The Poincaré-Lefschetz pairing viewed on Morse complexes
aa r X i v : . [ m a t h . G T ] J a n The Poincaré-Lefschetz pairing viewed on Morse complexesFrançois Laudenbach
Abstract.
Given a compact manifold with a non-empty boundary and equipped with a genericMorse function (that is, no critical point on the boundary and the restriction to the boundaryis a Morse function), we already knew how to construct two Morse complexes, one yieldingthe absolute homology and the other the relative homology. In this note, we construct a shortexact sequence from both of them and the Morse complex of the boundary. Moreover, we definea pairing of the relative Morse complex with the absolute Morse complex which induces theintersection product in homology, in the form due to S. Lefschetz. This the very first step inan ambitious approach towards A ∞ -structures buildt from similar data. Introduction
We are given an n -dimensional compact manifold M with a non-empty boundary ∂M and aMorse function f : M → R which is generic with respect to the boundary, meaning that f hasno critical point on the boundary and that the restriction f ∂ of f to ∂M is a Morse function.It is well-known that the set of critical points of f ∂ is divided into two types + and − :(1.1) critf ∂ = crit + f ∂ ⊔ crit − f ∂ . A point x belongs to crit + f ∂ (resp. crit − f ∂ ) if it is a critical point of f ∂ and the differential df ( x ) is positive (resp. negative) on a tangent vector at x pointing outwards. We have introduced in [3] the notion of quasi-gradients positively (resp. negatively ) adaptedto f . Such vector fields, noted respectively noted X + and X − , satisfy:– X + vanishes only in critf ∪ crit + f ∂ and h df, X + i > elsewhere;– X − vanishes only in critf ∪ crit − f ∂ and h df, X − i < elsewhere.The zeroes of both of them are assumed hyperbolic, implying the existence of local stableand unstable manifolds. The quasi-gradient X + (resp. X − ) is required to be tangent to theboundary near crit + f ∂ (resp. crit − f ∂ ). Globally, both X + and X − are nowhere pointingoutwards along ∂M . As a consequence, their flows are positively complete, and hence, globalunstable manifolds exist. By taking inverse images of the local stable manifolds by the positivesemi-flow, global stable manifolds are also defined (see Section 3).These invariant submanifolds are denoted by W s ( x, X ± ) and W u ( x, X ± ) respectively when x is a zero of the considered quasi-gradient. If x ∈ critf , the dimension of W s ( x, X + ) (resp. Mathematics Subject Classification.
Key words and phrases.
Morse theory, pseudo-gradient, manifolds with boundary, Poincaré-Lefschetz duality. Here, we choose to introduce notations which are more suggestive than in [3]. In [3], these vector fields are named pseudo-gradients though they vanish at points of ∂M where df does notvanish. So, we prefer to name them quasi-gradients . W u ( x, X − ) ) is equal to the Morse index of f at x . If x ∈ crit − f ∂ , the dimension of W u ( x, X − ) is equal to the Morse index of f ∂ at x ; but, if x ∈ crit + f ∂ , we have(1.2) dim W s ( x, X + ) = Ind x f ∂ + 1 . It makes sense to assume X ± Morse-Smale (mutual transversality of stable and unstablemanifolds); this property is open and dense. An orientation is chosen on each stable (resp.unstable) manifold arbitrarily when dealing with X + (resp. X − ). This makes the unstable(resp. stable) manifolds co-oriented and allows us to put a sign on the orbits in W s ( x, X + ) ∩ W u ( y, X + ) when the sum of the codimensions is equal to n − ; and similarly for X − .Thus, two Morse complexes C ∗ ( f, X + ) and C ∗ ( f, X − ) are built whose homologies are respec-tively isomorphic to H ∗ ( M, ∂M ; Z ) and to H ∗ ( M ; Z ) . By abuse of notation, we first neglectto mention the choice of orientations; this will be corrected in 3.4 for further need. For brevity,they are also noted C + ∗ and C −∗ .To be more precise, C + k is freely generated by crit k f ∪ crit + k − f ∂ (note the shift in the gradingdue to (1.2)) while C − k is freely generated by crit k f ∪ crit − k f ∂ . The differential ∂ + := ∂ X + evaluated on a generator x ∈ C + k is given by the algebraic counting of orbits of X + ending at x and starting from generators of C + k − . And similarly for the complex C −∗ . The present noteis aimed at proving two results which are stated below. Theorem 1.1.
