Functoriality and duality in Morse–Conley–Floer homology

In [J. Topol. Anal. 6 (2014), 305–338], we have developed a homology theory (Morse–Conley–Floer homology) for isolated invariant sets of arbitrary flows on finite-dimensional manifolds. In this paper, we investigate functoriality and duality of this homology theory. As a preliminary, we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known, but the proofs use isomorphisms to other homology theories. We give direct proofs by analyzing appropriate moduli spaces. The notions of isolated map and flow map allow the results to generalize to local Morse homology and Morse–Conley–Floer homology. We prove Poincaré-type duality statements for local Morse homology and Morse–Conley–Floer homology.


Introduction
In this paper, we address functoriality and duality properties of Morse homology, local Morse homology and Morse-Conley-Floer homology. The functoriality of Morse homology on closed manifolds is known [1,2,3,9,15], however no proofs are given through the analysis of moduli spaces. This analysis is done in Sections 2-4. These sections are of independent interest from the rest of the paper. In Section 5 we discuss isolation properties of maps, which are important for the functoriality in local Morse homology and Morse-Conley-Floer homology. This functoriality is discussed in Sections 6 and 7. In Section 8 we discuss Poincaré duality in these homology theories. Finally, in Appendix A we prove that the transverse maps that are crucial for defining the induced maps in Morse homology are generic. Below is a detailed description of the results in this paper.

Morse homology and local Morse homology
A Morse datum is a quadruple Q α = (M α , f α , g α , o α ), where M α is a choice of closed manifold and (f α , g α , o α ) is a Morse-Smale triple on M α . Thus f α is a Morse function on M α , and g α is a metric such that the stable and unstable manifolds of f α intersect transversely. Finally, o α denotes a choice of orientations of the unstable manifolds. The Morse homology HM * (Q α ) is defined as the homology of the chain complex of C * (Q α ) which is freely generated by the critical points of f α and graded by their index, with boundary operator ∂ * (Q α ) counting connecting orbits of critical points with Morse index difference of 1 appropriately with sign; cf. [5,15,16]. If Q β is another choice of Morse datum with M β = M α , there is a canonical isomorphism Φ βα * : HM * (Q α ) → HM * Q β . The canonical isomorphism is induced by continuation, i.e., a homotopy between the Morse data Q α and Q β ; see, for example, [7,16]. The Morse homology of the manifold M = M α is defined by where the inverse limit is taken over all Morse data with the canonical isomorphisms. 1 The Morse homology HM * (M ) is isomorphic to the singular homology H * (M α ). Important in what follows is that this construction can also be carried out locally. 2 Recall the following definitions from Conley theory. A subset S α ⊂ M α is called invariant for a flow φ α if φ α (t, S α ) = S α for all t ∈ R.
A compact neighborhood N α ⊂ M α is an isolating neighborhood for φ α if Inv N α , φ α = {x ∈ N α | φ α (t, x) ∈ N α for all t ∈ R}, is called the maximal invariant set in N α . An invariant set S α for which there exists an isolating neighborhood N α with S α = Inv(N α , φ α ), is called an isolated invariant set. A homotopy of flows is isolated if N α is an isolating neighborhood for each flow in the homotopy. Now suppose that (f α , g α ) is a Morse-Smale pair on N α ; cf. [14,Definition 3.5]. Thus N α is an isolating neighborhood of the gradient flow ψ α of (f α , g α ) such that all critical points of f α are nondegenerate on N α and the local stable and unstable manifolds intersect transversely. Then with o α a choice of orientation of the local unstable manifolds, is a local Morse datum. The local Morse homology HM * (P α ) is defined by a similar counting

Morse-Conley-Floer homology
We recall the definition of Morse-Conley-Floer homology, cf. [14], from a slightly different viewpoint. Let φ be a flow on M and S an isolated invariant set of the flow. A Lyapunov function f α φ for (S, φ) is a function that is constant on S and satisfies d dt t=0 f α φ φ(t, p) < 0 on N α \S, where N α is an isolating neighborhood for S α . Lyapunov functions always exist for a given isolated invariant set; cf. [14,Proposition 2.6]. We can compute the local Morse homology of a Lyapunov function with respect to the choice of a metric e α and N α .

