HOMOTOPICAL DYNAMICS: SUSPENSION AND DUALITY

OCTAVIAN CORNEA

Abstract. Flow type suspension and homotopy suspension agree for attractor-repellor homotopy data. The connection maps associated in Conley index theory to an attractor-repellor decomposition with respect to the direct flow and its inverse are Spanier-Whitehead duals in the stably parallelizable context and are duals modulo a certain Thom construction in general.

1. Introduction.

The Morse inequalities as well as their degenerate counterpart, the Lusternik-Schnirelmann inequality [15], are early examples of applications of homotopy theory to the study of smooth flows. More recently, the subject has developed along three directions:

-Various numerical homotopical invariants related to the Lusternik-Schnirelmann category have been investigated intensively.

-Particular atention has been given to Morse-Smale flows where the use of homotopical methods has had much success.

-The index theory of Conley [4] has introduced homotopical methods in the analysis of general flows.

The last few years have seen a certain convergence of these topics having at the center the notion of stabilization. For example, given a 2-connected manifold M, for k large enough, one can construct functions on the manifold M × D^{k} pointing inwards on the boundary and with no more than cat(M) + 2 critical points [6]. Attaining the best possible lower bound, cat(M) + 1, depends, for now, on showing that the L.S.-category and another homotopical invariant, the cone-length of M agree [5]. This seems likely for closed manifolds [9] but in general it is not true and recently have been constructed [10] new homotopical examples for which the two invariants differ.

Conversely, one can look for critical point estimates for functions that are defined on M × D^{k} with M closed and which are pointing out on M × D^{k−t−1} × S^{t} and in on the complement of this set in ∂(M × D^{k}). The lower bound in this case is cat(M, M × S^{t}) [8]. For any CW -complex, this invariant can be shown [17] to be strictly bigger than a stable version of the L. S.-category σ − cat (defined as the least n for which a suspension of the n-th filtration in Milnor’s classifying construction applied to ΩM admits a homotopy section [22]).

Date: December 1997.

1991 Mathematics Subject Classification. Primary 58F25, 55P25; Secondary 57R70, 58F09.

These functions are exactly the functions quadratic at infinity studied in relation to the Arnold conjecture in symplectic geometry. They are at the center of the ”stable” Morse and L.S. theory of Eliashberg and Gromov [11]. Because of the Arnold conjecture one does then expect that cat(M, M × S^{t}) ≥ cat(M) + 1. This inequality is known to hold in many cases and it implies [8] cat(M × S^{t}) = cat(M) + 1. In its turn, the validity of this equality for all CW -complexes is known as the Ganea conjecture. The conjecture has been recently disproved [14]. However, for symplectic, closed manifolds of symplectic form ω and with ω|_{π_{2}(M)} = 0 by [18] one has σ − cat(M) = cat(M) = cat(M, M × S^{t}) − 1 = dim(M).

Finally, again in relation to the Arnold conjecture and Floer homology, Cohen, Jones and Segal [3] have introduced a stable homotopy point of view in studying Morse-Smale flows (see also [13]).

All of this suggests that in the flow context suspending should be understood to mean taking the product of the given flow with the gradient flow of a quadratic form on R^{n}.

The present paper is concerned with the next step in the study of the homotopical properties of flows: attractor-repellor data. We address two main issues in this context.

• Understand the behaviour of attractor-repellor data under suspension.

• Determine in what measure and in what sense the homotopical information provided by a smooth flow and its inverse are redundant.

The key point is that suspension is central in dealing with the second matter.

Attractor-repellor data is best understood for Morse-Smale functions. Passage through a critical point of a Morse-Smale function corresponds to the attachement of a cell. Hence, for two consecutive critical points one has a relative attaching map d : S^{q−1} ⟶ S^{k} where k is the index of the first point and q that of the second. This map has been described by Franks in [12]. The negative of the original function being also Morse-Smale there is also a relative attaching map d′ : S^{n−k−1} ⟶ S^{n−q} with n being the dimension of the supporting manifold M. Franks has also shown that, if M is stably parallelizable, then d and d′ are stably the same up to sign.

Conley index theory allows one to define analogues of these maps for general attractor-repellor pairs of some general flow γ (which will be supposed here to be defined on a smooth manifold and to be itself smooth). Assume S is an isolated invariant set of γ and (A, A^{∗}) is an attractor-repellor decomposition of S. Then one has the map δ : c_{γ}(A^{∗}) ⟶ Σc_{γ}(A) called connection map of the attractor-repellor pair (A, A^{∗}). Let −γ be the inverse flow of γ, −γ_{t}(x) = γ_{−t}(x). As (A^{∗}, A) is an attractor-repellor decomposition of S with respect to −γ we also have the inverse connection map δ′ : c_{−γ}(A) ⟶ Σc_{−γ}(A^{∗}).

Here c_{γ}(−) is the Conley index of the respective invariant set. Specialized to the Morse-Smale case and consecutive critical points δ = Σd and δ′ = Σd′.

The results of the paper imply two main things:

• Flow type suspension and homotopy suspension agree for attractorrepellor homotopy data.

