Contemporary Mathematics

Spanier-Whitehead Duality and Critical Points

Octavian Cornea

Abstract. We study the relative attaching map associated to two consecutive critical points in a context considerably more general than the Morse one. We prove that this map behaves well when suspended and that the relative attaching maps with respect to the direct and reverse flows are Spanier-Whitehead duals modulo some twisting coming from the normal structure.

1. Introduction

Passage through a non-degenerate critical point of a smooth function f corresponds to the attachment of a cell. To a couple of consecutive non-degenerate critical points one associates a relative attaching map. There is a second attaching map corresponding to the same pair of critical points but with respect to the function −f. What is the relation between these two maps ?

This question has been considered by Franks [7] who showed that in the Morse-Smale case and if the manifold is stably parallelizable, then the stable classes of these maps coincide up to sign.

It turns out that a better understanding of this problem passes through a study of a more general situation than the Morse one.

Here we look to a class of relatively general, isolated, critical points that we call reasonable. Passage through such a critical point corresponds to the attachment of a cone over a space generally more complicated than a sphere. A relative attaching map is again associated to a pair of consecutive reasonable critical points.

In this context, let k and k′ be the homotopy classes of the attaching maps associated to f and, respectively, to −f for the same pair of reasonable consecutive critical points. We prove that there is a space P and a self-homotopy equivalence u of P such that (some suspension of) k factors through P as a composition k_{1} ∘ k_{2} and k′ and k_{1} ∘ u ∘ k_{2} are Spanier-Whitehead duals.

The map u induces the identity in homology (with Z/2 coefficients) and depends only on the normal structure coming from the embedding of our manifold in a highdimensional sphere. The construction of u is explicit and is shown to be related to the J-homomorphism.

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

This result is new even in the Morse case. Of course, if the manifold is stably parallelizable the two attaching maps are Spanier-Whitehead duals. This implies the result of Franks and shows that it is valid for Morse functions (and not only Morse-Smale ones).

The homological consequence of this duality is a special case of the result obtained by McCord [10] in the general context of the Conley index theory.

One key difference with respect to the Morse case is that, for a reasonable critical point, the stable and unstable ”manifolds” can be quite complicated sets. Because of this, to extend the results form the Morse case, one has to use instead certain thickenings (called cyllindrical neighborhoods) of the union of these two sets. This also forces the proof of the duality result to be different from an extension of the method used by Franks. On the other hand, Spanier-Whitehead duality appears in a natural way in this setting.

There are two main ideas that underline this paper. Firstly, from the homotopy angle, many results that seem specific to the non-degenerate context actually hold for a much larger class of critical points.

Secondly, much of the homotopical information one usually extracts from gradient flows gains in significance after passage to the stable category.

Here is the plan of the paper.

Section 2 starts by recalling the definition and some basic properties of reasonable critical points. We then construct a special type of neighborhood of the invariant set formed by a pair of consecutive critical points together with the flow lines that join them. The setting here is that of the Conley index theory [2]. This is used to define a connection map for a couple of reasonable, consecutive critical points in terms of local information on the flow. This map coincides in relevant cases with the homotopy class of the attaching map mentioned before but it has meaning even when such a natural identification is not possible.

The first result of section 3 concerns a product formula. It implies that the effect of adding a quadratic function to f is to suspend the connection map. Section 3 ends with the proof of the duality. This proof is based on the explicit constructions of the second section and on the product formula.

Acknowledgement. I am indebted to Mark Mahowald who helped me understand the significance of the normal structure.

2. The connection map.

From now on let M be a smooth, riemannian, compact, n-dimensional manifold with possibly non-void boundary. We fix a few basic notions below.

Reasonable critical points and the A-index. An isolated critical point P of a smooth function f : M ⟶ R is reasonable [3] if for a sufficiently small, open disk B around P the pair ((f^{−1}(f(P))−P) ⋂ B, P) is a Whitney stratification. Critical points that are reasonable are abundant: an isolated critical point P of a function analytic with respect to some chart around P is in this class. The homotopy type of the set f^{−1}(−∞, f(P)] ⋂ ∂B is called the A-index of P with respect to f, A_{f}(P). All finite CW -complexes can appear as the A-index of some reasonable critical point. The A-index of a non-degenerate critical point of index k is S^{k−1}.

The direct and inverse flows. The smooth function f : M ⟶ R induces two flows on M: the direct flow, induced by the vector field −∇f and the inverse flow, induced by ∇f. We denote by γ_{t}(x) the direct flow.