Functoriality in Morse homology
In Sections 2-4 we study functoriality in Morse homology on closed manifolds. Induced maps between Morse homologies are defined by counting appropriate intersections for transverse maps. W s (y).
The set of transverse maps is denoted by T (Q α , Q β ). We write for the moduli spaces.
Given Q α and Q β , the set of transverse maps T (Q α , Q β ) is generic; cf. Theorem A.1. The index of a critical point x ∈ Crit f α is denoted by |x|. The transversality assumption ensures that W h βα (x, y) is an oriented manifold of dimension |x| − |y|; cf. Proposition 2.1. Hence, for |x| = |y| we can compute the oriented intersection number n h βα (x, y) and define an induced map h βα * : C * (Q α ) → C * Q β by h βα * (x) = |y|=|x| n h βα (x, y) y.
Vol. 16 (2014) Functoriality in Morse-Conley-Floer homology 441 Functoriality in Morse-Conley-Floer homology 5 In Sections 2-4 we show that this defines a chain map, that homotopic maps induce chain homotopic maps, and that the composition of the induced maps is chain homotopic to the induced map of the composition. This implies that the induced map h βα * descends to a map h βα * : HM * (M α ) → HM * M β between the Morse homologies via counting, which is functorial and which does not depend on the homotopy class of the map h βα . The homotopy invariance and density of transverse maps allow for an extension to all smooth maps.
The isolation properties of flow maps are given in Proposition 5.4. If h βα is a flow map and N β is an isolating neighborhood, then is an isolating neighborhood. Moreover, h βα is isolated with respect to these isolating neighborhoods. Similar statements hold for compositions of flow maps.

Functoriality in local Morse homology
In Section 6 we define induced maps in local Morse homology. Due to the local nature of the homology, not all maps are admissible and the notion of isolated map becomes crucial. The maps are computed by the same counting procedure, but now done locally. We sum up the functorial properties of local Morse homology from Propositions 6.1 and 6.3.
(i) An isolated transverse map induces a chain map, hence descends to a map between the local Morse homologies. (ii) An isolated homotopy between transverse maps induces a chain homotopic map.  f, g, N) consisting of smooth functions f on M and metrics g such that the sets N are isolating neighborhoods for the associated gradient flows. A morphism These morphisms are then perturbed to transverse maps, from which the induced map can be computed; cf. Propositions 6.2 and 6.4. The local Morse homology functor generalizes the Morse homology functor. Any map h βα : M α → M β between closed manifolds equipped with flows is isolated homotopic to a flow map with since the flows on both manifolds are isolated homotopic to the constant flow, and any map is equivariant with respect to constant flows.
is a map that is isolated homotopic for some choice of isolating neighborhoods N α , N β to a flow map h βα such that N α = ( h βα ) −1 (N β

Chain maps in Morse homology on closed manifolds
In Sections 2-4, which discuss functoriality for Morse homology, we assume that the base manifolds are closed.