• The maps δ and δ′ are Spanier-Whitehead duals in the stably parallelizable context and are duals modulo a certain Thom construction in general.

The duality result implies the (co)-homological results of McCord [16]. In the stably parallelizable case the duality together with the suspension result imply that, up to flow suspension, the direct and inverse attractor-repellor homotopy data are redundant. Conley index theory works quite well for continuous flows on metrizable spaces, hence in singular or stratified contexts and parts of the theory extend even to semiflows. Our suspension result is likely to extend also to the most general setting. Not so the duality. This result seems to be specific to flows on manifolds and shows that Spanier-Whitehead duality is a key ingredient in understanding the related homotopical data. Possibly, the smootheness restriction might be relaxed.

Of course, the result of Franks on the stable equality up to sign of d and d′ is also extended (as Spanier-Whitehead duality of maps between spheres is stable equality up to sign). His paper [12] is closest in spirit to ours even if, due to the fact that for general flows one has no control on stable and unstable manifolds, our methods are forced to be quite different.

In trying to make the paper more easely accessible we have included, in the second section, a relatively detailed recall of Spanier-Whitehead duality and of the main properties of the Conley index.

The third section contains the precise statements and proofs and is organized as follows. We first discuss the Thom construction in the setting of the Conley index. This is needed to prove the suspension result and it also reduces proving the duality to the study of a flow defined on a sphere. For such a flow the duality of the two connection maps is obtained by identifying each of them to a connectant in a certain cofibration sequence and then showing that these two cofibration sequences are dual.

The paper is written in the language of connected simple systems for two reasons. Firstly, the constructions related to the Conley index contain many choices and in this way one is insured of the invariance of the result. Secondly, connected simple systems reflect precisely the homotopical information that can be extracted from the flow.

2. Recalls.

We recall a number of well-known facts, first on Spanier-Whitehead duality and then concerning the Conley index.

2.1. Spanier-Whitehead duality.

2.1.1. Definitions. Two pointed, connected CW -complexes X, X′ are Spanier-Whitehead dual [21],[20] if there is a map μ : X ∧ X′ ⟶ S^{m} such that the slant product μ^{∗}(i_{m})^{∗}/ : H_{q}(X′) ⟶ H^{m−q}(X) gives an isomorphism. The map μ is called an m-duality. This relation behaves well under suspension: if X and X′ are m-duals with duality map μ, then Σ^{k}X and Σ^{q}X′ are m + k + q-duals with respect to the obvious suspension of μ. Assume X, X′ are duals by an m-duality, μ, and Y, Y′ are duals by an m-duality ν. Two pointed maps f : X ⟶ Y, g : Y′ ⟶ X′ are duals if for some k,k′ the following diagram homotopy commutes:

[figure: Σ^{k}X ∧ Σ^{k′}Y′; 1∧g; Σ^{k}Y ∧ Σ^{k′}Y′; ν; f∧1; Σ^{k}X ∧ Σ^{k′}X′; S^{m+k+k′}; μ]

In general, only by allowing k and k′ to be sufficiently large this notion of duality gives a duality isomorphism between (stable) classes of maps D_{m}(μ, ν) : {X, Y } ≈ {Y′, X′}.

2.1.2. Construction of duality maps. Here is one way to obtain duality maps [21]. Assume A ⊂ S^{m+1}, B ⊂ S^{m+1} such that the inclusion induces an isomorphism H_{∗}(B) ⟶ H_{∗}(S^{m+1} − A), and A, B are connected and disjoint. Define a map in the following way : fix a point p ∈ S^{m+1} − (A ⋃ B) and identify S^{m+1} − {p} with R^{m+1}. We then define v : A × B ⟶ S^{m} by v(x, y) = (x − y)/||x − y||. Connectivity implies that this map is null-homotopic when restricted to A ∨ B. It defines a map A ∧ B ⟶ S^{m} which is a duality map. One can identify dual maps in this setting: if A′, B′ is another pair of subspaces of S^{m+1} that satisfies the conditions above and if A ⟶^{i} A′ and B′ ⟶^{i′} B are inclusions, then i and i′ are dual with respect to the duality maps constructed as above.

Spanier-Whitehead duality implies Poincaré duality via the Thom isomorphism.

2.2. The Conley index. We will use here and in the rest of the paper the conventions of [19].

2.2.1. Connected simple systems. Recall first that the pointed homotopy category of pointed topological spaces Ho − T has as objects pointed topological spaces and as morphisms homotopy classes of pointed (continuous) maps. A connected simple system is a subcategory C of Ho−T such that for any two objects X_{1}, X_{2} of C the set mor_{C}(X_{1}, X_{2}) consists of exactly one element. Any map X_{1} ⟶ X_{2} whose homotopy class belongs to mor_{C}(X_{1}, X_{2}) is called a comparison map. Clearly, all comparison maps are homotopy equivalences. A morphism between two connected simple systems X and Y associates to each pair X_{0} ∈ Ob(X) and Y_{0} ∈ Ob(Y) a unique homotopy class of maps X_{0} ⟶ Y_{0} with the obivous compatibility properties. We will sometimes call such a morphism a map of connected simple systems. Connected simple systems form also a category CS. Let X, Y ∈ Ob(CS).