If P is a reasonable critical point of f, then the A-indexes of P with respect to f and to −f are clearly represented by complements in S^{n−1}. In particular, when connected, they are Spanier-Whitehead duals [14,15].

Cylindrical neighborhoods. Suppose P is a reasonable critical point of f : M ⟶ R and let a = f(P). Denote by B(P, ϵ) the closed disk of radius ϵ around P.

For ϵ, δ, δ′ ∈ R^{+} sufficiently small let K(P; ϵ; δ, δ′) ⊂ M consist of those points x that satisfy:

1. a − δ ≤ f(x) ≤ a + δ′.

2. One of the following three possibilities holds:

(a) there is a t ∈ R such that γ_{t}(x) ∈ f^{−1}(a) ⋂ B(P, ϵ).

(b) lim_{t→∞}γ_{t}(x) = P.

(c) lim_{t→−∞}γ_{t}(x) = P.

Variants of this type of neighborhoods have been used often [3,4,5,6,11,16].

The Whitney stratification condition in the definition of a reasonable critical point implies [3] that for τ smaller than some constant the intersection of f^{−1}(a) and S(P, τ) = ∂B(P, τ) is transverse and by consequence f^{−1}(a) ⋂ B(P, τ) is the cone over its boundary. This shows that, when ϵ, δ, δ′ are smaller than some fixed constants, K(P; ϵ; δ, δ′) is a contractible neighborhood of P, all these neighborhoods are homeomorphic, each neighborhood of P contains one of this type, ∂(K(P; ϵ; δ, δ′)) ≃ S^{n−1} (≃ denotes homotopy equivalence). One also sees that the sets A_{K} = K(P; ϵ; δ, δ′) ⋂ f^{−1}(a − δ) are all homeomorphic. We also denote B_{K} = K(P; ϵ; δ, δ′) ⋂ f^{−1}(a + δ′). These sets have similar properties to the A_{K}’s. Let W_{K} be the ”vertical” walls of K(P; ϵ; δ, δ′): it consists of those points of K(P; ϵ; δ, δ′) situated on a flow line that passes through S(P, ϵ) ⋂ f^{−1}(a). Notice that A_{K} ⋂ W_{K} = ∂A_{K}, B_{K} ⋂ W_{K} = ∂B_{K} and W_{K} is homeomorphic to ∂A_{K} × [−1, 1] (when it is non-empty). We will always consider that the −1 end of this cobordism is in A_{K} and the +1 end in B_{K}.

The homotopy type of A_{K} coincides with the A-index of P relative to f, A_{f}(P) ≃ f^{−1}(−∞, a] ⋂ S(P, τ) (and similarly B_{K} ≃ A_{−f}(P) ). Indeed, the manifold (A_{K} ⋃ W_{K}) ⋂ f^{−1}(−∞, a] is invertibly cobordant (mod boundary) to f^{−1}(−∞, a] ⋂ S(P, τ). In particular they are homotopy equivalent [8]. On the other hand (A_{K} ⋃ W_{K}) ⋂ f^{−1}(−∞, a] ≃ A_{K}.

The pair (K(P; ϵ; δ, δ′), A_{K}) is an index pair (in the sense of Conley index theory [2,12]) for P viewed as an invariant set for the direct flow. Hence, the homotopy type of the quotient K(P; ϵ; δ, δ′)/A_{K} ≃ ΣA_{K} coincides with the Conley index of P.

Cone attachments. Assume that (W; V_{0}, V_{1}) is a smooth compact cobordism and that f : W ⟶ R is a smooth function that is minimal, regular and equal to a−δ on V_{0} and maximal, regular and equal to a+δ′ on V_{1}. Assume also that f has exactly one critical point P which is reasonable and of critical value a. Then, one sees immediately that W admits as a deformation retract V_{0} ⋃ K(P; ϵ; δ, δ′) for ϵ, δ and δ′ small enough. This space is homotopy equivalent to the homotopy cofibre of the inclusion A_{K} ⟶ V_{0}. Hence, we have a homotopy cofibration sequence A_{f}(P) ⟶ V_{0} ⟶ W. This means that, as mentioned in the introduction, the passage through a reasonable critical point corresponds to the attachment of a cone (over A_{f}(P)).

Consecutive critical points. Two critical points P and Q of f : M ⟶ R are called consecutive if f(P) < f(Q), all the flow lines (of the direct flow) connecting Q to P are not broken and there are no flow lines connecting Q to P that pass through the boundary of M.