Let X ∂ be a Morse-Smale descending pseudo-gradient of f ∂ on the boundary ∂M and let C ∗ ( f ∂ , X ∂ ) be the associated Morse complex. Then for suitable adapted quasi-gradients X − and X + , there exist a quasi-isomorphic extension b C ∗ ( f, X − ) of the complex C ∗ ( f, X − ) anda short exact sequence of complexes (1.3) −→ C ∗ ( f ∂ , X ∂ ) −→ b C ∗ ( f, X − ) −→ C ∗ ( f, X + ) −→ . The second result is stated right below. I should add that Theorem 1.2 corrects somethingwhich was poorly said at the end of [3].
Theorem 1.2.
Here, M is assumed oriented . For a generic choice of the adapted quasi-gradients X + and X − , there is a pairing at the chain level C k ( f, X + ) ⊗ C n − k ( f, X − ) → Z which induces the intersection pairing in homology ι : H ∗ ( M, ∂M ; Z ) ⊗ H n −∗ ( M ; Z ) → Z Intitially, this note was thought of as the beginning of an article on multiplicative structures,namely A ∞ -algebra structures, on Morse complexes [1]. It appeared that the pairing C + ∗ ⊗ C − n −∗ → Z was not of the same type in nature as the multiplications of these A ∞ -structures.Therefore, I decided to separate this piece from [1].2. A short exact sequence
We first describe the suitable adapted quasi-gradients X + and X − in Theorem 1.1. Let X ∂ be a vector field on ∂M which is a Morse-Smale descending pseudo-gradient of f ∂ and givesrise to the usual Morse complex of the boundary C ∗ ( f ∂ , X ∂ ) ; its differential is denoted by ∂ ∂M .By partition of unity, one easily constructs a quasi-gradient X of f which extends X ∂ . This For defining the differential of these complexes, only the local stable manifolds are needed. X is tangent to the boundary, and hence it is not an adapted quasi-gradient . But it satisfies X · f < everywhere except at the critical points of f and f ∂ , where it vanishes with some non-degeneracy condition. The flow of X is complete, positively and negatively as well. Therefore,one can make X Morse-Smale.When x ∈ crit − k f ∂ , the unstable W u ( x, X ) coincides with W u ( x, X ∂ ) ∼ = R k and is containedin the boundary. The stable manifold W s ( x, X ) is diffeomorphic to R n − k ≥ and is bounded by W s ( x, X ∂ ) . In the same way, when y ∈ crit + k f ∂ , the unstable manifold W u ( y, X ) coincides withthe unstable manifold W u ( y, X ∂ ) ∼ = R n − − k and is contained in the boundary. Moreover, theunstable manifold W s ( y, X ) is diffeomorphic to R k +1 ≥ and is bounded by W s ( y, X ∂ ) . Remark 2.1.
Since X is tangent to the boundary there are no connecting orbits of X descendingfrom x ∈ crit − f ∂ to y ∈ critf . Similarly, there are no connecting orbits of X descending from x ∈ critf to y ∈ crit + f ∂ . We now change X to X − = X + Y , which will be negatively adapted to f , just by adding asmall vector field Y which satisfies the following condition:(2.1) Y vanishes on a closed neighborhood U of crit − f ∂ in M ;2) Y points inwards along ∂M r U and satisfies Y · f ≤ Y vanishes away from a neighborhood of ∂M . Similarly, − X can be perturbed to X + , which will be positively adapted to f ; just take X + = − X + Z where Z is a small vector field vanishing on a neighborhood V of crit + f ∂ in M ,pointing inwards along ∂M r V and satisfying Z · f ≥ everywhere. The perturbations Y and Z are small enough so that Remark 2.1 still applies. So, X − and X + will be the desiredquasi-gradients of Theorem 1.1. Proposition 2.2.
Assume critf + ∂ is empty. Then the Morse complex C ∗ ( f ∂ , X ∂ ) embeds as asubcomplex of C ∗ ( f, X − ) . Moreover, one has the following short exact sequence: −→ C ∗ ( f ∂ , X ∂ ) i −→ C ∗ ( f, X − ) −→ C ∗ ( f, X + ) −→ . Proof.