Assume that we have a (non-void) set P of pairs (A, B) with A ∈ Ob(X) and B ∈ Ob(Y) and maps f_{A,B} : A ⟶ B that are defined for each such pair. We say that these maps induce a morphism f : X ⟶ Y if for any two pairs (A, B) and (A′, B′) in P the maps f_{A,B} and f_{A′,B′} commute, up to homotopy, with the obvious comparison maps and f is the unique morphism of connected simple systems represented by the maps f_{A,B}. Of course, defining f is possible even if P has a single element. However, in many of the constructions below the maps f_{A,B} are naturally defined for many pairs (A, B) and it is useful to know that the induced morphism does not depend on the particular choice of the pair. Hence, in the following we will speak of an induced map at the level of connected simple systems by always considering P to be as large as is allowed by the definition of the maps f_{A,B}.

2.2.2. Cofibration sequences. An obvious functor Σ : CS ⟶ CS is defined by taking the supension of spaces and maps. By definition, a cofibration sequence of connected simple systems is a triple of maps X ⟶^{i} Y ⟶^{p} Z ⟶^{δ} ΣX with X, Y, Z ∈ Ob(CS) with the property that there are X_{0} ∈ Ob(X), Y_{0} ∈ Ob(Y), Z_{0} ∈ Ob(Z), a cofibration sequence

X_{0} ⟶^{i_{0}} Y_{0} ⟶^{p_{0}} Z_{0} with [i_{0}] = i, [p_{0}] = p and if δ_{1} : Z_{0} ⟶ ΣX_{0} is the

obvious connectant, then δ = [δ_{1}]. The map δ is called the connection map of the cofibration sequence. The cofibration sequence X_{0} ⟶ Y_{0} ⟶ Z_{0} is called a representing cofibration sequence. The cofibration sequence is determined by the choice of i_{0} (however, because push outs in Ho − T do not coincide, in general, with homotopy push outs, the knowledge of i only does not suffice). For completeness we recall the definition of a cofibration sequence: the pair of maps A ⟶^{j} B ⟶^{r} C form a cofibration sequence if j is a cofibration, C = B/A and r is the obvious collapsing map. A cofibration sequence extends to the right giving a sequence of maps: A ⟶^{j} B ⟶^{r} C ⟶^{δ_{1}} ΣA ⟶^{Σj} ΣB ⟶^{Σr} · · ·.

2.2.3. The Conley index. We recall now a few elements of Conley index theory [4],[19] for the case of a continuous flow γ : N × R ⟶ N , N being a compact metric space. Let S ⊂ N be an isolated invariant set. A pair (N_{1}, N_{0}) of compact sets in N is an index pair for S in N if N_{0} ⊂ N_{1}, N_{1} − N_{0} is a neighborhood of S, S is the maximal invariant set in the closure of N_{1} − N_{0}, N_{0} is positively invariant in N_{1} and, if for x ∈ N_{1} there is some t ≥ 0 such that γ_{t}(x) ∉ N_{1}, then there exists a τ > 0 with γ_{t}(x) ∈ N_{1} for 0 ≤ t ≤ τ and γ_{τ}(x) ∈ N_{0}, [19].

The basic result in this theory is the existence of index pairs inside any isolating neighborhood. Moreover, for any two index pairs of S, (N_{1}, N_{0}) and (N′_{1}, N_{0}) there are flow induced comparison map N_{1}/N_{0} ⟶ N′_{1}/N′_{0}. In the following we need a uniform choice for the comparison maps. We will refer to the maps defined by the formula of Lemma 4.7 of [19] as the standard comparison maps. This gives the set of quotients N_{1}/N_{0} the structure of a connected simple system denoted by c_{γ}(S) and called the Conley index of S with respect to γ.

Assume that S admits an attractor-repellor decomposition, denoted (S; A, A^{∗}), of attractor A and repellor A^{∗} (this means that if x ∈ S − (A ⋃ A^{∗}) then lim_{t→∞} γ_{t}(x) ∈ A and lim_{t→−∞} −γ_{t}(x) ∈ A^{∗}). We have a cofibration sequence of connected simple systems denoted by

c_{γ}(S; A, A^{∗}): c_{γ}(A) ⟶^{i} c_{γ}(S) ⟶^{p} c_{γ}(A^{∗}) ⟶^{δ} Σc_{γ}(A) such that δ

as well as i and p are flow defined [19]. In this case δ is called the connection map of the given attractor-repellor decomposition.

Moreover, for any triple N_{0} ⊂ N_{1} ⊂ N_{2} with the inclusions being inclusions of NDR’s and such that (N_{2}, N_{1}) is an index pair of A^{∗}, (N_{1}, N_{0}) is an index pair of A and (N_{2}, N_{0}) an index pair of S the cofibration sequence N_{1}/N_{0} ⟶ N_{2}/N_{0} ⟶ N_{2}/N_{1} represents c_{γ}(S; A, A^{∗}).