The moduli space
For a transverse map, see Definition 1.1, the moduli spaces are smooth oriented manifolds.
is an oriented submanifold of W u (x). The orientation is induced by the exact sequence of vector bundles 4 where the latter is the normal bundle of W s (y  If |x| = |y|, the space W h βα (x, y) is zero dimensional and consists of a finite number of points carrying orientation signs ±1. Set n h βα (x, y) as the sum of these orientation signs. We define the degree zero map h βα * : Through compactness and gluing analysis of the moduli space W h βα (x, y) and related moduli spaces, we prove the following properties of induced maps.
(i) The induced map of the identity is the identity on chain level.
(ii) We show that h βα * is a chain map; i.e., (iii) In Section 3 we study homotopy invariance. Suppose that Q γ and Q δ are other Morse data on M α = M γ and M β = M δ . Suppose that Here Φ βα * denotes the isomorphism induced by continuation. (iv) In Section 4 we study compositions. If h γβ ∈ T (Q β , Q γ ) is such that then h γβ * h βα * and (h γβ • h βα ) * are chain homotopic; i.e., there exists a degree +1 map P γα The chain map property (ii) holds by the compactness properties of the moduli space W h βα (x, y ) with |x| = |y | + 1. This space is a one-dimensional manifold, but it is not necessarily compact. The noncompactness is due to breaking of orbits in the domain and in the codomain, which is the content of Proposition 2.2. In Figure 1 we depicted this breaking process in the domain. Functoriality in Morse-Conley-Floer homology 9 x y y p k with |x| = |x |. The space W h βα (x, y ) can be compactified by gluing in these broken orbits; cf. Proposition 2.4. Proposition 2.6 then states that the resulting object is a one-dimensional compact manifold with boundary. By counting the boundary components appropriately with sign, the chain map property is obtained; cf. Proposition 2.7.
Since M α is assumed to be compact, and any sequence p k ∈ W h βα (x, y ) has a subsequence converging to some point p ∈ M α . In the next proposition, see also Figure 1, it is stated to which points such sequences converge.
x ∈ Crit f α and y ∈ Crit f β with |x| = |y | + 1. Let p k ∈ W h βα (x, y ) be a sequence such that p k → p in M α . Then either one of the following is true: (ii) there exists y ∈ Crit f α , with |y| = |y | such that p ∈ W h βα (y, y ). The orbits through p k break to orbits (u 1 , . . . , u l ) as k → ∞ with u 1 ∈ M (x, y); T. O. Rot and R. C. A. M. Vandervorst The orbits through h βα (p k ) break to the orbits (u 1 , . . . , u l ) as k → ∞ with u l ∈ M (x , y ).
Proof. The spaces W h βα (x, y ), W h βα (y, y ) and W h βα (x, x ) are disjoint, and hence the three possibilities cannot occur at the same time. We choose a subsequence such that p k ∈ W (x, b) for some fixed b ∈ Crit f α , which is possible by the fact that there are only a finite number of such spaces by compactness. The broken orbit lemma, see for example [4,Lemma 2.5], states that p ∈ W (y, a) for some y, a ∈ Crit f α with |y| ≤ |x|, and equality if and only if x = y. Similarly, choosing a subsubsequence if necessary, we also assume that Assuming that p ∈ W h βα (x, y ), only two possibilities remain. If gives that |y | = |y|; see also Figure 1. The claim about the breaking orbits is the content of the broken orbit lemma.
The proposition generalizes to higher index difference moduli spaces. We do not need this, and this would clutter the notation without a significant gain.

Gluing the ends of
We compactify W h βα (x, y ) by gluing in the broken orbits described in Proposition 2.2. The following lemma is the technical heart of the standard gluing construction in Morse homology; see also Figure 2. We single this out because we have to construct several gluing maps, which all use this lemma. For a pair (f, g) of a function and a metric we denote by ψ its negative gradient flow.