The embedding i is induced by the inclusion critf ∂ = crit − f ∂ ֒ → (cid:0) critf ∪ crit − f ∂ (cid:1) . We have to prove that i is a chain morphism. This will follow from equalities (1) and (2) below.Let x ∈ crit k f ∂ = crit − k f ∂ . By Remark 2.1 applied to X − , for every y ∈ crit k − f we have (1) h ∂ − x, y i = 0 . If y ∈ crit − k − f ∂ , the intersection W u ( x, X − ) ∩ W sloc ( y, X − ) , which is transverse in M , can bepushed by an f -preserving isotopy to W u ( x, X ∂ ) ∩ W sloc ( y, X ∂ ) , which is a transverse intersectionin ∂M – note that W sloc ( y, X ∂ ) is the boundary of W sloc ( y, X − ) . Then, the signed number ofconnecting orbits is the same for both quasi-gradients and we have (2) h ∂ − x, y i = h ∂ ∂M x, y i . For the exactness of the sequence, observe that the complex C ∗ ( f, X + ) is generated by thecritical points of f . Both vector fields X + and X − are approximations of the Morse-Smale vector field X (up to sign). Therefore, for every x ∈ crit k f and y ∈ crit k − f , the signednumber of connecting orbits is the same when counted with X − or X + : h ∂ + x, y i = h ∂ − x, y i . The quotient kills crit − f ∂ , which generates the image of C ∗ ( f ∂ , X ∂ ) , and also the connectingorbits from critf to crit − f ∂ . The exactness follows. (cid:3) Proof of Theorem 1.1
It was shown in [3] (Lemma 2.4), that there is a C -small deformation, supported in aneighborhood U of crit − f ∂ , of the generic Morse function f to a new generic Morse function f ′ with the following property: each x ∈ crit − k f ∂ becomes a critical point of positive type andindex k . The degree of x as generator of C ∗ ( f ′ ) is k + 1 . This is obtained at the cost of a newcritical point x ′ ∈ int M for f ′ , of index k and close to x . The two critical points x and x ′ of f ′ are indeed linked by a unique gradient line; since x belongs to the boundary, this pair is notcancellable but its fusion cancels x ′ only and changes the type of x from + to − .Arguing this way with the function − f leads to the following. There exists a C -smalldeformation of f , supported in a neighborhood V of crit + f ∂ , to some generic function ˆ f havingthe following property: ˆ f ∂ = f ∂ and each x ∈ crit + k f ∂ becomes a critical point of negative typeand index k , that is, x ∈ crit − k ˆ f ∂ . This is made at the cost of a critical point ˆ x ∈ int M for ˆ f of index k + 1 and close to x and satisfying ˆ f (ˆ x ) > ˆ f ( x ) . The extension which is mentioned in Theorem 1.1 consists of adding to C −∗ a pair of newgenerators { x, ˆ x } for each x ∈ crit + f ∂ . More precisely, b C ∗ ( f, X − ) := C ∗ ( ˆ f , b X − ) for some quasi-gradient b X − negatively adapted to ˆ f . According to [3], the new complex isquasi-isomophic to the old one C ∗ ( f, X − ) . Since the restriction f ∂ = ˆ f | ∂M has no critical pointof positive type, Proposition 2.2 applies and there is an exact sequence −→ C ∗ ( f ∂ , X ∂ ) −→ C ∗ ( ˆ f , b X − ) −→ C ∗ ( ˆ f , b X + ) −→ , where b X + denotes a suitable vector field positively adapted to ˆ f . In order to identify thequotient in this exact sequence, it is necessary to specify this vector field b X + .In its support V , the modification from f to ˆ f is modelled similarly to the birth of a pair ofcritical points in usual Morse Theory. The model produces also a descending quasi-gradient b X of ˆ f from the quasi-gradient X of f , which coincides with X out of V and on ∂M . Then, − b X (which is tangent to the boundary) is changed to b X + by adding a vector field Z which is smallwith respect to b X and satisfies the condition (2.1) up to sign. Claim.
The bijection j : crit + f ∂ ∪ critf → crit ˆ f which maps x ∈ crit + f ∂ to ˆ x ∈ crit ˆ f andwhich is the identity on critf ⊂ crit ˆ f induces a chain isomorphism C ∗ ( ˆ f , b X + ) ∼ = C ∗ ( f, X + ) . After that [3] appeared, I was informed that a similar lemma exists in [5] in a setting where only the Morseinequalities are discussed.
Proof of the claim.