For M ⊂ N we use the notation I(M) for the maximal invariant set inside M.

2.2.4. Duality and connected simple systems. The definition of Spanier-Whitehead duality induces one for simple connected systems. Consider the connected simple system formed by a single space S^{m} and let X, X′ ∈ Ob(CS). We have X ∧ X′ ∈ Ob(CS). The two connected simple systems are Spanier-Whitehead duals if there is a morphism X ∧ X′ ⟶ S^{m} giving a duality map whenever restricted to X_{0} ∧ X′_{0}, X_{0} ∈ Ob(X), X′_{0} ∈ Ob(X′) . A similar definition applies to maps.

Let X ⟶^{i} Y ⟶^{p} Z ⟶^{δ} ΣX and Z′ ⟶^{p′} Y′ ⟶^{i′} X′ ⟶^{δ′} ΣZ′

be two cofibration sequences of connected simple systems. They are called dual cofibration sequences if the pairs of maps (i, i′), (p, p′) and (δ, δ′) are respectively dual. To insure this duality it is enough to find

cofibration sequences of spaces: X_{0} ⟶^{i_{0}} Y_{0} ⟶^{p_{0}} Z_{0} and Z′_{0} ⟶^{p′_{0}} Y′_{0} ⟶^{i′_{0}} X′_{0} representing respectively the two above and with (i_{0}, i′_{0}) and (p_{0}, p′_{0}) respectively duals. Indeed, in this case, by the basic properties of Spanier-Whitehead duality [20], we have that the associated connectant maps δ_{0} : Z_{0} ⟶ ΣX_{0} and δ′_{0} : X′_{0} ⟶ ΣZ′_{0} are Spanier-Whitehead duals. Of course, under these circumstances, one wants also to show that the duality relation obtained does not depend on the choice of representing cofibration sequences.

3. Bundles, suspension and duality.

Let ψ : F ⟶ E ⟶^{p} B be a locally trivial fibration of manifolds (possibly with boundary) with F and B compact, and let γ be a flow on B. A lift of γ to E is a flow γ′ on E such that p(γ′_{t}(x)) = γ_{t}(p(x)) for all x ∈ E and t ∈ R.

3.1. The Thom index. We fix a first construction with the next immediate fact.

Lemma 3.1. Let S be an isolated invariant set of the flow γ : B × R ⟶ B and let γ′ be a lift of γ to E. Then, S′ = p^{−1}(S) is an isolated, invariant set of γ′ and there is an induced morphism c_{γ′}(S′) ⟶ c_{γ}(S) depending only on ψ and γ. This morphism is natural with respect to attractor-repellor pairs.

Proof. Let (N_{1}, N_{0}) be an index pair of S. We first notice that S′ = I(f^{−1}(\overline{N_{1} − N_{0}})). Moreover, the pair (f^{−1}(N_{1}), f^{−1}(N_{0})) is an index pair for S′ (with respect to the flow γ′, of course). Denote N′_{1} = f^{−1}(N_{1}) and N′_{0} = f^{−1}(N_{0}). There is a map of pairs (N′_{1}, N′_{0}) ⟶ (N_{1}, N_{0}) which induces a map of quotients: p_{N} : N′_{1}/N′_{0} ⟶ N_{1}/N_{0}. Notice that this map induces one of connected simple systems in the following way.

First, let (K_{1}, K_{0}) be another index pair of S and let, as above, (K′_{1}, K′_{0}) be the index pair of S′ with K′_{i} = p^{−1}(K_{i}), i = 0, 1. There are standard, flow defined [19], comparison maps, N_{1}/N_{0} ⟶ K_{1}/K_{0} and N′_{1}/N′_{0} ⟶ K′_{1}/K′_{0}. By construction, these maps commute with the projections p_{N} and p_{K}.

Now assume that (N′′_{1}, N′′_{0}) is another index pair of S′ (and not necessarily one obtained as a preimage of one of S) and consider also an index pair of S, (T_{1}, T_{0}). We define a map N′′_{1}/N′′_{0} ⟶ T_{1}/T_{0} as the

composite N′′_{1}/N′′_{0} ⟶ N′_{1}/N′_{0} ⟶^{p_{N}} N_{1}/N_{0} ⟶ T_{1}/T_{0} where the first

and last maps in this composition are the standard comparison maps. The homotopy class of this map does not depend on the choice of the pair (N_{1}, N_{0}).