Remark 1. When does the existence of a cobordism (W^{n+1}, V_{0}, V_{1}) for which there is a homotopy cofibration sequence Z ⟶ V_{0} ⟶ W imply the existence of a smooth function f : W ⟶ R maximal, regular and constant on V_{1}, minimal, constant and regular on V_{0} and having a single critical point ? If the dimension of Z as a CW-complex is strictly smaller than half the dimension of V_{0} and if Z and V_{0} are both simply connected, then one can show [3] following a skeletal decomposition of Z that there is a thickening, Z′ of Z in V_{0} and a disk D^{n+1} in W^{n+1} such that D^{n+1} ⋂ V_{0} = Z′, ∂D^{n+1} −Int(Z′) is a neat submanifold of W and the topological manifold W −(Int(D^{n+1}) ⋃ Int(Z′)) is homeomorphic to V_{1} ×[0, 1]. Applying now the construction of Takens [16] we obtain the required function (a slight modification of his construction allows one to assume that the critical point constructed in this way is also reasonable).

Assume X is a 2-connected, k-dimensional CW -complex having Lusternik-Schnirelmann category m. Applying the argument above inductively following a cone decomposition of X one obtains that for any smooth, compact manifold M of the homotopy type of X, having simply connected boundary and of dimension ≥ 2k there is a smooth function f : M ⟶ R constant, maximal and regular on ∂M with at most m + 2 critical points.

The classical inequality of Lusternik and Schnirelmann shows that any function as above has at least m + 1 critical points. The relation between the L.S.-category and the attachment of cones is given by the fact that a space of L.S.-category m has the homotopy type of an m + 1 cone. If X is two connected one can even assume that the spaces over which the cones are attached are simply connected.

2.1. A special index pair for consecutive critical points. From now on let f : M ⟶ R be a fixed smooth function. If P is a reasonable critical point of f the pairs (K(P; ϵ; δ, δ′), A_{K}) are very useful index pairs (in the terminology of the Conley index theory) one advantage being that the set K(P; ϵ; δ, δ′) is homotopically simple. We describe below a similar construction for the case of a couple of reasonable critical points.

Given the critical points P, Q we denote by I(P, Q) the invariant set of the direct flow formed by P and Q together with all the points situated on the flow lines that join them.

Proposition 2.1. For any pair P, Q of consecutive, reasonable critical points there exists an index pair (N, E) of I(P, Q) with N being homeomorphic to a union K_{1} ⋃_{Z×{0}} ⋃ Z × [0, 1] ⋃_{Z×{1}} K′ with K_{1} a cylindrical neighborhood of P, K′ a cylindrical neighborhood of Q and Z a compact topological manifold. In particular, N is homotopy equivalent to a (non-reduced) suspension or to S^{0}.

Proof. It is natural to try to construct N as the union of three sets: one of the type of K(Q; − − −) the second of the type of K(P; − − −) and the third consisting of the points situated on the flow lines that join the first two. The construction is not so simple because we have to control better the properties of the third set. In fact, we will write it as Z × [0, 1] such that for all p ∈ Z, p × [0, 1] is a segment of a flow line with the 0 end in B_{K(P;..)} and the 1 end in A_{K(Q;..)}; with Z a compact, topological manifold with boundary ∂Z = ∂_{1}(Z) ⋃ ∂_{2}(Z) satisfying ∂_{1}(Z) × {1} ⊂ ∂A_{K(Q;−−−)} and ∂_{2}(Z) × {0} ⊂ ∂B_{K(P;−−−)}.

First, notice that if there are no flow lines connecting Q to P the disjoint union of one index pair for Q and one for P (both of type K(−; −; −, −)) gives the wanted index pair (Z = ∅ here). In particular, this covers the cases when P is a local maximum or Q a local minimum.

Supose that there are connecting flow lines.

To fix ideas we will first assume that P is not a local minimum.

Let K = K(P; ϵ; δ, δ′) and K′ = K(Q; η; μ, μ′) such that f(P) + δ′ < f(Q) − μ. Consider the set E = {x ∈ A_{K′} : there is a t(x) ∈ R^{+} such that γ_{t(x)}(x) ∈ B_{K}}.

The set E embeds in B_{K} by x ⟶ γ_{t(x)}(x). The key fact is that E is closed when ϵ and η are small enough.

