ( X ) isisomorphic to the Borel-Moore homology of X .Proof. Denote by h BM ∗ ( X ) the homology of the complex C BM ∗ ( X ) and by H BM ∗ ( X ) theBorel-Moore homology of X . By definition, the latter is given by equivalence classes ofclosed locally finite chains (cid:80) σ n σ σ , where σ : ∆ → X is a singular simplex. A map ψ : H BM ∗ ( X ) → h BM ∗ ( X ) is defined by [ (cid:80) σ n σ σ ] (cid:55)→ [ (cid:80) σ n σ (∆ , σ )]. Conversely, given agenerator ( P, f P ) of C BM ∗ ( X ), choose a triangulation of P which induces a triangulationof ∂P and so that if two components of ∂P are diffeomorphic, then the induced triangu-lations coincide. Since f is proper, the sum (cid:80) i f P | ∆ i =: (cid:80) i σ i over the top-dimensionalsimplices of the triangulation is a locally finite singular chain in X ; it is closed (resp.exact) if ( P, f P ) is closed (resp. exact) and its class in H BM ∗ ( X ) is independent of thechoice of the triangulation. Define φ : h BM ∗ ( X ) → H BM ∗ ( X ) by [ P, f P ] (cid:55)→ [ (cid:80) i σ i ]. It isimmidiate to check that φ and ψ are inverse to each other.In Section 3.3, we will use the fact that in two simple special cases the transferhomomorphism f ! BM ∗ : H BM ∗ ( Y ) → H BM ∗ ( X ) corresponding to a continuous map f : X → Y can be identified as the homomorphism induced by an explicit chain map ! CBM ∗ : C BM ∗ ( Y ) → C BM ∗ ( X ). Firstly, if f is the inclusion map i : X (cid:44) → Y of anopen subset X ⊂ Y , then the image under i ! CBM ∗ of a generator ( P, f P ) of C BM ∗ ( Y ) isgiven by ( f − P ( X ) , f P | f − P ( X ) ). Secondly, if f is the projection π : X → Y of a locallytrivial fibre bundle, then the image of a generator ( P, f P ) of C BM ∗ ( Y ) under π ! CBM ∗ isgiven by ( f ∗ P X, π ∗ f P ), where f ∗ P X → P is the pullback bundle and π ∗ f P the map whichmakes the following diagram commutative: f ∗ P X (cid:15) (cid:15) π ∗ f P (cid:47) (cid:47) X π (cid:15) (cid:15) P f P (cid:47) (cid:47) Y. We will also consider the complex C BM ∗ with coefficients in a local system. In thiscontext a generator is a pair ( P, f P ) as before, together with a section of the pullback ofthe local system under f P . If the local system is graded, then the degree of the chainis given as the sum of the dimension of P and the degree of the section. There is astraightforward analogue of Proposition 2.1 for homology with local coefficients.As was mentioned above, on any oriented manifold there is a Poincar´e duality isomor-phism H BM ∗ ( M ) (cid:39) H dim M −∗ ( M ). Over a field of characteristic zero, this is also true forthe space M Σ of complex structures. To explain this fact, we recall that by the ribbongraph decomposition of Riemann surfaces ([Har-88],[Pe-87], see also [Cos-07b]), there isa homeomorphism M Σ (cid:39) (cid:91) Γ M et (Γ) / ∼ . (8)The union on the right-hand side of (8) is over all ribbon graphs Γ whose associatedoriented surface is Σ and so that every internal vertex of the graph has valency at leastthree; the univalent vertices of Γ correspond to the marked points on ∂ Σ. The symbol
M et (Γ) denotes the space of metric structures on Γ, where we allow edges of length zero,subject to the condition that the sum of the lengths of the edges in any cycle remain pos-itive. The equivalence relation is generated by the following two identifications. Firstly,we identify a metric structure on a ribbon graph Γ with each metric structure obtainedas the pullback under a ribbon graph automorphism Γ → Γ which fixes each univalentvertex. Secondly, we identify two metric ribbon graphs if one is obtained from the otherby collapsing internal edges of zero length.As a simple example, consider the case of a disk with four marked points on the boundary(for simplicity, the real-valued labellings of the marked points are omitted). The mod-
Figure 2: The rib-bon graph decompo-sition of the mod-uli space of holomor-phic disks with fourmarked points on theboundary. uli space is a real line which is identified as the result of gluing two copies of R ≥ to a oint. The homeomorphism (8) means that each element of the moduli space is obtaineduniquely from the fully symmetric configuration by slicing open along one of the two axesof symmetry which do not contain marked points, and then gluing in a holomorphic strip.It follows from (8) that M Σ is an orbifold which admits a good cover in the sense of[AdLeRu-07], page 35. Thus over a field of characteristic zero, there is an isomorphism H ∗ c ( M Σ ; or ) (cid:39) H dim M Σ −∗ ( M Σ ) , (9)where H ∗ c ( M Σ ; or ) is the cohomology with compact support and coefficients in the orien-tation sheaf. By the universal coefficient theorem, H ∗ c ( M Σ ; or ) and H ∗ ( M Σ ) are the dualvector spaces to respectively H BM ∗ ( M Σ ; or ) and H ∗ ( M Σ ). We conclude the isomorphism H BM ∗ ( M Σ ; or ) (cid:39) H dim M Σ −∗ ( M Σ ) . (10) We now introduce spaces of flow graphs in a manifold and use them to define elements of C BM ∗ ( M Σ ). The discussion of transversality and the construction of natural compactifi-cations of the spaces follows a similar line of argument as in the classical setting of Morsetheory ([Sch-93]).Let Γ be a ribbon graph which appears on the right-hand side of the ribbon graph de-composition (8). Suppose that to every boundary marked point on ∂ Σ (or, equivalently,to every external edge e of Γ), a critical point p e of the Morse function f is associated.The partition of the marked points into incoming and outgoing points defines a partition p = ( p + , p − ) of the tuple p of the critical points p e . We will associate to this data aBanach manifold B Γ ( p + , p − ). We first introduce parametrizations of the edges of Γ.Let e be an external edge of Γ and v be the external vertex incident to e . If v is markedas incoming, then we fix a homeomorphism ψ e : ( −∞ , ∼ −→ e − { v } . In the case when v ismarked as outgoing, we fix ψ e as a homeomorphism ψ e : [0 , ∞ ) ∼ −→ e − { v } . In each casewe orient e by prescribing ψ e to be orientation-preserving. To choose parametrizations ofthe internal edges of Γ, we must take into account the fact that the choice of orientationsof these edges is non-canonical. Definition 2.2.
We denote by Γ (cid:48) → Γ the finite cover with fibre given by all choices oforientations of all the internal edges of Γ . Thus Γ (cid:48) is a graph whose points are given by pairs consisting of a point of Γ togetherwith a choice of orientations of all the internal edges of Γ. Every edge of Γ (cid:48) carries a nat-ural orientation and the projection π Γ : Γ (cid:48) → Γ preserves the orientations of the externaledges. The group T of covering transformations of Γ (cid:48) → Γ is generated by the involutions τ e given by reversing the orientation of an internal edge e of Γ.We fix for every internal edge e (cid:48) of Γ (cid:48) a continuous map ψ e (cid:48) : [0 , → e (cid:48) whose re-striction to (0 ,
1) is a homeomorphism and which induces the natural orientation of e (cid:48) .If e (cid:48) is an incoming internal edge of Γ (cid:48) which is mapped to e under π Γ , then a map ψ e (cid:48) : ( −∞ , → e (cid:48) is uniquely defined by requiring that the composition of π Γ ◦ ψ e (cid:48) coincide with ψ e . We define parametrizations ψ e (cid:48) : [0 , ∞ ) → e (cid:48) of the outgoing edges ofΓ (cid:48) analogously.If e (cid:48) is an incoming external edge of Γ (cid:48) and p the associated critical point of f , thenwe denote by H e (cid:48) the space of all elemenets γ ∈ W , loc (( −∞ , , M ) so that there exist > ξ ∈ W , ([ T, ∞ ) , T p M ) with γ ( t ) = exp p ( ξ ( t )) for all t ≥ T . Here exp p denotesthe exponential map at p , defined in a neighbourhood of the origin of T p M . In the casewhen e (cid:48) is outgoing, we define H e (cid:48) analogously, but with ( −∞ , T ] replaced by [ T, ∞ ). Foran internal edge e (cid:48) of Γ (cid:48) , we denote H e (cid:48) = W , ([0 , , M ). Definition 2.3.