To prove naturality with respect to attractor-repellor sequences consider an attractor-reppellor decomposition (S; A, A^{∗}) for an invariant set S and a triple N_{0} ⊂ N_{1} ⊂ N_{2} such that the pair (N_{1}, N_{0}) is an index pair for A, the pair (N_{2}, N_{1}) is an index pair for A^{∗} and (N_{2}, N_{0}) is an index pair for S. Following the notations above we denote by ′ the respective preimages by p of these sets. It is clear that A′ and (A^{∗})′ form an attractor-repellor pair. The result now follows from the commutativity of the diagram:

[figure: N′_{1}/N′_{0}; N′_{2}/N′_{0}; N′_{2}/N′_{1}; N_{1}/N_{0}; N_{2}/N_{0}; N_{2}/N_{1}]

The vertical arrows are given, as above, by projection. The rows are the cofibration sequences that represent the attractor-repellor cofibration sequences of connected-simple systems. Moreover, note that the right vertical map up is induced on cofibres by the left and middle projections and this implies that, by extending the cofibrations to the right, we will continue to get commutative squares.

For a pointed space X let C(X) be the (reduced) cone on X.

Definition 1. With the notations in the proof of Lemma 3.1, the Thom index of S with respect to ψ and γ is the connected simple system \overline{c}_{γ}^{ψ}(S) having as objects the spaces (N_{1}/N_{0}) ⋃_{p_{N}} C(N′_{1}/N′_{0}) and with the morphisms given by the homotopy classes of the maps (N_{1}/N_{0}) ⋃_{p_{N}} C(N′_{1}/N′_{0}) ⟶ (K_{1}/K_{0}) ⋃_{p_{K}} C(K′_{1}/K′_{0}) induced by the standard comparison maps.

Lemma 3.2. In the context of the lemma above and assuming that (S; A, A^{∗}) is an attractor-repellor decomposition there is a cofibration sequence of connected simple systems: \overline{c}_{γ}^{ψ}(A) ⟶ \overline{c}_{γ}^{ψ}(S) ⟶ \overline{c}_{γ}^{ψ}(A^{∗}) ⟶ Σ\overline{c}_{γ}^{ψ}(A).

Proof. The naturality given by the previous lemma implies the existence of the maps in the statement. These maps give a cofibration sequence because of the general fact that if in the diagram:

[figure: A′; B′; C′; A; B; C; A′′; B′′; C′′]

the top two horizontal rows as well as the columns are cofibration sequences, then the third horizontal row is also a cofibration sequence. In our case, the top row contains the spaces representing the Conley index of γ′ . The spaces in the middle row are obtained from the spaces giving the Conley indexes with respect to γ by pasting the reduced cylinders of the projections. The vertical maps are inclusions in the free ends of these cylinders.

Assume that (S; A, A^{∗}) is an attractor-repellor decomposition for the flow γ. Recall that we denote by c_{γ}(S; A, A^{∗}) the attractor-repellor cofibration sequence associated to this decomposition. We will denote by \overline{c}_{γ}^{ψ}(S; A, A^{∗}) the cofibration sequence given by Lemma 3.2.

Remark 1. Notice that this also applies to the limit case when the total space of ψ is void. In this case, \overline{c}_{γ}^{ψ}(S; A, A^{∗}) is isomorphic to c_{γ}(S; A, A^{∗}) (recall that X/∅ is the disjoint union of X and a point).

We discuss now the following particular case. Consider a riemannian fibre bundle η : R^{n} ⟶ T ⟶ B. As before let γ be a flow on B and let γ′′ be a lift of it to T (of course, such lifts always exist). Assume that there is a fixed quadratic form q : R^{n} ⟶ R and that there exist local charts U_{i} ⊂ B such that the restriction of η to each U_{i} is trivial and, moreover, with respect to these trivializations the vector field associated to γ′′ splits as a direct sum of the vector field associated to γ and −∇q; also assume that q is compatible with the metric.

There is a particular locally trivial fibration that is associated to this context. Consider the associated sphere bundle of η: S(η) ⟶^{p} B. Denote by e(q) the locally trivial fibration e(q) ⟶ B whose total space is the subset of S(η) where q is negative or null and whose projection is the restriction of p. On e(q) there is an obvious lift of γ obtained by projecting first γ′′ on S(η) and then by restriction.

Proposition 3.3. Let S be an isolated invariant set of γ. The Thom index \overline{c}_{γ}^{e(q)}(S) is isomorphic to c_{γ′′}(S) and this identification is natural with respect to attractor-repellor pairs. More precisely, if (S; A, A^{∗}) is an attractor-repellor decomposition, then \overline{c}_{γ}^{e(q)}(S; A, A^{∗}) is isomorphic to c_{γ′′}(S; A, A^{∗})