Assume that each intersection only consists of a single point, and write
Then there exist an R 0 > 0 and an injective map ρ :  Figure 2. The content of Lemma 2.3 is depicted. Discs that are transverse to the stable and unstable manifolds must intersect if they are flowed in forwards and backwards time. This intersection point is used to define the gluing map.
Finally there exist smaller discs D ⊂ D |y| and E ⊂ E m−|y| such that no or- Proof. We only sketch the proof; see also Since the discs are transverse to the stable and unstable manifolds of y, the λ-Lemma (cf. [12, Chapter 2, Lemma 7.2]) gives that for all t large, smaller discs are − C 1 close to B u and B s , respectively. It follows, through an application of the Banach Fixed Point Theorem, that there exists an R 0 > 0 such that D R and E −R intersect in a single point for each R > R 0 sufficiently large. Set The properties of ρ follow from the construction.
To prove that ∂ 2 = 0 in Morse homology, gluing maps are needed to compactify appropriate moduli spaces and these are constructed using the Morse-Smale condition and the previous lemma as follows. Let x, y and z be critical points with |x| = |y| − 1 = |z| − 2 and assume that M (x, y) and M (y, z) are nonempty. Then the Morse-Smale condition gives that there exists a disc D |y| in W u (x) that is transverse to an orbit in M (x, y). Because of T. O. Rot and R. C. A. M. Vandervorst transversality, one can also choose a disc E m−|y| in W s (z) that is transverse to an orbit in M (y, z). The gluing map is given by mapping R to the orbit through ρ(R) as in Figure 2. We now use similar ideas to compactify W h βα (x, y ) with |x| = |y | + 1.
. Then for critical points x, y ∈ Crit f α and x , y ∈ Crit f β , with |x| = |x | = |y| + 1 = |y | + 1, there exist R 0 > 0 and gluing embeddings Denote by γ u ∈ M (x, y) the orbit through u, and set Vol. 16 (2014)  . We show that the gluing map # 1 is compatible with the induced orientations, while # 2 reverses the orientations. Consider the notation of the proof of Proposition 2.4. By W (x, y)| u denote the connected component of W (x, y) containing u ∈ W (x, y), with similar notation for other moduli spaces. If |x| = |y|+1, the moduli space W (x, y) is one dimensional and can be oriented by the negative gradient vector field.
The gluing map # 1 induces a map of orientations Similarly, the gluing map # 2 induces a map of orientations given by Proposition 2.5. Let the notation be as above. Then σ 1 preserves the orientation, and σ 2 reverses the orientation induced by o α and o β .
Proof. We first treat the gluing map σ 1 . By the transversality assumptions we have the following exact sequences of oriented vector spaces: 3) The following isomorphisms of oriented vector spaces are induced by parallel transport and the fact that the stable and unstable manifolds are contractible: Analogously, from (2.3), (2.4) and (2.5) we get Combining the last two formulas, we see that The orientation on the right-hand side is ab [u]. The tangent vector to the orbit γ u through u has a well-defined limit as t → ∞, which we denote by [u(∞)], and similarly the tangent vector to γ v has a well-defined limit as t → −∞, which we denote by [v(−∞)]; cf. [ v with the gradient flow, cf. Figure 3, which we denote by Note that y is in the closure of this space. Then the orientations agree on this space (here we extend the manifold to its closure). Quotienting out the R action, we see that the orientation The orientation map σ 1 preserves the orientation. The proof that σ 2 is orientation reversing is analogous. Again we use the notation of Proposition 2.4, with u ∈ W h βα (x, x ) and v ∈ W (x , y ). We have the following exact sequences of oriented vector spaces: By isomorphisms induced by parallel transport, analogous to the isomorphisms (2.4), (2.5), and (2.6), and from (2.8) and (2.7) we get that Similarly, from (2.9) and the isomorphisms induced by parallel transport we get which gives The orientation on the right-hand side is ab [v]. Locally around u, h βα is injective when restricted to W h βα (x, y )| u . We can therefore take this image and flow with ψ β , which we denote by h βα (W h βα ( and agree. By quotienting out the flow, it follows that The map σ 2 is orientation reversing; cf. Theorem 2.6. Let h βα be transverse with respect to Q α and Q β . For each x ∈ Crit f α and y ∈ Crit f β with |x| = |y | + 1, the space has a natural structure as a compact oriented manifold with boundary given by the gluing maps. Note that this proposition does not state that exactly one half of the boundary components correspond to M (x, y)×W h βα (y, y ) and the other half to W h βα (x, x ) × M (x , y ). Still it does follow that h βα * is a chain map.
Proof. Let x ∈ Crit f α . Using Proposition 2.6 we compute Because the oriented count of the boundary components of a compact oriented one-dimensional manifold is zero, h βα * is a chain map.