Indeed, suppose that x_{n} is a sequence in E that converges to some y. We intend to show y ∈ E. Consider x′_{n} = γ_{t(x_{n})}(x_{n}). For compacity reasons we may assume that x′_{n} converges also to some point y′.

At this moment it is useful to assume that M is included in a second manifold M′ wich is just M extended by a collar of its boundary and f is defined on all of M′ (it is easy to see that this is not restrictive as P and Q are consecutive with respect to the restriction of f to M − { a sufficiently small collar of ∂M}, in case ∂M = ∅ this argument is not necessary). As B_{K} is closed the following three possibilities are open to y: first, for some T, γ_{T} (y) ∉ M; secondly lim_{t→∞}γ_{t}(y) = R for some critical point R; thirdly, y ∈ E and γ_{t(y)}(y) = y′. Notice that the first can be eliminated. Indeed, if y exits M (before reaching the f(P) + δ′ critical level) there is an entire neighborhood of it that does the same. This contradicts the fact that x_{n} converges to y.

Hence, we are left with the second case. There is a critical point R, different from P and Q, and such that lim_{t→∞} γ_{t}(y) = R. For a very small ν assume that x′′_{i} ∈ f^{−1}(f(R) − ν) are the images of x_{i} by the flow induced embedding. We may assume that x′′_{i} converges to some point y′′. It is easy to see that y′′ is situated on a flow line that originates at R. Reapplying the same reasoning for the sequence x′′_{n} and y′′ we see that the points y and y′ are connected by a broken flow line. Now, if for arbitrarily small neighborhoods K and K′ we can find pairs of points y ∈ K′, y′ ∈ K that are connected by a broken flow line, then, the same type of arguments as above show that, Q and P are connected by a broken flow line. This is not possible as P and Q are consecutive.

We assume in the following that ϵ and η are as small as required by the above argument. Moreover, because of the requirement relative to ∂M in the definition of consecutive critical points, we may assume that ϵ, η are small enough such that no point in A_{K′} is connected to a point in B_{K} by a flow line that passes through the boundary of M.

Let S = E ⋂ ∂A_{K′}. Denote by S′ the image of S by the embedding e : x ⟶ γ_{t(x)}(x). Consider a neighborhood of S in ∂A_{K′}, S_{1}. If this neighborhood is close enough to S it is embedded in f^{−1}(f(P) + δ′) by e as a codimension 1 submanifold S′′. We now consider another neighborhood K_{1} = K(P; ϵ′; δ, δ′) with ϵ′ < ϵ. By possibly pushing a bit the vertical wall of K_{1} we may assume that S′′ and ∂B_{K_{1}} intersect transversely, we denote by I this intersection. As I ⊂ B_{K_{1}} ⊂ B_{K} we have I ⊂ S′.

We will refer in the following to this transversality condition as condition Ψ. Let Z = {x ∈ A_{K′} : there is a t(x) ∈ R^{+} such that γ_{t(x)}(x) ∈ B_{K_{1}}}. Similarly

to E this is also closed. Let ∂_{1}(Z) = Z ⋂ ∂A_{K′}. Of course, Z and ∂_{1}(Z) embedd in B_{K_{1}} (in the same way as E and S do). We denote the image of these embeddings by Z′ and respectively ∂_{1}(Z′). Notice that Z′ is a topological manifold with boundary. In fact, ∂Z′ is equal to the union of ∂_{1}(Z′) and ∂_{2}(Z′) over I with ∂_{2}(Z′) = ∂B_{K_{1}} ⋂ Z′. Indeed, suppose that x ∈ (Z − ∂A_{K′}) and is not an interior point of Z. The fact that there are no flow lines connecting points in A_{K′} to points in B_{K_{1}} by passing through ∂M implies that there is a neighborhood of x in A_{K′} that is embedded by the flow γ in f^{−1}(f(P) + δ′). The image of any such neighborhood by this embedding is not contained in B_{K_{1}}. Hence, x ∈ ∂B_{K_{1}}. We denote by ∂_{2}(Z) the image of ∂_{2}(Z′) by the inverse embedding.

Consider the set W′ = {x ∈ M : f(P) + δ′ ≤ f(x) ≤ f(Q) − μ; there is a t ∈ R such that γ_{t}(x) ∈ Z′}. It is homeomorphic to Z × [0, 1] (with Z′ being identified to Z × {0}). Using this parametrization let ∂_{2}(W′) = ∂_{2}(Z′) × [0, 1] ⊂ W′. We now let N = K_{1} ⋃ W′ ⋃ K′. The exit set is E = A_{K_{1}} ⋃ W_{K_{1}} ⋃ ∂_{2}(W′) ⋃(A_{K′} − Z).