Proof. Suppose (N_{1}, N_{0}) is an index pair for S with respect to the flow γ. Let N′_{i} be the total space of the restriction of the unit disk bundle associated to η restricted to N_{i}, i = 0, 1. Let e(N_{i}) be the total space of the restriction to N_{i} of e(q). It is easy to see that the pair (N′_{1}, N′_{0} ⋃ e(N_{1})) is an index pair for S with respect to the flow γ′′. Notice that if D^{n} is the unit disk in R^{n} and S^{n−1} its boundary, then the region where q is negative or null in D^{n} has the structure of a cone over the boundary. This implies that the region L ⊂ N′_{1} where q is negative or null is homeomorphic to N_{1} ⋃(e(N_{1}) × [0, 1]) the identification used making x × 0 correspond to the point p(x) for each x ∈ e(N_{1}). On the other hand we have a map of pairs (L, (N′_{0} ⋂ L) ⋃ e(N_{1})) ⊂ (N′_{1}, N′_{0} ⋃ e(N_{1}). This induces a map between the respective quotients: t : L/((N′_{0} ⋂ L) ⋃ e(N_{1})) ⟶ N′_{1}/(N′_{0} ⋃ e(N_{1})). Because of the description of L above the domain of this map is a space in \overline{c}_{γ}^{e(q)}, the target is a space in c_{γ′′}(S). This map is a homotopy equivalence as both inclusions L ⊂ N′_{1} and N′_{0} ⋂ L) ⋃ e(N_{1}) ⊂ N′_{0} ⋃ e(N_{1}) are homotopy equivalences (the first because the domain and the target are homotopy equivalent to N_{1} and the second for a similar reason).

It is easy to verify, as in the lemma above, that the map t induces an isomorphism of connected simple systems and that this morphism is natural with respect to attractor-repellor pairs.

Remark 2. a. The proposition implies that the Thom index and the corresponding cofibration sequence are completely determined by the homotopical data extracted from the flow γ′′.

b. It is easy to see that the statement and proof above can be extended in many ways. For example, to the case when q is a more complicated germ of an isolated but still ”reasonable” singularity (see [6] for a definition of such reasonable critical points).

c. Assume that B is connected and that the projection of γ′′ on each fibre of η has 0 as single compact, invariant set. The continuation properties of the Conley index imply that the Conley index of 0 with respect to the projection of γ′′ on a given fibre does not depend on the choice of fibre. Moreover, it is to be expected that one has an induced locally trivial fibration of base B and of fibre the Conley index of 0 for some projection of γ′′ on some fibre. This may also be used to estimate the Conley index of S with respect to γ′′.

Corollary 3.4. In the context of the proposition above, if η is trivial in a neighborhood of S, then we have c_{γ′′}(S) ≃ \overline{c}_{γ}^{e(q)}(S) ≃ Σ^{k}c_{γ}(S) and if (S; A, A^{∗}) is an attractor-repellor decomposition c_{γ′′}(S; A, A^{∗}) ≃ \overline{c}_{γ}^{e(q)}(S; A, A^{∗}) ≃ Σ^{k}c_{γ}(S; A, A^{∗}) where k is the index of the quadratic form q.

Proof. Let K = {x ∈ D^{n} : q(x) ≤ 0} and let H = K ⋂ ∂D^{n}. Notice that H ≃ S^{k−1} and K is contractible. With the notations in the proof of the proposition, we have L = N_{1} × K, N′_{0} ⋂ L = N_{0} × K, e(N_{i}) = N_{i} × H. The statement is an immediate consequence of the construction of the Thom index and of the fact that the cofibre of the projection A×S^{k−1} ⟶ A is, in a natural way, homotopy equivalent to Σ^{k}A ∨ S^{k}. With this identification we have the commutative diagram below:

[figure: N_{0} × H; N_{1} × H; N_{0} × K; N_{1} × K; Σ^{k}N_{0} ∨ S^{k}; Σ^{k}N_{1} ∨ S^{k}]

the columns being (homotopy) cofibration sequences. By definition the Thom index of S is represented by the cofibre of the bottom horizontal map which is Σ^{k}(N_{1}/N_{0}).

Remark 3. The above corollary is well-known at the space level (in that case being a consequence of the product formula for the Conley index [4]). For our purposes the key consequence of this corollary is the description of the connection map of γ′′ in terms of that of γ. In particular, when the fibration is trivial they are related by suspension.

3.2. Duality. Assume now that B is a smooth, compact manifold of dimension n that is embedded in a sphere S^{n+k}. Let η : R^{k} ⟶ T ⟶ B be its normal bundle. Assume a riemannian metric fixed on the total space of η. Let S(η) be the induced unit sphere bundle. We recall that we denote by −γ the inverse flow of γ: −γ_{t}(x) = γ_{−t}(x).

Theorem 3.5. Let (S; A, A^{∗}) be an attractor-repeller pair for γ. The cofibration sequences \overline{c}_{γ}^{S(η)}(S; A, A^{∗}) and c_{−γ}(S; A^{∗}, A) are Spanier-Whitehead duals by a duality map that depends only on the embedding of B in S^{n+k}.

Proof. We will construct a duality between two cofibration sequences representing those of the statement. Afterwards we show that the induced duality of cofibrations of connected simple systems does not depend on the choice of the representing sequences. The proof has four steps.

Reduction to a flow on S^{n+k}. Let U be a tubular neighborhood of B. In the following we identify it to T. Let q : B ⟶ R be a smooth function which locally has the form q(x, y) = −||x||^{2} where x is the coordinate along the fibre of η and y the coordinate along the base. Consider on U the flow γ′′ induced by the sum of the canonical horizontal lift of γ and of the flow induced by −∇q. Notice that e(q) = S(η) and e(−q) = ∅. Applying Proposition 3.3 to γ′′ we obtain isomorphisms \overline{c}_{γ}^{S(η)}(S; A, A^{∗}) ≃ c_{γ′′}(S; A, A^{∗}), c_{−γ′′}(S; A^{∗}, A) ≃ \overline{c}_{−γ}(S; A^{∗}, A) ≃ c_{−γ}(S; A^{∗}, A). This reduces the proof to showing that c_{γ′′}(S; A, A^{∗}) and c_{−γ′′}(S; A^{∗}, A) are Spanier-Whitehead duals.