Say x ∈ crit + k f ∂ . On the one hand, each X -orbit descending from x to y ∈ crit k f gives rise to an b X -orbit from ˆ x to y and hence, an b X + -orbit from y to ˆ x . Similarly,each X -orbit on ∂M descending from x to y ∈ crit + k − f ∂ gives rise to an b X + -orbit from ˆ y to ˆ x .And conversely. Making j an identification, this proves the following: ∂ X + x = ∂ b X + ˆ x . On the other hand, we have to consider y ∈ crit k +2 f and compute its two differentialswith respect to X + and b X + and evaluate them at x (recall that x has degree k + 1 in C + ∗ ).When x and y have consecutive critical values, as a consequence of Remark 2.1, there are no X + -connecting orbits from x to crit k +2 f .But, if their critical values are not consecutive, one could have a broken X -orbit from y to x made of an orbit from y to z ∈ crit − k +1 f ∂ and an orbit from z to x on ∂M . By using thedeformation formula X + = − X + Z , such a broken orbit gives rise to an X + -orbit from x to y , and hence to an b X + -orbit from ˆ x to y . Then, such connecting orbits may exist. Conversely,by looking at the fusion of the pair ( x, ˆ x ) we get that every b X + -orbit from ˆ x to y is producedby an X + -orbit from x to y . Then, via j the following equality holds true: h ∂ X + y, x i = h ∂ b X + y, ˆ x i . This finishes the proof of the claim and Theorem 1.1 follows. (cid:3) Global stable manifolds and application to intersection pairing
We now discuss the question of global stable manifolds for adapted quasi-gradients. We onlyconsider X − in the definition below; there is a similar definition for X + . If x ∈ ∂M is a criticalpoint of negative type, so far we have only considered its local stable manifold W sloc ( x, X − ) .If x is of index k , it is a small half-disc D n − k − whose planar boundary lies in a level set of f and spherical boundary lies in ∂M . Since the flow of X − , noted X − t at time t , is positivelycomplete, the following definition makes sense: Definition 3.1.
For x ∈ critf ∪ crit − ( f ∂ ) , the global stable manifold of x with respect to X − is defined as the union W s ( x, X − ) = [ t> (cid:0) X − t (cid:1) − (cid:0) W sloc ( x, X − ) (cid:1) . Under mild assumptions, it is a (non-proper) submanifold with boundary and its closure is astratified set. The following assumption (Morse-Model-Transversality) is made in what follows.(MMT)
For every x ∈ critf ∪ crit − f ∂ and y ∈ crit − f ∂ , the neighborhood U y of y in ∂M where X − is tangent to the boundary is mapped by the flow transversely to W sloc ( x, X − ) . Notice that if X − is Morse-Smale, the transversality condition is satisfied along a small neigh-borhood U of the local unstable manifold W uloc ( y, X − ) . Then, after some small perturbation of X − on U y r U destroying the tangency of X − to ∂M , condition (MMT) is fulfilled for the pair ( y, x ) . Thus, condition (MMT) is generic among the negatively adapted vector fields. Proposition 3.2.
If the negatively quasi-gradient X − is Morse-Smale and fulfils condition (MMT) then the following holds: The global stable manifold W s ( x, X − ) is a submanifold with boundary (non-closed in general);its boundary lies in ∂M . If z lies in the frontier of W s ( x, X − ) in M , then it belongs to the stable manifold of somecritical point y in critf ∪ crit − f ∂ such that dim W s ( y, X − ) < dim W s ( x, X − ) . This statement also holds for stable manifolds of critical points in critf ∪ crit + f ∂ with respectto positively adapted vector fields. Proof.
1) According to the Implicit Function Theorem, the conclusion is clear near any pointwhere X − is transverse to the boundary. Near a point z of U y , it follows from (MMT).2) This fact is well known in the case of closed manifolds. It is an easy consequence of theMorse-Smale assumption. The proof is alike if the boundary is non-empty. (cid:3) Remark 3.3.