If P is a local minimum the same construction works without any modification except for the definition of the exit set which now becomes E = A_{K′} − Z. This definition makes sense as in this case Z is a union of connected components of A_{K′}. Indeed, it is easy to see that Z is both open and closed in A_{K′}.

Remark 2. There is an index pair of the type described inside any neighborhood of I(P, Q).

Properties of N. For further use we collect the properties of the neighborhood N constructed above : N = K_{1} ⋃ W′ ⋃ K′; W′ = Z × [0, 1], Z × {0} ⊂ B_{K_{1}}, Z × {1} ⊂ A_{K′}; for all z ∈ Z, z × [0, 1] is a segment of a flow line. Moreover, Z is a compact, topological manifold whose boundary is the union of two submanifolds ∂_{1}(Z) and ∂_{2}(Z) such that ∂Z has corners along ∂_{1}(Z) ⋂ ∂_{2}(Z) = ∂(∂_{1}(Z)) = ∂(∂_{2}(Z)) and ∂_{1}(Z) × {1} ⊂ ∂A_{K′}, ∂_{2}(Z) × {0} ⊂ ∂B_{K_{1}}.

We call the set Z as above the incidence set of K′ and K_{1}.

2.2. Construction of the connection map. To define the connection map we use the explicit form of the index pair (N, E) of I(P, Q) described in Proposition 2.1. This map is not defined if Q is a local minimum or P is a local maximum.

The idea of the construction is simple. We consider two neighborhoods K′ = K(Q; η; μ, μ′) and K_{1} = K(P; ϵ′; δ, δ′) as in 2.1 and consider the map that sends a point of A_{K′} that flows to a point in B_{K_{1}} to its impact point viewed as an element of K′/(A_{K_{1}} ⋃ W_{K_{1}}); a point that misses B_{K_{1}} is sent to the basis point in K_{1}/(A_{K_{1}} ⋃ W_{K_{1}}).

We proceed with a formal definition below. We assume that condition Ψ is satified.

We denote by k(K′, K_{1}) the homotopy class of the map A_{K′} ⟶ ∗ if Z is void and, if this is not the case, the homotopy class of the map A_{K′} ⟶ ΣA_{K_{1}} that is given by the composition: A_{K′} ⟶ A_{K}/\overline{(A_{K} − Z)} = Z/∂_{2}(Z) ⟶ Z′/∂_{2}(Z′) ⟶ ⟶ K_{1}/(A_{K_{1}} ⋃ W_{K_{1}}) ⟶ K_{1}/A_{K_{1}} ⟶ ΣA_{K_{1}}. Here, Z′ = Z × {0}, the first map is the collapsing, the second map is the flow induced homeomorphism, the third is induced by the inclusion (Z′, ∂_{2}(Z′)) ↪ (K_{1}, (A_{K_{1}} ⋃ W_{K_{1}})), the fourth map is induced by a homeomorphism that pushes W_{K_{1}} to a collar of the boundary in A_{K_{1}}. Notice that as K_{1} is contractible the homotopy type of the quotient K_{1}/A_{K_{1}} is that of ΣA_{K_{1}}. The fifth map is the homotopy inverse of the map induced by (CA_{K_{1}}, A_{K_{1}}) ⟶ (K_{1}, A_{K_{1}}) (which restricts to the identity on A_{K_{1}}; CX is the cone over the space X; C∅ ≃ ∗; Σ∅ ≃ S^{0}).

We now want to compare the different maps k(K′, K_{1}) constructed as above. Assume R is a reasonable critical point of f. For ϵ < η, δ < τ, δ′ < τ′ there is

a canonical inclusion (and homotopy equivalence)

Y = K(R; ϵ; δ, δ′) ↪ Y′ = K(R; η; τ, τ′)

Moreover, there is a homeomorphism i(Y, Y′) : Y ⟶ Y′ which is homotopic to this inclusion. It is defined by noting that K(R; η; τ, τ′) − Int(K(R; ϵ; δ, δ′) is homeomorphic to ∂(K(R; ϵ; δ, δ′)) × [0, 1]. We then use a small interior collar of ∂(K(R; ϵ; δ, δ′)) that we can ”inflate” as to cover K(R; η; τ, τ′) − Int(K(R; ϵ; δ, δ′). We can do this such that the homeomorphism i(Y, Y′) restricts to a homeomorphism of the boundaries of the two neigborhoods and moreover to the A_{−}, B_{−} and W_{−} parts of these boundaries.