We define B Γ ( p + , p − ) as the space of contunuous maps γ : Γ (cid:48) → M ,so that for every edge e (cid:48) of Γ (cid:48) , the composition γ e (cid:48) = γ ◦ ψ e (cid:48) is an element of H e (cid:48) andmoreover for any pair of edges e (cid:48) , e (cid:48)(cid:48) of Γ (cid:48) , where e (cid:48) is an internal edge, γ τ e (cid:48) ( e (cid:48)(cid:48) ) ( t ) = (cid:40) γ e (cid:48)(cid:48) ( t ) if e (cid:48) (cid:54) = e (cid:48)(cid:48) ,γ e (cid:48)(cid:48) (1 − t ) if e (cid:48) = e (cid:48)(cid:48) . (11)The space B Γ ( p + , p − ) is a Banach manifold. The tangent space T γ B Γ ( p + , p − ) at apoint γ is the closed subspace of ⊗ e (cid:48) ∈ E (Γ (cid:48) ) W , ( γ ∗ e (cid:48) T M ) consisting of all the elements s = ( s e (cid:48) ) e (cid:48) ∈ E (Γ (cid:48) ) which define a continuous section of γ ∗ T M and so that for each pair e (cid:48) , e (cid:48)(cid:48) of edges of Γ (cid:48) , where e (cid:48) is an internal edge, s τ e (cid:48) ( e (cid:48)(cid:48) ) ( t ) = (cid:40) s e (cid:48)(cid:48) ( t ) if e (cid:48) (cid:54) = e (cid:48)(cid:48) , − s e (cid:48)(cid:48) (1 − t ) if e (cid:48) = e (cid:48)(cid:48) . (12)Here W , ( γ ∗ e T M ) denotes the space of sections of class W , and E (Γ (cid:48) ) is the set ofedges of Γ (cid:48) .We define a Banach bundle E →
M et (Γ) × B Γ ( p + , p − ) as the pullback of T B Γ ( p + , p − )under the projection M et (Γ) × B Γ ( p + , p − ) → B Γ ( p + , p − ) to the second factor. There isa well-defined section S = ( s e (cid:48) ) e (cid:48) ∈ E (Γ (cid:48) ) ∈ L ( E ) (13)determined by the condition that s e (cid:48) ( γ ) = ddt γ e (cid:48) ( t ) (14)for every edge e (cid:48) of Γ (cid:48) .We want to consider maps in B Γ ( p + , p − ), so that the restriction of the map to everyedge is a piece of a trajectory of the flow of a given one-parameter family of vector fields.We now introduce the setup for the construction of these vector field data.We choose for each edge e (cid:48) of Γ (cid:48) a one-parameter family of vector fields on M . For-mally, consider the vector bundle E → M et (Γ) × [0 , × M given as the pullback of T M under the projection to the last factor. Let X Γ denote the space of all sections of class W , of E . Definition 2.4.
We define X Γ to be the space of all elements x = ( x e (cid:48) ) e (cid:48) ∈ E (Γ (cid:48) ) ∈ X ⊗| E (Γ (cid:48) ) | Γ (15) which satisfy the following conditions:1. For any two edges e (cid:48) , e (cid:48)(cid:48) of Γ (cid:48) , where e (cid:48) is an internal edge, and each t ∈ [0 , , x τ e (cid:48) ( e (cid:48)(cid:48) ) ( · , t, · ) = (cid:40) x e (cid:48)(cid:48) ( · , t, · ) if e (cid:48) (cid:54) = e (cid:48)(cid:48) , − x e (cid:48)(cid:48) ( · , − t, · ) if e (cid:48) = e (cid:48)(cid:48) . (16) . There is a constant C > , so that for every edge e (cid:48) of Γ (cid:48) , the estimate (cid:107) x e (cid:48) ( l , t, · ) (cid:107) W , ( T M ) < C (17) holds true for all ( l , t ) ∈ M et (Γ) × [0 , . The first condition will be used to associate to every element of X Γ a well-definedsection of E . The second condition is essential for the construction of compactificationsof the spaces of flow graphs.To every element x ∈ X Γ , we associate a section F x = ( F e (cid:48) ) e (cid:48) ∈ E (Γ (cid:48) ) of E as follows.Let σ : R → R be a smooth function with σ ( t ) = 1 for | t | ≤ σ ( t ) = 0 for | t | ≥
2. If e (cid:48) is an external edge of Γ (cid:48) , then F e (cid:48) ( l , γ )( t ) = ∇ g f ( γ e (cid:48) ( t )) + σ ( t ) x e (cid:48) ( l , | t | , γ e (cid:48) ( t )) . (18)If e (cid:48) is an internal edge of Γ (cid:48) which is mapped to e ∈ E (Γ) under the projection Γ (cid:48) → Γ,then we denote F e (cid:48) ( l , γ )( t ) = l e x e (cid:48) ( l , t, γ e (cid:48) ( t )) , (19)where l e is the length of e in the metric structure l .With this notation in place, we can now define the spaces of flow graphs. Definition 2.5.
For x ∈ X Γ , let S x ∈ L ( E ) denote the section given as the difference S x = S − F x , (20) where S is defined as in (14). We define the space M Γ , x ( p + , p − ) ⊂ M et (Γ) × B Γ ( p + , p − ) (21) of flows over Γ subject to the vector field datum x as the zero locus of S x . Definition 2.5 associates to each graph Γ which appears in the ribbon graph decom-position (8) of M Σ a corresponding space M Γ , x ( p + , p − ) of flow graphs in a manifold.This is illustrated by the following simple example. Example 2.6.
Consider the case when Σ is an annulus with two marked points on thesame boundary component. In this example, the moduli space M Σ is homeomorphic to an open disk (for sim-plicity, we omit the real-valued labels at the marked points). There are five distinctisomorphism classes of ribbon graphs which appear on the right-hand side of (8). Threetwo-dimensional cells corresponding to the three ribbon graphs with two internal verticesof valency three are glued together along two one-dimensaion cells which correspond tothe two graphs with a single internal vertex of valency four. There are no non-trivial au-tomorphisms. The ribbon graph decomposition and the spaces of flow graphs associatedto the cells are illustrated in Figure 3. The shaded parts of the graphs indicate the vectorfield data.The next Proposition summarizes the arguments used to equip the space M Γ , x ( p + , p − )with the structure of a manifold. Proposition 2.7.