Construction of some special index pairs. Extend γ′′ to a flow on S^{n+k} which is constant outside a neighborhood of U. We continue to denote this flow by γ′′. Let S_{1} be the maximal attractor of γ′′ that does not contain S , let R_{1} be the complementary repellor. Let S_{2} be the attractor of γ′′ obtained as the union of S_{1} with the set of the points of A or situated on flow lines originating in A and let R_{2} be the complementary repellor. Similarly, let S_{3} be given by the union of S_{2} with the set of the points of A^{∗} or situated on flow lines originating at A^{∗}, let R_{3} be the complementary repellor. For each pair (S_{i}, R_{i}) for 1 ≤ i ≤ 3 we may construct by classical techniques (see, for example, [1]) smooth Lyapounov functions, f_{i} : S^{n+k} ⟶ R such that f_{i}(S_{i}) = 1 and f_{i}(R_{i}) = 0. By Sard’s theorem we may find regular values a and b of respectively f_{1} and f_{3} such that f_{1}^{−1}(a) and f_{3}^{−1}(b) intersect transversely. Notice that S is the maximal invariant set inside N_{2} = {x : f_{1}(x) ≥ a, f_{3}(x) ≤ b}. Moreover let U_{0} = f_{1}^{−1}(a) ⋂ N_{2} and V_{0} = f_{3}^{−1}(b) ⋂ N_{2}. Then ∂N_{2} = U_{0} ⋃ V_{0} and the pairs (N_{2}, U_{0}) and (N_{2}, V_{0}) are index pairs for S for, respectively, the flows γ′′ and −γ′′. We now choose a regular value c of f_{2} such that f_{2}^{−1}(c) cuts transversely f_{3}^{−1}(b) and such that the inequality f_{1}(x) ≤ a implies f_{2}(x) < c (this is possible because R_{2} ⊂ R_{1}). Let U_{1} = f_{2}^{−1}(c) ⋂ N_{2}, N_{1} = f_{2}^{−1}((−∞, c]) ⋂ N_{2}, L_{1} = N_{2} − (Int(N_{1}) ⋃ U_{0}), G_{1} = \overline{N_{2} − N_{1}} and V_{1} = ∂G_{1} ⋂ V_{0}, U_{2} = U_{1} ⋃(V_{0} − V_{1}). With these notations the pairs (N_{2}, N_{1}), (L_{1}, U_{2}) and (G_{1}, U_{1}) are index pairs of A^{∗} for the flow γ′′; (L_{1}, V_{0}) and (G_{1}, V_{1}) are index pairs for A^{∗} with respect to the flow −γ′′; (N_{1}, U_{0}) is an index pair for A with respect to γ′′; (N_{2}, L_{1}), (N_{1}, U_{2}) are index pairs for A with respect to −γ′′.

Duality. The basic argument that will be used is very classical in nature and is a variant of one that appears in [2]. For completeness we will formulate it as a lemma and we will indicate the idea of the proof.

Lemma 3.6. Assume that M is an n + k manifold with boundary emdedded in S^{n+k} such that ∂M admits a decomposition ∂M = M_{0} ⋃ M_{1} with M_{0} and M_{1} being n + k − 1-dimensional manifolds such that M_{1} ⋂ M_{0} = ∂M_{1} = ∂M_{0}. Then M/M_{0} and M/M_{1} are n + k-Spanier-Whitehead duals.

Proof of the lemma. Embedd S = S^{n+k} in S^{n+k+1} as the boundary of a disk D = D^{n+k+1}. Denote by D′ the complementary disk. Consider a copy of M, M′, in the exterior of D and paste it, in S^{n+k+1}, to D following M_{0}. (We use here and below the existence of collared neighborhoods of M, M_{0}, M_{1} and M_{0} ⋂ M_{1} .) The result of this pasting, M^{∗}, is homotopy equivalent to M/M_{0}. As D′ is the cone over the boundary of D it is easy to see that M′′ = S^{n+k+1} − M^{∗} deforms to the union M′′′ ⋃_{M_{1}} D′′ with D′′ a slightly smaller disk than D′ and M′′′ a copy of M that does not touch M′. Hence, M′′ ≃ M/M_{1}.

Notice that the argument of the lemma can be also used to prove the duality of some maps. Indeed assume that K is an n + k-dimensional submanifold of M such that ∂K decomposes, similarly to ∂M, as K_{0} ⋃ K_{1} and such that K_{0} ⊂ M_{0}, K_{1} separates and is an NDR in M, K_{0} and K_{1} ⋂ ∂M are NDRs in ∂M. Then the maps K/K_{0} ⟶ M/M_{0} and M/M_{1} ⟶ K/K_{1} are Spanier-Whitehead duals. The reason is that, as K_{0} ⊂ M_{0} and K/K_{1} = M/((M − K) ⋃ K_{1}), by the construction of the lemma the two maps are identified to a pair of dual inclusions.