Proposition 2.2. Assume that P and Q are two consecutive, reasonable, critical points with P different from a local extremum and Q from a local minimum. We consider two neighborhoods K = K(Q; ϵ; δ, δ′), K′ = K(P; μ; τ, τ′) that satisfy the transversality condition Ψ. For any other pair L = K(Q; ϵ_{1}; δ_{1}, δ′_{1}), L′ = K(P; μ_{1}; τ_{1}, τ′_{1}) that also satisfies condition Ψ and with the parameters for the first neighborhoods strictly bigger than the corresponding ones for L and L′, we have the following homotopy commutative diagram:

[figure: A_{L}; i(L,K); ΣA_{L′}; i(L′,K′); k(L,L′); A_{K}; ΣA_{K′}; k(K,K′)]

Proof. We denote by Z = {x ∈ A_{K} : there is a t(x) ∈ R^{+} such that γ_{t(x)}(x) ∈ B_{K′}}, Z′ is the image of Z in B_{K′} by the flow induced homeomorphism; ∂Z′ = ∂_{1}(Z′) ⋃ ∂_{2}(Z′) with ∂_{2}(Z′) ⊂ ∂B_{K′}, ∂_{1}(Z′) being the image of ∂_{1}(Z) by the flow induced homeomorphism (see the proof of Proposition 2.1). The corresponding sets for the pair L, L′ will be indicated by the _{1} subscript. By using the flow induced deformations we may assume that δ = δ_{1}, δ′ = δ′_{1}, τ = τ_{1}, τ′ = τ′_{1}. Then we have Z_{1} ⊂ Z and Z′_{1} ⊂ Z′ and these inclusions commute with the flow induced homeomorphisms. We may also assume that K and L′ satisfy condition Ψ. We will indicate the sets constructed as above for the pair K, L′ by a _{2} subscript. Notice first the relations : A_{L} ⊂ A_{K}, Z_{1} ⊂ A_{L}, Z_{2} ⊂ A_{K}, Z_{1} = Z_{2} ⋂ A_{L}, Z ⊂ A_{K}, Z_{2} ⊂ Z − ∂_{2}(Z), ∂_{2}(Z_{1}) ⊂ ∂_{2}(Z_{2}), Z/\overline{(Z − Z_{2})} = Z_{2}/∂_{2}(Z_{2}). They induce the following commutative diagram. The vertical compositions are equal to the composition of the first three maps in the definition of the corresponding connection maps.

[figure: A_{L}; A_{K}; A_{K}; id; Z_{1}/∂_{2}(Z_{1}); Z_{2}/∂_{2}(Z_{2}); Z/∂_{2}(Z); L′/(A_{L′} ⋃ W_{L′}); K′/((K′ − Int(L′)) ⋃ A_{K′}); K′/(A_{K′} ⋃ W_{K′}); id]

The proof ends by remarking that the following diagram is homotopy commutative (the horizontal maps are the number four maps appearing in the composition defining the connection maps).

[figure: K′/(A_{K′} ⋃ W_{K′}); K′/A_{K′}; K′/((K′ − Int(L′)) ⋃ A_{K′}); id; L′/(A_{L′} ⋃ W_{L′}); L′/A_{L′}]

The homotopy class of any of the maps A_{f}(Q) ⟶ ΣA_{f}(P) constructed above will be denoted by k_{f}(Q, P) (or k(Q, P) when there is no ambiguity) and will be called the connection map of Q and P with respect to f.

Remark 3. 1. It is obvious that Σ(k(Q, P)) is the usual connection map of an attractor-repeller pair that appears in Conley index theory.

2. Assume that (P, Q) and (Q, R) are two pairs of consecutive, reasonable, critical points of f and suppose ∂M = ∅. Let v_{1}, v_{2}, v_{3}, v_{4} be four regular values of f that satisfy v_{1} < f(P), f(P) < v_{2} < f(Q), f(Q) < v_{3} < f(R), v_{4} > f(R). Assume also that there are no other critical points of f whose corresponding critical value belongs to the interval [v_{1}, v_{4}]. Let W_{i} = f^{−1}(−∞, v_{i}]. We have homotopy cofibration sequences A_{f}(P) ⟶ W_{1} ⟶ W_{2}, A_{f}(Q) ⟶ W_{2} ⟶ W_{3}, A_{f}(R) ⟶ W_{3} ⟶ W_{4}. It is immediate that k(Q, P) and k(R, Q) are identified with the resulting attaching maps: A_{f}(Q) ⟶ ΣA_{f}(P) and A_{f}(R) ⟶ ΣA_{f}(Q). Because of this we also have Σk(Q, P) ∘ k(R, Q) ≃ ∗.