1. For every x ∈ X Γ , the map S x : M et (Γ) × B Γ ( p + , p − ) → E (22) is Fredholm of index ind ( S x ) = | p − | − | p + | + dχ (Σ) − dn − + | E (Γ) | . (23) Here | p + | and | p − | denote the sum of all the Morse indices of the critical pointscorresponding to the incoming and to the outgoing marked points respectively. Thesymbol | E (Γ) | stands for the number of edges of Γ .2. There is a subset X Γ ,reg ⊂ X Γ of second category so that for each x ∈ X Γ ,reg , S x istransverse to the zero section of E .3. Suppose that for every graph (cid:101) Γ obtained from Γ by collapsing internal edges (whereas before no cycle is collapsed), an element x (cid:101) Γ ∈ X (cid:101) Γ ,reg is fixed. Then there existsa subset of second category of X Γ , so that for every element x of that subset, theconclusion of 2. holds true and, in addition, the restriction of x to M et ( (cid:101) Γ) ⊂ M et (Γ) coincides with x (cid:101) Γ .Proof. The arguments are analogous to the classical case of spaces of flow trajectoriesand we will thus stay brief. Denote by D S x the linearization of S x . For fixed l ∈ M et (Γ)and s ∈ T γ B Γ ( p + , p − ), we can write D S x (0 , s ) in the form( D S x (0 , s )) e (cid:48) ( t ) = ddt s e (cid:48) ( t ) − A ( t ) s e (cid:48) ( t ) , (24)where A ( t ) ∈ End ( T γ e (cid:48) ( t ) M ) are endomorphisms such that if e (cid:48) is an external edge with γ e (cid:48) ( t ) → p ∈ Crit ( f ) for | t | → ∞ , then A ( t ) → Hess f ( p ). Using the non-degeneracy ofthe Hessian, one concludes from (24) the inequality (cid:107) s e (cid:48) (cid:107) W , ≤ c ( (cid:107) s e (cid:48) (cid:107) L + (cid:107) ( D S x (0 , s )) e (cid:48) (cid:107) L ) (25)for a positive constant c . Using the compactness of the embedding W , ( T γ B Γ ( p + , p − )) (cid:44) → L ( T γ B Γ ( p + , p − )), it follows from (25) that the map ( D S x ) : s (cid:55)→ D S x (0 , s ) has finite-dimensional kernel and closed image. A standard computation using partial integrationshows that each element r ∈ L ( E ) so that (cid:104) r , D S x (0 , s ) (cid:105) = 0 for all s ∈ T γ B Γ ( p + , p − ) isweakly differentiable and satisfies ddt r e (cid:48) ( t ) + A T ( t ) r e (cid:48) ( t ) = 0 . (26)Together with the Sobolev embedding W , loc ( R , M ) (cid:44) → C ( R , M ) and using uniquenessof solutions of an ordinary differential equation, it follows that the cokernel of ( D S x ) isfinite-dimensional and thus ( D S x ) is Fredholm. Since M et (Γ) is finite-dimensional, we onclude that D S x is Fredholm aswell. The index formula (23) is straightforward.To prove the second part of the Proposition, we consider the map S X : X Γ × M et (Γ) × B Γ ( p + , p − ) → E , ( x , l , γ ) (cid:55)→ S x ( l , γ ) . (27)It suffices to show that S X is transverse to the zero section of E . To this end, we mustcheck that if S X ( x , l , γ ) = 0, then every element r ∈ L ( E ), so that (cid:104) r , D S X ( y , m , s ) (cid:105) = 0for all ( y , m , s ) ∈ X Γ ⊕ T l M et (Γ) ⊕ T γ B Γ ( p + , p − ), vanishes identically. By the proof ofthe first part of the Proposition, each component r e (cid:48) , e (cid:48) ∈ E (Γ (cid:48) ), is continuous. Thus itsuffices to show that r e (cid:48) ( t ) = 0 for t ∈ (0 , e (cid:48) is an internal edge, the case when e (cid:48) is an external edge being similar.From (19), we have ( D S X ( y , , e (cid:48) ( t ) = y e (cid:48) ( t, γ e (cid:48) ( t )) . (28)Suppose that r e (cid:48) ( t ) (cid:54) = 0, t ∈ (0 , < ε < min( t , − t ) and y e (cid:48) ∈ X Γ , so that (cid:104) r e (cid:48) ( t ) , y e (cid:48) ( l , t, γ e (cid:48) ( t )) (cid:105) g > | t − t | < ε (29)and y e (cid:48) ( l , t, · ) ≡ | t − t | ≥ ε. (30)Using y e (cid:48) , we define y ∈ X Γ as follows. Let e be the edge of Γ corresponding to e (cid:48) under theprojection Γ (cid:48) → Γ. If e (cid:48)(cid:48) is an edge of Γ (cid:48) obtained from e (cid:48) by a covering transformation ofΓ (cid:48) → Γ which preserves the orientation of e , then we define y τ e (cid:48)(cid:48) ( e (cid:48) ) ( l , t, · ) = y e (cid:48) ( l , t, · ). Inthe case when e (cid:48)(cid:48) is obtained from e (cid:48) by applying a covering transformation which reversesthe orientation of e , we put y τ e (cid:48) ( e (cid:48) ) ( l , t, · ) = − y e (cid:48) ( l , − t, · ). Finally, define y e (cid:48)(cid:48) ≡ e (cid:48)(cid:48) of Γ (cid:48) . Then y = ( y e (cid:48) ) e (cid:48) ∈ E (Γ (cid:48) ) is a well-defined element of X Γ . Using (12),we compute: (cid:104) r , D S ( y , , (cid:105) = 2 | E int (Γ) | ε (cid:90) t =0 (cid:104) r e (cid:48) ( t ) , y e (cid:48) ( t, γ e (cid:48) ( t )) (cid:105) g > r . This completes the proof of the second part of theProposition. The proof of the third part is analogous. Corollary 2.8.
1. For every element x ∈ X Γ ,reg , the space M Γ , x ( p + , p − ) is a mani-fold whose dimension is given by the right-hand side of the index formula (23). Thespace is empty in the case when the right-hand side of (23) is negative.2. Let X Σ denote the vector space X Σ = ⊕ Γ X Γ , (32) where the direct sum is over all the ribbon graphs Γ in the ribbon graph decomposition(8) of M Σ . There is a subset X Σ ,reg ⊂ X Σ of second category, so that X Σ ,reg ⊂ ⊕ Γ X Γ ,reg (33) and, in addition, every element y = ( x Γ ) Γ ∈ X Σ , reg satisfies the following condition:if the graph (cid:101) Γ is obtained from Γ by collapsing internal edges, then x (cid:101) Γ coincides withthe restriction of x Γ to M et ( (cid:101) Γ) ⊂ M et (Γ) . he second part of this Corollary means that the vector field data for different ribbongraphs can be chosen consistently with the attachments of the corresponding cells in theribbon graph deocmposition. Proof.
The first part of the Corollary follows from the first two parts of Proposition 2.7.To prove the second part, start by associating an arbitrary vector field datum in X Γ ,reg toeach graph Γ labelling a top-dimenional cell in (8) and use the third part of Proposition2.7 successively to extend over the remaining cells.We will consider certain partial compactifications of the spaces M Γ , x ( p + , p − ). Themain idea is as follows. We observe that there are the following three sources of non-compactness of the spaces:1. Breaking of the trajectory corresponding to an external edge of the graph intoseveral trajectories connecting critical points.2. Convergence to zero of the length l e of an internal edge e of the graph.3. Breaking of a trajectory corresponding to an internal edge of the graph. Figure 4: Threetypes of bound-ary strata of nat-ural compactifica-tions of the spacesof flow graphs.
We will consider the partial compactification M Γ , x ( p + , p − ) of M Γ , x ( p + , p − ) obtainedby adding the strata of the first and the second (but not the third) type. We now givethe formal definition.Given two critical points p and p (cid:48) of f , denote by M ( p, p (cid:48) ) = { γ : R → M, ˙ γ ( t ) = ∇ g f ( γ ( t )) , lim t →−∞ γ ( t ) = p, lim t →∞ γ ( t ) = p (cid:48) } / R (34)the space of flow trajectories emanating at p and converging to p (cid:48) . Denote by M ( p, p (cid:48) ) = M ( p, p (cid:48) ) ∪ (cid:91) m ≥ q ,...,q m M ( p, q ) × · · · × M ( q m , p (cid:48) ) (35)the corresponding space of broken trajectories. Given two n -tuples p = ( p , . . . , p n ) and p (cid:48) = ( p (cid:48) , . . . , p (cid:48) n ) of critical points, we write M ( p , p (cid:48) ) = M ( p , p (cid:48) ) × · · · × M ( p n , p (cid:48) n ) . (36) Definition 2.9.