Homotopy-induced chain homotopies
The main technical work, showing that h βα * is a chain map, is done. To show that homotopic maps induce the same maps in Morse homology, we could again define an appropriate moduli space and analyze its compactness failures. However, a simpler method is to construct a dynamical model of the homological cone. This trick is used in Morse homology to show that Morse homology does not depend on the choice of function, metric and orientation. Using the homotopy we build a higher-dimensional system-the dynamical cone-where we use the fact that an induced map is a chain map to prove homotopy invariance.
and assume the maps are homotopic. Then h δγ * Φ γα * and Φ δβ * h βα * are chain homotopic. That is, there exists a degree +1 map P δα Proof. Let h λ : M α → M β be a smooth homotopy between h βα and h δγ . Let g γα λ be a smooth homotopy between g α and g γ , and let g δβ λ be a smooth homotopy between g β and g δ . Similarly, let f γα λ be a smooth homotopy between f α and f γ , and let f δβ λ be a smooth homotopy between f β and f δ . Choose 0 < < 1/4 and let ω : R → [0, 1] be a smooth, even and 2-periodic function and We identify S 1 with R/2Z. Under the identification of S 1 with R/2Z, the function ω descends to a smooth function S 1 → [0, 1], which is also denoted by ω.
Let r > 0. We define the functions F α on M α × S 1 and F β on M β × S 1 by . For r sufficiently large, the functions F α , F β are Morse, cf. [14,Lemma 4.5], and the critical points can be identified by The connections of the gradient flow at µ = 0 and µ = 1 are transverse, and the map H restricted to the neighborhoods at µ = 0 and µ = 1 also satisfies the required transversality properties. Hence we can perturb H while keeping it fixed in the neighborhoods of µ = 0 and µ = 1, as well as the metrics G α = g γα ω(µ) (x) ⊕ dµ 2 and G β = g δβ ω(µ) (x) ⊕ dµ 2 outside − < µ < and 1 − < µ < 1 + to obtain Morse-Smale flows on M α × S 1 and M β × S 1 , such that the map H is transverse everywhere. We orient the unstable manifolds in M α × S 1 by Similarly for (y, 0) ∈ C k (F β ), i.e., y ∈ C k−1 (f β ), we have Finally for (y, 1) ∈ C k (F β ), i.e., y ∈ C k (f δ ), we have By Propositions 2.2 and 2.4 and the construction of H it is clear 5 that we can restrict the count of the induced map H * to W H (x, y) ∩ M α × [0, 1]; see also Proposition 6.1. Note that so n H ((x, 0), (y, 0)) = n h βα (x, y) as oriented intersection numbers. Similarly for |(x, 1)| = |(y, 1)|, we find that For |(x, 1)| = |(y, 0)| we compute that n H ((x, 1), (y, 0)) = 0. Finally, we define a map P δα k : C k (f α ) → C k+1 (f δ ) by counting the intersections of W u ((x, 0)) and H −1 (W s ((y, 1))) with sign. Thus where the Φ's are the maps that induce isomorphisms in Morse homology. We know that H k is a chain map, i.e., The lower left corner of this matrix equation gives the desired identity.

Composition-induced chain homotopies
To show that the compositions of a map induce the same map as the compositions of the induced maps in Morse homology, we take in spirit also a homotopy between h γβ and h βα and h γβ • h βα . This is not directly possible as the maps have different domains and codomains, but we have an approximating homotopy, which is sufficient. For a flow φ : We assume, up to possibly a perturbation of h γβ , that the map for all x ∈ Crit f α and z ∈ Crit f γ . We then have the moduli spaces of dimension |x| − |z| + 1. The compactness issues if R → ∞ are due to the breaking of orbits in M β , which is described in the following proposition. Proposition 4.1 (Compactness). Assume the above situation, with |x| = |z|. Let (p k , R k ) ∈ W h γβ ,h βα (x, z) be a sequence with R k → ∞ as k → ∞. Then there exist y ∈ Crit f β , with |x| = |y| = |z|, and a subsequence (p k , R k ) such that p k → p ∈ W h βα (x, y), and ψ β R k • h βα (p k ) → q ∈ W h γβ (y, z). Proof. By compactness of M α we can choose a subsequence of (p k , R k ) such that p k → p, and choose a subsubsequence such that also By similar arguments as in Proposition 2.2, using the transversality, p ∈ W h βα (x , y), with |x| ≥ |x | ≥ |y|, and q ∈ W h γβ (y , z ), with |y | ≥ |z | ≥ |z|, where the equality holds if and only if x = x , z = z. Moreover, since q k and h βα (p k ) are on the same orbit, we must have that |y | ≥ |y| with equality if and only if y = y. Since |x| = |z|, it follows that x = x and z = z, and therefore also y = y. Proposition 4.2 (Gluing). Assume the above situation. Let x ∈ Crit f α , y ∈ Crit f β and z ∈ Crit f γ , with |x| = |y| = |z|. Then there exist an R 0 > 0 and a gluing embedding Vol. 16 (2014) then the sequence (p k , R k ) lies in the image of the embedding for k sufficiently large.
Proof. Let u ∈ W h βα (x, y) and v ∈ W h γβ (y, z). By transversality of h βα , we choose a disc D |y| ⊂ W u (x) such that h βα | D |y| is injective and the image h βα (D |y| ) intersects W s (y) transversely in h βα (u). We also choose a disc E m β −|y| ⊂ M β , intersecting W u (y) transversely in v, whose image is contained in W s (z); cf. Figure 5. By Lemma 2.3, we get an R 0 > 0 and a map ρ : Here we use the fact that h βα D |y| is bijective to h βα (D |y| ). The properties stated follow from Lemma 2.3.