Due to the transversality assumptions, a small perturbation of X − (resp. X + )moves each of stable and unstable manifolds by a small isotopy, and hence, preserves thecomplex C −∗ (resp. C + ∗ ) up to a canonical isomorphism. As a consequence, without changing the above-mentioned complexes, we are allowed to as-sume that the X − -unstable manifolds of critf ∪ crit − f ∂ intersect the global X + -stable manifoldsof critf ∪ crit + f ∂ transversely. If the orientation of some of the unsta-ble manifolds is changed then the differential of the considered Morse complex (absolute orrelative) is changed by a non-trivial isomorphism. So, to understand the role of the orientabil-ity of M in what follows, it will be better to replace C ∗ ( f, X − ) with C ∗ ( f, X − , ε − f ) where ε − f denotes the chosen orientation map which associates an orientation of W u ( x, X − ) with each x ∈ critf ∪ crit − f ∂ . Note that ε − f orients the unstable manifolds regardless of the quasi-gradientsince they all have isotopic germs at the critical points. And similarly for C ∗ ( f, X + ) . Actually,we will only apply this change of notation at the places where it will be crucial. At the homology level, this isomorphism is aisomorphism P : H ∗ ( M, ∂M ; Z ) → H n −∗ ( M ; Z ) , We wish to describe it by means of our Morse complexes in order to deduce a Morse theoreticaldescription of the homological intersection. There are several steps to achieve.1) First, we recall that there is a natural isomorphism at the homology level I ∗ ( f, X − ) : H ∗ (cid:0) C ∗ ( f, X − ) (cid:1) → H ∗ ( M ; Z ) . Indeed, we have described in [3] a canonical process for removing the critical points of f ∂ ofnegative type. Once this is done, the unstable manifolds of X − emerging from crit ( f ) yield a cell decomposition of M (see [2] ) whose homology is canonically isomorphic to the singularhomology of M (see [4], p. 90).We now explain the naturality of this isomorphism. Let ( g, Y − ) be another pair of genericMorse function and negatively adapted quasi-gradient. The choice of a generic path γ from ( g, Y − ) to ( f, X − ) gives rise to some simple homotopy equivalence (3.1) γ ∗ : C ∗ ( g, Y − ) → C ∗ ( f, X − ) well defined up to the orientations. At each time that γ crosses a stratum corresponding toa codimension-one defect of genericity of the pair ( function, negatively adapted quasi-gradient )this yields an elementary modification of the Morse complex, indeed a quasi-isomorphism [3].One checks at each occurrence that this quasi-isomorphism is compatible to the isomorphismwith H ∗ ( M ; Z ) . Finally, γ ∗ is the composition of all these quasi-isomorphisms. It induces anisomorphism [ γ ∗ ] in homology making the next diagram commute:(3.2) H ∗ ( C ∗ ( g, Y − )) [ γ ∗ ] / / I ∗ ( g,Y − ) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ H ∗ ( C ∗ ( f, X − )) I ∗ ( f,X − ) (cid:15) (cid:15) H ∗ ( M ; Z ) By taking the transpose of all morphisms of chain complexes we get a similar diagram incohomology made of isomorphisms: H ∗ ( C ∗ ( g, Y − )) I ∗ ( g,Y − ) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ H ∗ ( C ∗ ( f, X − )) [ γ ∗ ] o o I ∗ ( f,X − ) (cid:15) (cid:15) H ∗ ( M ; Z ) Note that a change of orientations of some unstable manifolds has the same effect on [ γ ∗ ] andon I ∗ ( − , − ) . So, the commutativity of the above diagrams is not affected.2) We can do the same for the Morse complex C ∗ ( f, X + ) which calculates the relative homology.Here, we will use the stable manifolds of X + that we introduced in the beginning of Section3. More precisely, there is a canonical process, similar to the one above-mentioned for thecomplex C ∗ ( f, X − ) , which removes the positive type critical points of f ∂ . After removingthem, the stable manifolds of X + associated with critf give rise to a filtration of M startingfrom ∂M : ∂M ⊂ M [1] ⊂ . . . ⊂ M [ k ] ⊂ . . . ⊂ M [ n ] = M .