We define M Γ , x ( p + , p − ) = (cid:91) q + , q − , (cid:101) Γ ≺ Γ M ( p + , q + ) × M (cid:101) Γ , x ( q + , q − ) × M ( q − , p − ) , (37) where the union is over all q + ∈ Crit ( f ) × n + , q − ∈ Crit ( f ) × n − and all ribbon graphs (cid:101) Γ obtained by collapsing edges of Γ , so that no cycle is collapsed. e now establish the properties of the spaces M Γ , x ( p + , p − ) which will be used inthe proof of Theorem 1.1. Proposition 2.10.
1. For every x ∈ X Γ as in the third part of Proposition 2.7, thespace M Γ , x ( p + , p − ) is a manifold with corners.2. The boundary ∂ M Γ , x ( p + , p − ) is given by the disjoint union ∂ M Γ , x ( p + , p − ) = ( (cid:97) | q + |−| p + | =1 M ( p + , q + ) × M Γ , x ( q + , p − )) (cid:97) ( (cid:97) | p − |−| q − | =1 M Γ , x ( q + , p − ) × M ( q − , p − )) (cid:97) ( (cid:97) e ∈ E int (Γ) − L (Γ) M Γ /e, x ( p + , p − )) , (38) where the last union is over all the internal edges e of Γ which are not loops andwhere Γ /e denotes the ribbon graph obtained from Γ by collapsing e .3. The projection π Γ : M Γ , x ( p + , p − ) → M Σ defined by forgetting γ is proper.Proof. In order to equip the space M Γ , x ( p + , p − ) with the structure of a manifold withcorners, we will identify it as a transverse intersection of a manifold and a manifold withcorners.Assume that a connected component Γ (cid:48) of Γ (cid:48) is fixed or, equivalently, an orientation of ev-ery internal edge of Γ is chosen. For an internal edge e (cid:48) of Γ (cid:48) , denote by Φ e (cid:48) l , x ( t, · ) : M → M the flow of the vector field on the right-hand side of (19). We define M e (cid:48) = M e (cid:48) ( l , x ) ⊂ M × M as the subspace M e (cid:48) = { ( q, Φ e (cid:48) l , x (1 , q )) ∈ M × M : q ∈ M } . (39)Thus M e (cid:48) is a manifold diffeomorphic to M .Suppose now that e (cid:48) is an external edge of Γ (cid:48) with the corresponding critical point p e (cid:48) . Denote by Φ e (cid:48) l , x ( t, · ) the flow of the vector field on the right-hand side of (18). If e (cid:48) ismarked as incoming, then we define W u ( p e (cid:48) ) = { q ∈ M : lim t →−∞ Φ e (cid:48) l , x ( t, q ) = p e (cid:48) } , (40) W u ( p e (cid:48) ) = (cid:91) p (cid:48) ∈ Crit ( f ) M ( p e (cid:48) , p (cid:48) ) × W u ( p (cid:48) ) (41)and M e (cid:48) = { ( q, q ) : q ∈ W u ( p e (cid:48) ) } ⊂ M × M. (42)Similarly, if e (cid:48) is marked as outgoing, then we consider W s ( p e (cid:48) ) = { q ∈ M : lim t →∞ Φ e (cid:48) l , x ( t, q ) = p e (cid:48) } , (43) W s ( p e (cid:48) ) = (cid:91) p (cid:48) ∈ Crit ( f ) W s ( p (cid:48) ) × M ( p (cid:48) , p e (cid:48) ) (44)and M e (cid:48) = { ( q, q ) : q ∈ W s ( p e (cid:48) ) } ⊂ M × M. (45) t is well known that W u ( p e (cid:48) ) and W s ( p e (cid:48) ) are manifolds with corners, whose boundariesare given by respectively ∂W u ( p e (cid:48) ) = (cid:97) ind f ( p e (cid:48) ) − ind f ( p )=1 M ( p e (cid:48) , p ) × W u ( p ) (46)and ∂W s ( p e (cid:48) ) = (cid:97) ind f ( p ) − ind f ( p e (cid:48) )=1 W s ( p ) × M ( p, p e (cid:48) ) . (47)We refer to [We-12] for a detailed study of trajectory spaces in Morse theory.We have thus far associated to every edge e (cid:48) of Γ (cid:48) a manifold with corners M e (cid:48) . Asa submanifold of M × M , M e (cid:48) depends on the choice of l ∈ M et (Γ) and of the vectorfield datum x , however the diffeomorphism type of M e (cid:48) is independent of these choices.Let us define N = M et (Γ) × ( M × M ) ×| E (Γ (cid:48) ) | (48)and denote by L x the subset L x = (cid:91) l ∈ Met (Γ) ( { l } × ( × e (cid:48) ∈ E (Γ (cid:48) ) M e (cid:48) ( l , x ))) ⊂ N. (49)As a product of manifolds with corners, L x is again a manifold with corners.Next we consider the submanifold L (cid:48) ⊂ N defined as follows. Recall that each edgeof Γ (cid:48) is oriented. We assign to each factor M appearing on the right-hand side of (48) avertex of Γ (cid:48) by the following rule: if e (cid:48) ∈ E (Γ (cid:48) ) is incident to the vertices v and v (cid:48) in thisorder, then to the two factors of the copy of M × M corresponding to e (cid:48) the pair v and v (cid:48) respectively are associated. Write each element of N as a ( l , q ), where q is a tuple whoseentries are points of M and denote for each entry q by v ( q ) the vertex of Γ (cid:48) assigned tothe corresponding factor of M . Then L (cid:48) ⊂ N is defined as the subset of all those tuples( l , q ), such that if for two entries q and q (cid:48) of q , v ( q ) = v ( q (cid:48) ) is the same internal vertexof Γ (cid:48) , then q = q (cid:48) . Thus L (cid:48) ⊂ N is a fat diagonal which corresponds to the incidencerelations of the internal vertices of the graph.We identify the space M Γ , x ( p + , p − ) as the intersection L x ∩ L (cid:48) : points of L x are tuplesof flow lines associated to the edges of the graph, while intersecting with L (cid:48) correspondsto imposing the incidence relations that stem from continuity at the internal vertices.It follows from Proposition 2.7 that the intersection L x ∩ L (cid:48) is transverse. This estab-lishes the first part of the Proposition. Formula (38) follows from (46) and (47). Thethird claim is a consequence of the above discussion of compactifications of the spaces M Γ , x ( p + , p − ).Proposition 2.10 implies that the pair ( M Γ , x ( p + , p − ) , π Γ ), together with a choiceof orientation of M Γ , x ( p + , p − ), defines an element of the chain complex C BM ∗ ( M Σ )introduced in Section 2.1. This Section contains the discussion of orientations of the spaces of flow graphs as wellas the proofs of Theorem 1.1 and of the gluing axiom. .1 Orientations of the Spaces of Flow Graphs The identification given in the proof of Proposition 2.10 of the space M Γ , x ( p + , p − ) asa transverse intersection of submanifolds can be used in the discussion of orientations:to orient M Γ , x ( p + , p − ), it suffices to fix orientations of the manifolds L x , L (cid:48) and N . Itfollws from the definition of these manifolds that their orientations can be determined bychoosing orientations of the internal edges of Γ as well as linear orderings of the verticesand of the edges of Γ. Moreover, it is straightforward to determine how the orientationsof L x , L (cid:48) and N , and thus the orientation of M Γ , x ( p + , p − ), change when we reordervertices and edges or reverse the orientation of an edge. The result of this discussion issummarized in the following Proposition. Proposition 3.1.