Proposition 4.3. Consider the above situation. Then there exists an
has a compactification as a smooth oriented manifold with boundary The space only has two noncompact ends. One is counted by for which we have constructed a gluing map. The other noncompact end is counted by where the gluing map is given by sending

Proof. By Proposition 4.3 we have
The homotopy between induces a chain homotopy by Proposition 3.1. That is, there is a degree +1 map P * such that Combining the last two equations gives the chain homotopy.

Isolation properties of maps
For the remainder of this paper, we do not assume that the base manifolds are necessarily closed. We localize the discussion on functoriality in Morse homology on closed manifolds, and we study functoriality for local Morse homology, as well as functoriality for Morse-Conley-Floer homology. For this, we need the isolation properties of maps. For a manifold equipped with a flow φ, denote the forwards and backwards orbits as follows: satisfies the property that for all p ∈ S h βα , we have Compositions of isolated maps need not be isolated, however the condition is open in the compact-open topology.  Proof. Define the sets which are compact. Then Note that a map being isolated is equivalent to the following two properties of points on the boundary of the isolating neighborhoods in the domain and the codomain.

In each of these cases we construct open sets
, these properties remain true for points on the boundary of the isolating neighborhoods. By compactness of N α , N β we can choose a finite number of U α/β p that still cover ∂N α/β . Then is the required open set.
Let p ∈ ∂N α and assume that property (ai) holds. Then there exists a t < 0 such that φ α (t, p) ∈ M α \N α . By continuity there exists an open neighborhood U p p such that φ α (t, This is by definition open in the compact-open topology. Then for all q ∈ U p and all (φ α ,h βα ,φ β ) ∈ A α p there exists a t < 0 such thatφ α (t, q) ∈ M α \ N α . Now let p ∈ ∂N α and assume that property (aii) holds. Choose t ∈ [0, T ] and s > 0 such that Then for all q ∈ U β p and all (φ α ,h βα ,φ β ) ∈ A β p there exists a t > 0 such that φ β (t, q) ∈ N β .
Finally, let p ∈ ∂N β and assume that property (bii) holds. Thus Since both sets are compact, there exists an open set Let us return to the codomain. By continuity of the flow and the choice of T there exists an > 0 such that Using compactness, choose a finite number of times t j such that and for all t ∈ T q we have φ β (t, q) ⊂ V . Define Vol. 16 (2014) Functoriality in Morse-Conley-Floer homology 463

Functoriality in Morse-Conley-Floer homology 27
Then for all q ∈ U β p and all (φ α ,h βα ,φ β ) ∈ A β p , Hence we can choose a finite number of p j such that for all p ∈ ∂N β and all (φ α ,h βα , φ β ) ∈ A β = j A β pj either property (bi) or (bii) holds.
Set A = A α ∩A β . Then, by construction of the set A, properties (ai) and (aii) hold for all (φ α ,h βα ,φ β ) ∈ A and all p ∈ N α . Similarly, properties (bi) and (bii) hold for all p ∈ ∂N β . Thush βα is isolated with respect to the flows φ α ,φ β , and the isolating neighborhoods N α , N β .
Moreover, h βα is isolated with respect to N α and N β . If h γβ is another flow map, and N γ is an isolating neighborhood, then Proof. We follow McCord [10]. Since h βα is proper, N α is compact. If p ∈ Inv(N α ), then φ α (t, p) ∈ N α for all t ∈ R. By equivariance, and since N β is an isolating neighborhood, h βα (p) ∈ int(N β ). Thus If p ∈ S α , then for all t we have that φ α (t, p) ∈ N α , and thus By equivariance it follows that φ β (t, h βα (p)) ∈ N β for all t. Hence h βα (p) ∈ S β . Analogously, if p ∈ (h βα (p)) −1 (S β ), then φ β (t, h βα ) ∈ S β for all t. By equivariance it follows that h βα (φ α (t, p)) ∈ S β , and this implies that We finally show that h βα is isolated. If p ∈ S h βα , then φ α (t, p) ∈ N α for all t < 0, and φ β (t, h βα (p)) ∈ N β for all t > 0. By equivariance, The proof of the latter statement follows along the same lines.
The previous proposition states that we can pull back isolated invariant sets and neighborhoods along flow maps. The same is true for Lyapunov functions. Then Again by equivariance, Thus f α is a Lyapunov function. We now prove that h βα is an isolated map with respect to the isolating neighborhoods N α and N β for the gradient flows ψ α and ψ β . The neighborhoods are isolating for the gradient flows by [14,Lemma 3.3]. Let P h βα be the set of equation ( The function b p is continuous, smooth outside zero and by the Lyapunov By [14,Lemma 3.1] we know that Crit Hence b p is constant and it follows that p ∈ S α . Thus the full orbit through p is contained in int N α and the full orbit through h βα (p) is contained in int N β . Thus h βα is isolated with respect to the gradient flows.