We return now to the proof of the theorem. We intend to show that the cofibration sequences: N_{1}/U_{0} ⟶^{i} N_{2}/U_{0} ⟶^{p} N_{2}/N_{1} and L_{1}/V_{0} ⟶^{i′} N_{2}/V_{0} ⟶^{p′} N_{2}/L_{1} are Spanier-Whitehead duals. Consider first the pair of maps (i, p′). We have N_{2}/L_{1} = N_{1}/U_{2} and the duality is immediate by the above. Similarly, for the pair (p, i′) we have L_{1}/V_{0} = G_{1}/V_{1} and N_{2}/N_{1} = G_{1}/U_{1}.

Duality for connected simple systems. The proof is completed by showing that the duality induced by the two cofibration sequences described above does not depend on the choice of the Lyapounov functions or on the choice of the constants a, b and c. It is easely seen that this statement reduces to proving that if N′_{2}, U′_{0} and V′_{0} are constructed in the same way as, respectively, N_{2}, U_{0} and V_{0} and we also have

N′_{2} ⊂ Int(N_{2}), then the map N′_{2}/U′_{0} ∧ N′_{2}/V′_{0} ⟶^{μ′} S^{n+k} is homotopic to

the composition N′_{2}/U′_{0} ∧ N′_{2}/V′_{0} ⟶^{c∧c^{∗}} N_{2}/U_{0} ∧ N_{2}/V_{0} ⟶^{μ} S^{n+k}

Here μ and μ′ are duality maps (here and above all duality maps given by the lemma are constructed with respect to the same embedding of S^{n+k} in S^{n+k+1}); the maps c : N′_{2}/U′_{0} ⟶ N_{2}/U_{0} and c^{∗} : N′_{2}/V′_{0} ⟶ N_{2}/V_{0} are the standard comparison maps given, respectively, by the connected simple systems c_{γ′′}(S) and c_{−γ′′}(S).

This can be seen as follows. Let N′′_{2} = {x ∈ N_{2} : ∃t ≥ 0, γ′′_{−t}(x) ∈ N′_{2}\}, U′′_{0} = N′′_{2} ⋂ U_{0} and V′′_{0} = \overline{∂N′′_{2} − U′′_{0}}. Notice that U′_{0} and U′′_{0} are homeomorhpic as well as the pairs V′_{0}, V′′_{0} and N′_{2}, N′′_{2}. The comparison map c factors using these homeomorphisms N′_{2}/U′_{0} ⟶ N′′_{2}/U′′_{0} ⟶^{c′′} N_{2}/U_{0} with c′′ being induced by the inclusions. We apply the general duality argument in the lemma to obtain that c′′ is dual to the map N_{2}/V_{0} ⟶ N′′_{2}/V′′_{0}. This map is clearly homotopic to the comparison map c′′′ : N_{2}/V_{0} ⟶ N′′_{2}/V′′_{0}.

Hence, if we denote by c_{1}^{∗} the comparison map which is the homotopy inverse of c^{∗} we obtain that c_{1}^{∗} and c are duals and this implies the homotopy we look for.

Corollary 3.7. The two connection maps \overline{δ} : \overline{c}_{γ}^{S(η)}(A^{∗}) ⟶ Σ\overline{c}_{γ}^{S(η)}(A) and δ^{∗} : c_{−γ}(A) ⟶ Σc_{−γ}(A^{∗}) are Spanier-Whitehead duals. In particular, if η is trivial, then δ^{∗} is dual to the connection map δ : c_{γ}(A^{∗}) ⟶ Σc_{γ}(A).

Remark 4. This clearly extends the relation, discovered by Franks [12], between the two connection maps in the case when A and A^{∗} are two succesive critical points of a Morse-Smale flow. The corollary also implies, via the Thom isomorphism, the (co)-homological results of McCord [16]. It also provides an extension of the duality result for succesive reasonable critical points of [7]. However, in the Morse-Smale case the understanding of the geometry of the problem is better because of the explicit description of the stable and unstable manifolds of the critical points. Also, more precise information can be deduced in the case of succesive reasonable critical points with η non trivial. In that case η is trivial in small enough neighborhoods of A and of A^{∗}, hence Σ^{k}c_{γ}(A) ≃ \overline{c}_{−γ}^{S(η)}(A) and similarly for A^{∗}. One can then prove that δ and δ^{∗} are Spanier-Whitehead duals modulo some twisting depending in a precise way on η.

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Université de Lille 1, U.F.R. de Mathématiques & U.R.A. 751 au CNRS, 59655 Villeneuve D’Ascq, France

E-mail address: cornea@gat.univ-lille1.fr
URL: http://www-gat.univ-lille1.fr/~cornea/octav.html