Hence, if (P, Q), (Q, R) are two pairs of consecutive, reasonable critical points and Σk(Q, P) ∘ k(R, Q) is not trivial, then there are some other critical points in f^{−1}([f(P), f(R)]) besides P, Q and R.

3. When f is Morse-Smale it is easy to see that our description of the connection map is close to that of Franks [7]. When index(P) + 1 = index(Q) it implies the usual description given in terms of counting (with sign) the flow lines connecting Q to P (for example, see the proof given to this fact in [13]). In this case we can choose the neighborhoods K′ and K_{1} from the definition of our map in such a way that each such flow line will correspond to one component of Z, each component of this type being a disk. It is then easy to identify ∂_{2}(Z) and ∂_{1}(Z) and the result follows.

3. Two properties of the connection map.

In this section we study the connection map of a product flow and the relation between the connection maps of the direct and reverse flows.

These two questions are interdependent: to compare the direct and reverse connection maps one has to pass in the stable category; to understand what suspending means in this context one needs a product formula.

We continue to discuss only the case of functions with reasonable, isolated critical points. However, it seems likely that both results admit generalizations to much more general invariant sets.

3.1. A product formula. Suspension. As before, we assume that f : M ⟶ R is smooth, M compact and that P and Q are consecutive reasonable, critical points of f. Assume also that g : D^{m} ⟶ R has just one critical point, 0 ∈ D^{m}, which is reasonable. We consider f ⊕ g : M × D^{m} ⟶ R, f ⊕ g(x, y) = f(x) + g(y). We assume that P is not a local maximum and that Q is not a local minimum.

Theorem 3.1. The points (P, 0) and (Q, 0) are consecutive, reasonable critical points of f ⊕ g and the respective connection map k_{f⊕g}((Q, 0), (P, 0)) can be identified with k_{f}(Q, P) ∗ 1_{A_{g}(0)}.

Remark 4. 1. The join A ∗ B is defined as the push out of the two inclusions A × B ⟶ A × CB and A × B ⟶ CA × B where CA, CB are the cones over A and B respectively (by a slight abuse we use ∅ ∗ B ≃ B). This construction is functorial. If u : A ⟶ G and t : B ⟶ D are two maps their join is denoted u ∗ t : A ∗ B ⟶ G ∗ D.

2. At the beginning of the proof below we will make explicit the identifications of A_{f}(P) ∗ A_{g}(0) with A_{f⊕g}((P, 0)) and the corresponding one for Q that were refered to in the statement.

Proof. The strategy of the proof is to replace in this case the usual connection map with one defined in terms of product neighborhoods, identify the incidence set in this case and then point out the needed join decompositions.

The facts that (P, 0), (Q, 0) are reasonable and consecutive are immediate. The space level part of the statement is Corollary 2.7 in [3] (and will be recalled below). We first assume that 0 is not a local minimum of g.

Products of cylindrical neighborhoods of Q and of 0. Consider two neighborhoods K = K(Q; ϵ; δ, δ) and N = K(0; ϵ; δ, δ). By modifying conveniently the different constants involved one can find inside any neighborhood T = K((Q, 0); α; β, β′) a product of type K × N and inside any such product a neighborhood like T. This allows one to use an invertible cobordism argument [3] to show that the set P (K, N) = {z ∈ ∂(K × N) : (f ⊕ g)(z) ≤ 0} has the same homotopy type as A_{f⊕g}((Q, 0)). On the other hand, it has the same homotopy type as A_{K} ∗ A_{N}. Indeed, the join of A_{K} and A_{N} is given by the union L(K, N) = A_{K} ×N ⋃ K ×A_{N} union that is taken over the product A_{K} × A_{N} ⊂ K × N. It is easy to see that both L(K, N) and P (K, N) are deformation retracts of C(K, N) = ((A_{K} ⋃ W_{K}) × N) ⋃(K × (A_{N} ⋃ W_{N})) ≃ A_{K} ∗ A_{N}. This is the closure of the complement of J(K, N) = B_{K} × N ⋃ K × B_{N} in ∂(K × N). Moreover, on L(K, N) the vector field −∇(f ⊕ g) points out of K × N and it points in on J(K, N).