Suppose that for every critical point p of f , an orientation of theunstable submanifold W uf ( p ) is fixed.1. An orientation of M Γ , x ( p + , p − ) is uniquely defined by choosing orientations of allthe internal edges of Γ and linear orderings of all the edges and of the internalvertices of Γ .2. Reversing the orientation of an internal edge or interchanging two consecutive in-ternal vertices changes the orientation of M Γ , x ( p + , p − ) by a ( − d , where d is thedimension of M .3. Interchanging two consecutive edges e i , e j changes the orientation of M Γ , x ( p + , p − ) by ( − k i k j , where the integers k i are given by the following rule. If e i is an internaledge of the graph, then k i = d + 1 . If e i is an external edge with the correspoindingcritical point p i , then k i = (cid:40) d − ind f ( p i ) + 1 , if e i is marked as incoming ,ind f ( p i ) + 1 , if e i is marked as outgoing. (50)We can now explain the local systems det and or on M Σ which appear in Theorem1.1. A local trivialization of det on M et (Γ) /Aut (Γ) ⊂ M Σ is defined by a choice of ori-entations of the internal edges as well as of linear orderings of the vertices, of the internaledges and of those external edges of Γ, which are marked as incoming. Changing theorientation of an internal edge or interchanging two consecutive vertices or edges changesthe sign of the trivialization. The fibre of det can be identified with the determinant lineof the cohomology H ∗ (Γ , O − ) of Γ relative the vertices corresponding to the outgoingmarked points. The local system det is graded by assigning to each section the degree χ (Σ) − n − .The local system or is the orientation sheaf of M Σ . Explicitly, a local trivializationof or on M et (Γ) /Aut (Γ) ⊂ M Σ is defined by a choice of orientations of the internaledges and of linear orderings of all the vertices and edges of Γ. As before, reversing theorientation of an edge or interchanging consecutive vertices or edges reverses the sign ofthe trivializazion. The fibre of the local system or can be identified with the determinantline of H ∗ (Γ). The grading of the system or is trivial, i. e. every section has degree zero. Lemma 3.2.
Suppose that every vertex of the ribbon graph Γ has odd valency. Then thepair ( M Γ , x ( p + , p − ) , π Γ ) defines an element of C BM ∗ ( M Σ ; det ⊗ d ⊗ or ) .Proof. We first note that since every external edge of Γ is marked as either incoming oroutgoing and thus has a natural orientation, we could equivalently define the local system det by considering linear orderings of all the vertices instead of only the internal ones.Indeed, if e is an external edge with the incident vertices v and v (cid:48) , where v is an internal nd v (cid:48) an external vertex, then we insert v (cid:48) into a given ordering of the internal verticesas either the predecessor or the successor of v , according to the orientation of e .We must show that an orientation of M Γ , x ( p + , p − ) is the same as a trivialization ofthe pullback under π Γ of the local system det ⊗ d ⊗ or . Comparing the definitions of det and or with the result of Proposition 3.1, it suffices to check that a trivialization of or isgiven by a choice an orientation of the vector space spanned by the edges of the graph.This follows from the observation going back to J. Conant and K. Vogtmann ([CoVo-03],Corollary 1) that if all vertices of Γ have odd valency, then there is a natural orientationof the vector space spanned by the vertices and the half-edges. We can now complete the proof of Theorem 1.1. Recall that the top-dimensional cellsin the ribbon graph decomposition (8) are labelled by the ribbon graphs whose internalvertices have valency three. By Proposition 2.10 and Lemma 3.2, to each such graph Γand each choice of a vector field datum x ∈ X Γ ,reg is associated a geometric chain Z Γ , x = ( M Γ , x ( p + , p − ) , π Γ ) ∈ C BM ∗ ( M Σ ; det ⊗ d ⊗ or ) . (51)Moreover, we may assume that the vector field data for different graphs Γ are chosen asin the second part of Corollary 2.8.To prove that F M Σ : ( C ∗ ( f )) ⊗ n + → C BM ∗ ( M Σ ; det ⊗ d ⊗ or ) ⊗ ( C ∗ ( f )) ⊗ n − , p + (cid:55)→ (cid:88) Γ , p − Z Γ , x ( p + , p − ) ⊗ p − is a cochain map, we compute ( (cid:80) Γ , p − ∂Z Γ , x ( p + , p − ) ⊗ p − ). By the second part ofProposition 2.10, the latter expression is a sum of terms of two types: the first typecorresponds to breaking of trajectories at external edges (see the expressions in the firsttwo lines of (38)), while terms of the second type correspond to collapsing an internaledge of Γ (see the expression in the third line of (38)). The summands of the first typeyield (cid:88) Γ , p − Z Γ , x ( d p + , p − ) ⊗ p − − (cid:88) Γ , p − ( − dim M Γ , x ( p + , p − ) Z Γ , x ( p + , p − ) ⊗ d p − . (52)We must show that the sum of all the terms of the second type is zero. These are of theform ± ( M (cid:101) Γ , x ( p + , p − ) , π (cid:101) Γ ), where (cid:101) Γ is obtained by collapsing a single internal edge in aribbon graph Γ, all of whose internal vertices have valency three. The sign is determinedas follows: M (cid:101) Γ , x ( p + , p − ) is oriented as a boundary component of M Γ , x ( p + , p − ) and thetrivialization of the pullback to M (cid:101) Γ , x ( p + , p − ) of the local system det ⊗ d ⊗ or is inducedby the trivialization of the pullback to M Γ , x ( p + , p − ).We observe that in the sum of the terms of second type exactly those ribbon graphs (cid:101) Γ appear, where there is a unique internal vertex of valency four and all the remaininginternal vertices have valency three. For each such (cid:101)
Γ, there are exactly two distinct pairs(Γ , e ) and (Γ , e ), so that (cid:101) Γ is obtained from Γ and Γ by collapsing the internal edges e ∈ E (Γ ) and e ∈ E (Γ ) respectively: Γ and Γ arise from the two different ways ofexpanding the four-valent vertex of (cid:101) Γ into two trivalent vertices.To complete the proof of the first part of the Theorem, it suffices to show:
Lemma 3.3.
The two copies of ( M (cid:101) Γ , x ( p + , p − ) , π (cid:101) Γ ) corresponding to boundary compo-nents of (Γ , e ) and of (Γ , e ) enter the sum with the opposite sign.Proof. Assume first that d is even. In this case by Proposition 3.1, an orientation of M Γ , x ( p + , p − ) is the same as an orientation of the vector space W E (Γ ) spanned by theedges of Γ . The boundary orientation of M (cid:101) Γ , x ( p + , p − ) ⊂ ∂ M Γ , x ( p + , p − ) is determinedby requiring that the projection W E (Γ ) (cid:39) R e ⊕ W E ( (cid:101) Γ) → W E ( (cid:101) Γ) (53)to the second factor be orientation-preserving. Here the symbol R e denotes the directsummand of R corresponding to the edge e . The orientation of M (cid:101) Γ , x ( p + , p − ) as aboundary component of M Γ , x ( p + , p − ) is defined analogously.It follows from the definition that a trivialization of the pullback of the local system or to M Γ , x ( p + , p − ) is given by an orientation of the vector space W E (Γ ) ⊕ W V (Γ ) ⊕ W H (Γ ) ,where W V (Γ ) and W H (Γ ) denote the vector spaces generated by the vertices and by thehalf-edges of Γ respectively. The projection W E (Γ ) → W E ( (cid:101) Γ) was