Functoriality in local Morse homology
We are interested in the functorial behavior of local Morse homology. Let us recall the definition of local Morse homology more in depth. Suppose that N α is an isolating neighborhood of the gradient flow of f α and g α . The local stable and unstable manifolds of critical points inside N α are defined by We write We say that the gradient flow is Morse-Smale on N α , cf. [14,Definition 3.5], if the critical points of f α inside N α are nondegenerate, and for each p ∈ W N α (x, y), we have that T p W u (x) + T p W s (y) = T p M α . The intersection is said to be transverse and we write where n N α , loc (x, y) denotes the oriented count of points in M N α (x, y). The differential satisfies ∂ 2 (P α ) = 0, hence we can define local Morse homology. This is not an invariant for N α but crucially depends on the gradient flow. If h βα is isolated with respect to P α and P β , we say that it is transverse (with respect to P α and P β ), if for all The oriented intersection number is denoted by n h βα , loc (x, y). Proposition 6.1. Let h βα ∈ T (P α , P β ) and suppose that h βα is isolated with respect to R α and R β . Then h βα * (x) := |x|=|y| n h βα , loc (x, y)y is a chain map. Suppose that P γ and P δ are different local Morse data with such that the gradient flow of P α is isolated homotopic to the gradient flow of P γ , the gradient flow of P β is isolated homotopic to the gradient flow of P δ , and h βα ∈ T (P α , P β ) and h δ,γ ∈ T (P γ , P δ ) are isolated homotopic through these isolated homotopies. Then Φ δβ * h βα * and h δγ * Φ γα are chain homotopic.
Proof. We argue that the gluing maps constructed in Proposition 2.4 restrict to the local gluing maps y) and v ∈ W h βα , loc (y, y ). Geometric convergence implies that the backwards orbit O − (γ u # 1 R v) lies arbitrary close, for R sufficiently large, to the images of O(u) and O − (v). The latter are contained in int N α , hence Recall that the local Morse homology of the isolating neighborhood of any pair (f, g), which is not necessarily Morse-Smale, is defined by where the inverse limit runs over all local Morse data P α whose gradient flows are isolated homotopic to the gradient flows of (f, g) on N , with respect to the canonical isomorphisms. HM * (f α , g α , N α ) → HM * f β , g β , N β . Suppose that h δγ is isolated homotopic to h βα through isolated homotopies of gradient flows between (f α , g α , N α ) and (f γ , g γ , N γ ) and (f β , g β , N β ) and (f δ , g δ , N δ ). Then the following diagram commutes: O. Rot and R. C. A. M. Vandervorst Figure 6. All isolated maps induce chain maps in local Morse homology, but they are not necessarily functorial, as they do not capture the dynamical content. The gradient flows of f α (x) = x 2 , f β = (x−3) 2 and f γ (x) = x 2 on R, with isolating neighborhoods N = [−1, 1] are depicted. The identity maps are isolated. We compute that id βα * = 0, id γβ * = 0, but id γα * is the identity. The problem is that id γβ •ψ β R • id βα is not isolated for all R > 0. and h βα is isolated for the isolated homotopies connecting (f α , g α ) with (f γ , g γ ) and (f β , g β ) with (f δ , g δ ), where h βα ∈ T (P γ , P δ ). This is possible by Proposition 5.2, and [14,Corollary 3.12]. By the density of transverse maps (Theorem A.1), there exists a small perturbation h δγ which is isolated homotopic to h βα by this homotopy. By Proposition 6.1 we get a map h δγ * : HM * P γ → HM * P δ . Moreover, given different local Morse data P and P ζ as above, we can construct an isolated map h ζ which is isolated homotopic to h βα and hence also to h δβ by concatenation. From Proposition 6.1 it follows that the following diagram commutes: Which means that we have an induced map h βα * : HM * (f α , g α , N α ) → HM * f β , g β , N β between the inverse limits. The arguments for the homotopy of the maps is analogous.
The chain maps defined above are not necessarily functorial. Consider, for example, Figure 6. The problem is that, in the proof of functoriality for Morse homology, we need the fact that h γβ • ψ β R • h βα is isolated for all R ≥ 0 to establish functoriality. If we require this almost homotopy to be isolated, the proof of functoriality follows mutatis mutandis. maps with respect to the gradient flows. Let N γ be an isolating neighborhood of the gradient flow of (f γ , g γ ). Set Then these are isolating neighborhoods, the maps h γβ , h βα and h γβ • h βα are isolated and the following diagram commutes: Proof. The proposition follows by combining Propositions 5.4, 6.2 and 6.3. Theorem 1.5 now follows from the fact that isolated homotopic maps induce the same maps in local Morse homology.