Here, M [ k ] is the union of ∂M and the closure of the stable manifolds of X + converging to crit k f . The cellular homology associated with this filtration gives a canonical isomorphism I ∗ ( f, X + ) : H ∗ (cid:0) C ∗ ( f, X + ) (cid:1) → H ∗ ( M, ∂M ; Z ) . In this reference, a stronger assumption is made on the vector field which implies this cell decompositionto be a CW -complex. Without this assumption, the cell decompositon has only the homotopy type of a CW -complex. This is sufficient for our discussion. The creation/cancellation times of pair of critical points along γ do not allow us to carry orientations alongthe path. Moreover, this isomorphism is natural with respect to change of function and quasi-gradientin the same sense as it is detailed in 1) above.3) Here comes the important point for orientations. Let ε + f be a choice of orientations ofthe stable manifolds of X + . Since M is oriented, the unstable manifolds of X + are not onlyco-oriented but they are also oriented. The latter orientations are denoted by ε ⊥ f .We recall that X + is a negatively adapted quasi-gradient of − f ; we denote it by Y − := X + when it is considered as a descending quasi-gradient of − f . So, we have a chain complex C ∗ ( − f, Y − , ε −− f ) where ε −− f is determined by ε + f by the rule(3.3) ε −− f = ε ⊥ f . By applying the functor
Hom ( − , Z ) we have its dual, a co-chain complex, C ∗ ( − f, Y − , ε ⊥ f ) .By construction of C + ∗ , we have(3.4) η ∗ : C ∗ ( f, X + , ε + f ) = −→ C n −∗ ( − f, Y − , ε ⊥ f ) . This equality means same generators and same differential; only the grading is reversed. Itinduces the equality H ∗ (cid:0) C ∗ ( f, X + , ε + f ) (cid:1) = H n −∗ (cid:0) C n −∗ ( − f, Y − , ε ⊥ f ) (cid:1) and by combining it withthe isomorphims I ∗ ( f, X + ) and I n −∗ ( − f, Y − ) we get a description at the Morse complex levelof the Poincaré-Lefschetz isomorphism: P : H ∗ ( M, ∂M ; Z ) → H n −∗ ( M ; Z ) . We are interested in describing a pairing atthe chain level σ : C k ( f, X + ) ⊗ C n − k ( f, X − ) → Z which induces the intersection pairing in homology. This is achieved in the following way.At the homology level the Poincaré-Lefschetz isomorphism P carries the intersection product ι : H ∗ ( M, ∂M ; Z ) ⊗ H n −∗ ( M, Z ) → Z to the evaluation map ev : H n −∗ ( M ; Z ) ⊗ H n −∗ ( M ; Z ) → Z .After what was done in the previous subsection, we only have to understand this evaluationmap in the setting of Morse homology. First, there is a canonical evaluation map ev = < − , − > : C n −∗ ( − f, Y − ) ⊗ C n −∗ ( − f, Y − ) → Z which on the basis elements is the Kronecker product. A more sophisticated way to say thesame thing is to count the transverse intersection W s ( x, Y − ) ∩ W u ( y, Y − ) for every pair ofcritical points of the same degree, that is both in crit k f ∪ crit + k − f ∂ for some integer k . Here,it is essential Y − to be Morse-Smale for avoiding undesirable orbits connecting points of thesame degree. Here, some convention has to be used, for instance: co-or(-) ∧ or(-)= or( M ). We choose a generic path Γ from ( f, X − ) to ( − f, Y − ) which yields a quasi-isomorphism Γ ∗ : C n −∗ ( f, X − , , ε − f ) → C n −∗ ( − f, Y − , ε ⊥ f ) . Thanks to (3.4), the desired evalution map is givenby(3.5) σ = ev ◦ ( η ∗ ⊗ Γ ∗ ) . If necessary, by Remark 3.3 we may approximate X − in order to make mutually transverse W s ( x, X + ) and W u ( y, X − ) for every x ∈ crit k f ∪ crit + k − f ∂ and y ∈ crit n − k f ∪ crit − n − k f ∂ . Claim.
For every pair of cycles α ∈ C k ( f, X + ) and β ∈ C n − k ( f, X − ) , the geometric formulafor σ ( α, β ) is given by counting the signed intersection number of the respective stable and un-stable manifolds entering in the linear combinations forming α and β . Indeed, by (3.2) the cycles β and Γ ∗ ( β ) are homologous in M . Therefore, they have thesame algebraic intersection with the cycle α . Notice that the frontier of the involved invariantmanifolds does not appear in this counting since it is made of invariant manifolds of lessdimension. Corollary 3.7.
The pairing σ induces the homological intersection. References [1] Abbaspour H., Laudenbach F.,
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Laboratoire de mathématiques Jean Leray, UMR 6629 du CNRS, Faculté des Sciences etTechniques, Université de Nantes, 2, rue de la Houssinière, F-44322 Nantes cedex 3, France.
E-mail address : [email protected] First, choose a generic path ( f t ) in the space of functions; then, complete with a path of quasi-gradients.For this second step, use the convexity of the set of quasi-gradients adapted to f t for a given t . If the function f t has a codimension-one singularity, the involved critical point needs to be a zero of X tt