described above, whilethe projection W V (Γ ) ⊕ W H (Γ ) → W V ( (cid:101) Γ) ⊕ W H ( (cid:101) Γ) is given as follows. Denote by h and h (cid:48) the half-edges of e , by v and v (cid:48) the corresponding vertices of Γ and by v ∈ V ( (cid:101) Γ)the vertex to which e is collapsed. We identify W V (Γ ) −{ v (cid:48) ,v (cid:48) } (cid:39) W V ( (cid:101) Γ) −{ v } (54)as well as W H (Γ ) −{ h ,h (cid:48) } (cid:39) W H ( (cid:101) Γ) (55)and consider the map W V (Γ ) ⊕ W H (Γ ) (cid:39) ( R h ⊕ R h (cid:48) ⊕ R v ) ⊕ ( R v (cid:48) ⊕ W V (Γ ) −{ v ,v (cid:48) } ⊕ W H (Γ ) −{ h ,h (cid:48) } ) → R v ⊕ W V ( (cid:101) Γ) −{ v } ⊕ W H ( (cid:101) Γ) (cid:39) W V ( (cid:101) Γ) ⊕ W H ( (cid:101) Γ) , (56)where the arrow denotes projection to the second factor. A trivialization of or over M Γ , x ( p + , p − ) induces a trivialization over M (cid:101) Γ , x ( p + , p − ) by requiring that the projec-tion W E (Γ ) ⊕ ( W V (Γ ) ⊕ W H (Γ ) ) → W E ( (cid:101) Γ) ⊕ ( W V ( (cid:101) Γ) ⊕ W H ( (cid:101) Γ) ) (57)whose components are given by (53) and (56) be orientation-preserving. In the same waya trivialization over M (cid:101) Γ , x ( p + , p − ) is induced by a trivialization over M Γ , x ( p + , p − ). ecall from the proof of Lemma 3.3 that since Γ and Γ have vertices of odd valency, theribbon structures of Γ and of Γ define orientations of the vector spaces W V (Γ ) ⊕ W H (Γ ) and W V (Γ ) ⊕ W H (Γ ) respectively. It is immidiate to check using the explicit descriptiongiven by (56) that the two orientations on W V ( (cid:101) Γ) ⊕ W H ( (cid:101) Γ) induced by the projections W V (Γ ) ⊕ W H (Γ ) → W V ( (cid:101) Γ) ⊕ W H ( (cid:101) Γ) and W V (Γ ) ⊕ W H (Γ ) → W V ( (cid:101) Γ) ⊕ W H ( (cid:101) Γ) are opposite. Thus if the orientations of M (cid:101) Γ , x ( p + , p − ) as boundary of M Γ , x ( p + , p − )and as boundary of M Γ , x ( p + , p − ) coincide, i. e. the projection (53) and the cor-responding projection for Γ are both orientation-preserving, then the orientations of W E ( (cid:101) Γ) ⊕ W V ( (cid:101) Γ) ⊕ W H ( (cid:101) Γ) induced by the projection (57) and by the corresponding pro-jection for Γ are opposite. In this case the two trivializations of or over M (cid:101) Γ , x ( p + , p − )differ by a sign. This completes the proof of the Lemma in the case when d is even. Thecase of odd d follows by the same argument, but with W E (Γ) replaced everywhere by W det ⊕ W E (Γ) , where W det is the vector space whose orientation corresponds to a trivi-alization of det (explicitly, W det = W E (Γ) − E − (Γ) ⊕ W V (Γ) ⊕ W H (Γ) , where E − (Γ) denotethe outgoing external edges of Γ).We now turn to the proof of the second part of the Theorem. By the first part,Proposition 2.1 and the isomorphism (10), F M Σ induces a linear map HF M Σ : ( H ∗ ( M )) ⊗ n + → H ∗ ( M Σ ; det d ) ⊗ ( H ∗ ( M )) ⊗ n − . (58)Comparing the definition of the grading of the local system det with the index formulaof Proposition 2.7, one finds that the degree of HF Σ is zero (note that if every internalvertex of Γ has valency three, then the number | E (Γ) | of edges of Γ coincides with thedimension of M Σ ).The proof of the independence of the operations HF M Σ of the choices relies on a flowgraph version of the continuation argument, which is classically used to prove the in-variance of Morse homology. Given two triples ( f, g, x ) and ( f (cid:48) , g (cid:48) , x (cid:48) ), we fix a smoothone-paramater family ( f t , g t , x t ) t ∈ R so that( f t , g t , x t ) = (cid:40) ( f, g, x ) if t ≤ − f (cid:48) , g (cid:48) , x (cid:48) ) if t ≥ . (59)Recall that the classical continuation principle consists of the following: one observes thatfor suitable choice of the one-parameter family ( f t , g t ) t ∈ R , each of the spaces N ( p, p (cid:48) ) = { γ : R → M, ˙ γ ( t ) = ∇ g t f t ( γ ( t )) , lim t →−∞ γ ( t ) = p, lim t →∞ γ ( t ) = p (cid:48) } , (60)where p ∈ Crit ( f ), p (cid:48) ∈ Crit ( f (cid:48) ), is a compact oriented manifold. One then shows thatthe count of the elements of the zero-dimensional spaces yields a quasi-isomorphismΨ : C ∗ ( f, g ) → C ∗ ( f (cid:48) , g (cid:48) ) , Ψ : p (cid:55)→ (cid:88) ind f ( p (cid:48) )= ind f ( p ) |N ( p, p (cid:48) ) | p (cid:48) . (61) ow denote byΦ M Σ : ( C ∗ ( f (cid:48) , g (cid:48) )) ⊗ n + → C BM ∗ ( M Σ ; det d ⊗ or ) ⊗ ( C ∗ ( f (cid:48) , g (cid:48) )) ⊗ n − the map associated by the construction of the first part of the Theorem to the triple( f (cid:48) , g (cid:48) , x (cid:48) ). We will construct a chain homotopy Θ between Φ M Σ ◦ Ψ ⊗ n + and ( Id ⊗ Ψ n − ) ◦ F M Σ . This chain homotopy will be obtained by studying spaces N Γ ( p + , p (cid:48)− ) which wenow introduce.Recall from Definition 2.5 that M Γ , x ( p + , p − ) is the zero locus of S − F x , where S and F x are sections of a Banach bundle E over M et (Γ) × B Γ ( p + , p − ) (see Definition 2.3). Given p + ∈ Crit ( f ) × n + and p (cid:48)− ∈ Crit ( f (cid:48) ) × n − , we define N Γ ( p + , p (cid:48)− ) ⊂ M et (Γ) × B Γ ( p + , p (cid:48)− )as the union N Γ ( p + , p (cid:48)− ) = ∪ T ∈ R N Γ ,T ( p + , p (cid:48)− ) , where N Γ ,T ( p + , p (cid:48)− ) is the zero locus of the section S − F T of E , with S is defined as in(14) and F T given as follows. If e (cid:48) is an external edge of Γ (cid:48) , then F e (cid:48) ,T ( l , γ )( t ) = ∇ g t + T f t + T ( γ e (cid:48) ( t )) + σ ( t ) x e (cid:48) ,t + T ( l , | t | , γ e (cid:48) ( t )) . (62)If e (cid:48) is an internal edge of Γ (cid:48) which is mapped to e ∈ E (Γ) under the projection Γ (cid:48) → Γ,then F e (cid:48) ,T ( l , γ )( t ) = l e x e (cid:48) ,T ( l , t, γ e (cid:48) ( t )) (63)(compare with (18) and (19) respectively). We denote by N Γ ( p + , p (cid:48)− ) the partial com-pactification of N Γ ( p + , p (cid:48)− ) given by N Γ ( p + , p (cid:48)− ) = (cid:91) q + , q (cid:48)− , (cid:101) Γ ≺ Γ M ( p + , q + ) × N (cid:101) Γ , x ( q + , q (cid:48)− ) × M (cid:48) ( q (cid:48)− , p (cid:48)− ) (cid:91) ( (cid:91) | q (cid:48) + | = | p + | N ( p + , q (cid:48) + ) × M Γ , x (cid:48) ( q (cid:48) + , p (cid:48)− )) (cid:91) ( (cid:91) | q − | = | q (cid:48)− | M Γ , x ( p + , q − )) × N ( q − , p (cid:48)− )) , (64)where N ( p , p (cid:48) ) denotes the product N ( p , p (cid:48) ) = N ( p , p (cid:48) ) × · · · × N ( p n , p (cid:48) n ) (65)(note that if | p | = | p (cid:48) | , then the space N ( p , p (cid:48) ) is empty unless ind f ( p j ) = ind f (cid:48) ( p (cid:48) j ) for j = 1 , . . . , n .)The meaning of the terms of the right-hand side of (64) is as follows. The expressionin the first line corresponds to allowing internal edges of length zero as well as break-ing of trajectories along the external edges of Γ for fixed T ∈ R . Thus if we denote by N Γ ,T ( p + , p (cid:48)− ) the partial compactification of N Γ ,T ( p + , p (cid:48)− ) defined as in 2.9, then theunion in the first line of (64) can be identified with ∪ T ∈ R N Γ ,T ( p + , p (cid:48)− ). The terms in thesecond and third line correspond to partial compactifications at T → ∞ and T → −∞ respectively.By a slight abuse of notation, we will again denote by π Γ : N Γ ( p + , p (cid:48)− ) → M Σ thenatural projection. emma 3.4.