Functoriality in Morse-Conley-Floer homology
We use the induced maps of local Morse homology to define induced maps for flow maps in Morse-Conley-Floer homology. h βα * : HI * (S α , φ α ) → HI * S β , φ β , which is functorial: The identity is mapped to the identity and the following diagram commutes: Proof. The induced map is defined as follows. Let f β be a Lyapunov function 8 with respect to the isolating neighborhood N β of (S β , φ β ). Then Now let Q α = {M α , f α , g α , o α } be a Morse datum on an oriented closed manifold M α . We have the following exact sequence of vector spaces: Because T x W u (x) is oriented by o α and T x M is also oriented, this sequence orients N x W u (x) ∼ = T x W s (x). The stable manifolds are therefore also oriented. The stable manifolds of (f α , g α ) are precisely the unstable manifolds of (−f α , g α ) which we orient by the above exact sequence. We denote this choice of orientation of the unstable manifolds of (−f α , g α ) by o α Definition 8.1. Let Q α be a Morse datum on an oriented manifold. The dual Morse datum Q α is defined by Under our compactness assumptions the Morse complex is finitely generated. Hence the dual complex is finitely generated. A basis of C * (Q α ) is the dual basis given, for x ∈ Crit f α , by η x (y) = 1, x = y, 0, x = y.
Note that a critical point of f α of index |x| is a critical point of −f α of index m α − |x|. Define the Poincaré duality map PD k : C k (Q α ) → C m α −k ( Q α ) by PD k (x) = η x .
The negative gradient flow of Q α is minus the negative gradient flow of Q α . Quotienting out the induced R actions, we see that the minus sign disappears and that n(x, y; Q α ) = n(y, x; Q α ). Then n(x, y; Q α ) η y = PD k−1 ∂ k x.
Vol. 16 (2014) Since the duality map commutes with the canonical isomorphisms, and the gradient flows of Q α and Q α are isolated homotopic, this gives duality of the Morse complex of the manifold; i.e., there exists a Poincaré duality map PD k : HM k (M α ) → HM m α −k (M α ).

Duality in local Morse homology
Recall that a closed subset C of a manifold M is orientable if there exists a continuous section in the orientation bundle over M ; cf. [6, Chapter VI.7]. Let P α be a local Morse datum. The local Morse datum is orientable if S α = Inv(N α , ψ α ) is orientable. A choice of a section of the orientation bundle is an orientation of P α . If M α is an oriented manifold, all closed subsets are oriented, thus on an orientable manifold all local Morse data are orientable.
If a local Morse datum P α is oriented, we can define the dual local Morse datum P α = M α , −f α , g α , N α , o α as before. Again we have Poincaré duality isomorphisms PD k : HM k (P α ) → HM m α −k P α .
A crucial difference is now that the gradient flow of P α is in general not isolated homotopic to the gradient flow of P α , thus HM m α −k Q α ∼ = HM m α −k (Q α ).
Theorem 8. 3. Let (f, g, N ) be a triple of a function, a metric and an isolating neighborhood of the gradient flow. Assume that S = Inv(N, ψ) is oriented. Then there exist Poincaré duality isomorphisms