1. There exists a one-parameter family ( x t ) t ∈ R , so that every pair Y Γ ( p + , p (cid:48)− ) = ( N Γ ( p + , p (cid:48)− ) , π Γ ) (66) defines an element of C BM ∗ ( M Σ , det ⊗ d ⊗ or ) .2. Define Θ : ( C ∗ ( f )) ⊗ n + → C BM ∗ ( M Σ ; det ⊗ d ⊗ or ) ⊗ ( C ∗ ( f (cid:48) , g (cid:48) )) ⊗ n − , p + (cid:55)→ (cid:88) Γ , p (cid:48)− Y Γ ( p + , p (cid:48)− ) ⊗ p (cid:48)− . (67) Then d ◦ Θ + Θ ◦ d = ( Id ⊗ Ψ ⊗ n − ) ◦ F M Σ − Φ M Σ ◦ (Ψ ⊗ n + ) (68)This Lemma completes the proof of the second part of Theorem 1.1: it follows from(68) that the maps HF M Σ and H Φ M Σ defined by ( f, g, x ) and ( f (cid:48) , g (cid:48) , x (cid:48) ) respectively coin-cide up to the isomorphism H ∗ ( M ) → H ∗ ( M ) induced by Ψ. Proof.
The first part of the Lemma follows by analogous arguments as in the proofs ofPropositions 2.7 and 2.10. From (64), the boundary of N Γ ( p + , p (cid:48)− ) may be identified asthe disjoint union ∂ N Γ ( p + , p (cid:48)− ) = (cid:97) | q + |−| p + | =1 M ( p + , q + ) × N Γ , x ( q + , p (cid:48)− ) (cid:97) ( (cid:97) | p (cid:48)− |−| q (cid:48)− | =1 N Γ , x ( q + , p (cid:48)− ) × M (cid:48) ( q (cid:48)− , p (cid:48)− )) (cid:97) ( (cid:97) | q (cid:48) + | = | p + | N ( p + , q (cid:48) + ) × M Γ , x (cid:48) ( q (cid:48) + , p (cid:48)− )) (cid:97) ( (cid:97) | q − | = | q (cid:48)− | M Γ , x ( p + , q − ) × N ( q − , p (cid:48)− )) (cid:97) ( (cid:97) e ∈ E int (Γ) − L (Γ) N Γ /e ( p + , p (cid:48)− )) . (69)The sum over all Γ of the geometric chains corresponsing to the terms on the left-handside and in the first two lines of the right-hand side of (69) yields the left-hand side of(68), while the expressions in the third and fourth lines correspond to the right-hand sideof (68). Finally, using Lemma 3.3, the sum of the chains corresponding to the terms inthe last line of (69) vanishes. The goal of this Section is to show that the operations constructed in Theorem 1.1 arecompatible with the gluing of surfaces and thus form what is called a homological con-formal field theory.Let Σ and Σ be two surfaces as in Theorem 1.1 and assume that the number n − of the outgoing marked points of Σ coincides with the number n of the incomingmarked points of Σ . Denote this common number by n and denote by Σ the compactorieted surface obtained by attaching Σ to Σ along closed disjoint intervals around the utgoing marked points of Σ respectively around the incoming marked points of Σ .Using the homeomorphism (8) of the ribbon graph decomposition, there is a mapΞ : M Σ × M Σ → M Σ (70)defined by first attaching the outgoing edges of a metric ribbon graph corresponding toΣ to the incoming edges of a metric ribbon graph corresponding to Σ and then erasingthe bivalent vertices from the resulting graph. Denote by π : M Σ × M Σ → M Σ andby π : M Σ × M Σ → M Σ the projection to the first respectively to the second factor.The local system det is compatible with gluing, i. e. Ξ ∗ det (cid:39) π ∗ det ⊕ π ∗ det . Thus Ξinduces a mapΞ ∗ : H ∗ ( M Σ ; det ⊗ d ) → H ∗ ( M Σ ; det ⊗ d ) ⊗ H ∗ ( M Σ ; det ⊗ d ) . (71)The next Proposition establishes the compatibility of the operations constructed in The-orem 1.1 with the gluing maps (71). Proposition 3.5.
The following diagram commutes: ( H ∗ ( M )) ⊗ n + HF M Σ (cid:47) (cid:47) HF M Σ1 (cid:15) (cid:15) H ∗ ( M Σ ; det ⊗ d ) ⊗ ( H ∗ ( M )) ⊗ n − Ξ ∗ ⊗ Id (cid:15) (cid:15) H ∗ ( M Σ ; det ⊗ d ) ⊗ ( H ∗ ( M )) ⊗ n Id ⊗ HF M Σ2 (cid:47) (cid:47) H ∗ ( M Σ ; det ⊗ d ) ⊗ H ∗ ( M Σ ; det ⊗ d ) ⊗ ( H ∗ ( M )) ⊗ n − . Before giving the proof of this Proposition, we gather some preliminary arguments.First, let us express the gluing homomorphism (71) using geometric homology (we willfor simplicity leave out the local coefficient systems in the notation). Via the Poincar´eduality isomorphisms (10) on M Σ , M Σ and M Σ , one obtains from Ξ ∗ the correspondingtransfer homomorphismΞ ! BM ∗ : H BM ∗ ( M Σ ) → H BM ∗ + n ( M Σ × M Σ ) . (72)The remarks made in section 2.1 allow to identify Ξ ! BM ∗ as the homomorphism inducedby an explicit chain mapΞ ! CBM ∗ : C BM ∗ ( M Σ ) → C BM ∗ + n ( M Σ × M Σ ) . (73)To this end, we observe that the imageΞ( M Σ × M Σ ) ⊂ M Σ is an open subset: it consists of the equivalence classes of all the metric ribbon graphsΓ which can be obtained by gluing together graphs Γ and Γ from the ribbon graphdecompositions of M Σ and M Σ respectively. There is a homeomorphism M Σ × M Σ (cid:39) Ξ( M Σ × M Σ ) × R n + (74)so that Ξ can be identified as the composition of the projection π to the first factorof Ξ( M Σ × M Σ ) × R n + with the inclusion i : Ξ( M Σ × M Σ ) (cid:44) → M Σ . This yieldsthe following map (73): given a genereator ( P, f P ) of C BM ∗ ( M Σ ), we first intersect withΞ( M Σ × M Σ ) to obtain ( Q, f Q ) = i ! CBM ∗ ( P, f P ) ∈ C BM ∗ (Ξ( M Σ × M Σ )). ThenΞ ! CBM ∗ ( P, f P ) = ( Q × R n , f Q × R n ) = π ! CBM ∗ ( Q, f Q ) is obtained from ( Q, f Q ) by puttinga bivalent vertex on each of the edges of Γ which originate from attaching an outgoingedge of Γ to an incoming edge of Γ . roof of Proposition 3.5. To prove the Proposition, it suffices to show that the followingdiagram becomes commutative in homology:( C ∗ ( f )) ⊗ n + F M Σ (cid:47) (cid:47) F M Σ1 (cid:15) (cid:15) C BM ∗ ( M Σ ) ⊗ ( C ∗ ( f )) ⊗ n Ξ ! CBM ∗ ⊗ Id (cid:15) (cid:15) C BM ∗ ( M Σ ) ⊗ ( C ∗ ( f )) ⊗ n Id ⊗ F M Σ2 (cid:47) (cid:47) C BM ∗ ( M Σ × M Σ ) ⊗ ( C ∗ ( f )) ⊗ n − . We will find a chain homotopy between the compositions (Ξ ! CBM ∗ ⊗ Id ) ◦ F M Σ and( Id ⊗ F M Σ ) ◦ F M Σ .Denote Φ = (Ξ ! CBM ∗ ⊗ Id ) ◦ F M Σ and Φ n = ( Id ⊗ F M Σ ) ◦ F M Σ . Let us construct cochainmaps Φ , . . . , Φ n − : ( C ∗ ( f )) ⊗ n + → C BM ∗ ( M Σ × M Σ ) ⊗ ( C ∗ ( f )) ⊗ n − (75)and show that for k = 0 , . . . , n −
1, there is a chain homotopy between Φ k and Φ k +1 .Denote by Σ( k ) the surface obtained by gluing Σ to Σ along disjoint closed intervalsaround the first n − k outgoing marked points on Σ resp. the first n − k incoming markedpoints on Σ . Thus Σ(0) = Σ, while Σ( n ) is the disjoint union of Σ and Σ .We define F k : ( C ∗ ( f )) ⊗ n + → C BM ∗ ( M Σ( k ) ) ⊗ ( C ∗ ( f )) ⊗ n − p + (cid:55)→ (cid:88) Γ( k ) , q Z Γ( k ) , x ( k ) (( p + , q ) , ( q , p − )) ⊗ p − , (76)where the sum is over all the graphs Γ( k ) in the ribbon graph decomposition (8) of Σ( k )and over all tuples q ∈ Crit ( f ) × k . Attaching the last k outgoing edges of Γ( k ) to the last k incoming edges yields a ribbon graph Γ whose associated surface is Σ. We will assumethat the vector field data x and x ( k ) for Σ and Σ( k ) are chosen so that for each k andall Γ k , x and x ( k ) coincide on the complement of these edges.The homomorphism Φ k is given asΦ k = (Ξ k, ! CBM ∗ ⊗ Id ) ◦ F k , (77)where Ξ k : M Σ × M Σ → M Σ( k ) (78)is the gluing map from (70). Lemma 3.6.
For each k = 0 , . . . , n − , there is a chain homotopy ∆ k between Φ k and Φ k +1 .Proof. To simplify terminology, we will refer to a pair consisting of an outgoing edge ofΓ and the corresponding incoming edge of Γ as an attachment pair .The above formal definition of the homomorphism Φ k means the following. The map( Id ⊗ F M Σ ) ◦ F Σ is given by a count of chains corresponding to pairs of graphs in theribbon graphs decompositions of M Σ and M Σ , i. e.( Id ⊗ F M Σ ) ◦ F M Σ : p + (cid:55)→ (cid:88) Γ , Γ , q , p − Z Γ , x ( p + , q ) ⊗ Z Γ , x (cid:48) ( q , p − ) ⊗ p − , (79) here the sum is over ribbon graphs Γ and Γ in the ribbon graph decompositions of Σ and of Σ respectively and over all tuples of critical points ( q , p − ) ∈ ( Crit ( f )) × ( n + n − ) .The map Φ k is obtained by counts of geometric chains as on the right-hand of (79), butwhere instead of a broken trajectory of the gradient flow, to the first n − k attachmentpairs, a finite piece of a flow trajectory is associated, namely to these attachment pairssolutions of (19) are associated, where the parameter l e is given as the sum of the lengths ofthe two edges of the pair. We denote the spaces of such flow graphs, partially compactifiedas in Definitioin 2.9 by allowing breaking of trajectories along external edges as well ascollapsing of internal edges, by M Γ , Γ ,k ( p − , q , p + ) and write π Γ , Γ : M Γ , Γ ,k ( p − , q , p + ) → M Σ × M Σ (80)for the projection which maps each graph flow to the underlying metric structures on Γ and on Γ . Using this notation, Φ k is given byΦ k ( p + ) = (cid:88) Γ , Γ , q , p − Z Γ , Γ ,k ( p + , q , p − ) ⊗ p − , (81)where Γ and Γ are as in (79), q ∈ Crit ( f ) × k and Z Γ , Γ ,k ( p + , q , p − ) = ( M Γ , Γ ,k ( p − , q , p + ) , π Γ , Γ ) . (82)The construction of a chain homotopy between Φ k and Φ k +1 relies on the study of spaces L Γ , Γ ,k ( p − , q , p + ) which we now introduce.Elements of L Γ , Γ ,k ( p − , q , p + ) are pairs consisting of a positive number T togetherwith a flow graph of the same form as in the case of M Γ , Γ ,k ( p − , q , p + ), however theequation (19) corresponding to the ( n − k )-th attachment pair is changed to F e (cid:48) ( l , γ )( t ) = T ∇ g ( f ( γ e (cid:48) ( t ))) + y e (cid:48) ( T, l , t, γ e (cid:48) ( t )) , (83)subject to the following:1. T is greater or equal to the sum l e + l e of the lengths of the two edges of theattachment pair.2. For l e + l e ≤ T ≤ l e + l e + 1, y e (cid:48) ( T, l , t, · ) = x e (cid:48) ( l , t, · ) (84)for all t ∈ [0 , T > l e + l e + 2, the vector field y e (cid:48) ( T, l , t, · ) satisfies y e (cid:48) ( T, l , t, · ) = (cid:40) σ ( T t )( x ( k )) e (cid:48) ( l , T t, · ) for 0 ≤ t ≤ ,σ ( T (1 − t ))( x ( k )) e (cid:48) ( l , T t, · ) for ≤ t ≤ , (85)where e (cid:48) and e (cid:48) are the elements of the ( n − k )th attachment pair and e (cid:48) the edge obtainedby gluing e (cid:48) and e (cid:48) and erasing the resulting bivalent vertex. Here l and l are the met-ric structures on Γ and Γ respectively (we also note that the orientations of e (cid:48) and e (cid:48) define an orientation of e (cid:48) ). This space is again partially compactified as in Definition 2.9by allowing broken trajectories at the edges corresponding to the boundary marked pointsof Σ( k ) and collapsing of the remaining edges, but in addition we allow breaking along e (cid:48) (i. e. we take the union with the spaces M Γ , Γ ,k +1 ( p − , q (cid:48) , p + ) for q (cid:48) ∈ ( Crit ( f )) × ( k +1) ). enoting again by π Γ , Γ : L Γ , Γ ,k ( p − , q , p + ) → M Σ × M Σ the natural projection, itfollows as in Proposition 2.7 that X Γ , Γ ,k ( p − , q , p + ) = ( L Γ , Γ (cid:48) ,k ( p − , q , p + ) , π Γ , Γ ) is awell-defined element of C BM ∗ ( M Σ × M Σ ). We denote∆ k : ( C ∗ ( f )) ⊗ n + → C BM ∗ ( M Σ × M Σ ) ⊗ ( C ∗ ( f )) ⊗ n − , p + (cid:55)→ (cid:88) Γ , Γ , q , p − X Γ , Γ ,k ( p + , q , p − ) ⊗ p − , (86)where the sum is as in (81).The boundary of ( L Γ , Γ ,k ( p − , q , p + ) , π Γ , Γ ) is a disjoint union of components of thefollowing form. Firstly, corresponding to the case T = l e + l e , we have the componentsof M Γ , Γ ,k ( p − , q , p + ). Secondly, corresponding to the case T → ∞ , we have bound-ary components of the form M Γ , Γ ,k +1 ( p − , q (cid:48) , p + ), where q (cid:48) ∈ ( Crit ( f )) k +1 . Thirdly,there are the boundary components corresponding to the breaking of trajectories alongthe incoming edges of Γ and the outgoing edges of Γ . Finally, we have the boundarycomponents corresponding to collapsing an internal edge of Γ or of Γ .Summing up all the boundary components of the first two types yieldsΦ k +1 − Φ k , while the sum of the boundary components of the third type and all the expressions( ∂X Γ , Γ ,k ( p + , q , p − )) ⊗ p − yields d ◦ ∆ k + ∆ k ◦ d. Finally, using Lemma 3.3, the sum of all the components of the last type is zero.This completes the Proof of Proposition 3.5.
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