The connection map in terms of product neighborhoods. Suppose that T is such that it includes the product K × N. We have two flow induced maps that are homotopy equivalences: i : L(K, N) ⟶ A_{T} and h : ΣA_{T} = T/(A_{T} ⋃ W_{T}) ⟶ (K × N)/C(K, N). The first is given by just leting points flow down and stop once we reach A_{T}; the second does not move points belonging to the interior of K × N as for the exterior points, the ones that reach in forward time Int(J(K, N)) have as image the impact point, the others as well as C(K, N) are sent to the base point.

Suppose now that K′ is a neighborhood of P similar to K and N′ one of 0 that is similar to N and that contains N. We are going to use the above notations for the product K′ × N′.

There is a connection map k : L(K, N) ⟶ (K′ × N′)/C(K′, N′): all the points from L(K, N) that flow down to some points of Int(J(K′, N′)) have the impact point as image, all the others are sent to the base point. Assume T′ is a neighborhood of (Q, 0) similar to T and that it contains K′ × N′.

Obviously, the diagram below is homotopy commutative:

[figure: L(K, N); i; (K′ × N′)/C(K′, N′); k; A_{T}; ΣA_{T′}; h; k(T,T′)]

Therefore, what we need to show is k ≃ k(K, K′) ∗ 1_{A_{N}} (obviously, i and h are both homotopy equivalences).

It is clear that, in fact, we do not need to use for the definition of the map k above the same constants ϵ, δ, δ for both K and N (and similarly for K′ and N′).

The incidence set for the product neighborhoods. Choose K and K′ satisfying condition Ψ and let Z ⊂ A_{K} be their incidence set as in Proposition 2.1. We pick N and N′ such that the time needed for all the points in Z to reach B_{K′} (under the flow γ induced by −∇f) is smaller than the time needed for any of the points in N to exit N′ (under the flow γ^{1} induced by −∇g). We may also choose these neighborhoods such that the time nedeed for all the points in A_{N} to reach A_{N′} is strictly smaller than that nedded for any of the points in B_{K} to reach B_{K′} (this is possible by choosing ϵ in the definition of K sufficiently small).

Denote the flow induced by −∇(f ⊕ g) by γ^{2}.

Let Z_{p} = {(x, y) ∈ L(K, N) : there exists T ∈ R^{+}, γ_{T}^{2} (x, y) ∈ J(K′, N′)}. As in the proof of proposition 2.1 we see that Z_{p} is closed. The points in L(K, N) can not reach (in forward time) K′ × B_{N′} so Z_{p} = {(x, y) ∈ L(K, N) : there are t, k ≥ 0, γ_{t}(x) ∈ Z ⊂ A_{K}, γ_{t+k}(x) ∈ B_{K′} and for 0 ≤ τ ≤ t+k, γ_{τ}^{1}(y) ∈ N′}. We see that Z × N ⊂ Z_{p}. In fact, Z_{p} is identified with (Z × N) ⋃(Z × A_{N}) × [0, 1] where we consider Z × {0} = Z ⊂ A_{K} ⊂ K. With this identification (Z × A_{N}) × {1} consists exactly of those points (x, y) of L(K, N) such that for some T ∈ R^{+} we have γ_{T}^{2} (x, y) ∈ B_{K′} × ∂(N′). There is a deformation of Z_{p} to Z × N given by (x, y) ⟶ (γ_{r}(x), y) where r ∈ R^{+}.

We now consider ∂_{2}(Z_{p}) = (∂_{2}(Z)×N) ⋃(∂_{2}(Z)×A_{N} ×[0, 1]) ⋃(Z ×A_{N} ×{1}). Notice that ∂_{2}(Z_{p}) consists exactly of the elements of Z_{p} that flow by γ^{2} to points in ∂(J(K′, N′)). This is a consequence of the facts that γ sends exactly the points of ∂_{2}(Z) in ∂(B_{K′}) and that no other points of Z_{p} besides those of Z × A_{N} × {1} reach (in forward time) B_{K′} × ∂(N′). Moreover, ∂Z_{p} ⋂ Int(L(K, N)) ⊂ ∂_{2}(Z_{p}), ∂_{2}(Z_{p}) ⊂ ∂Z_{p}, ∂_{2}(Z_{p}) ⋂ ∂L(K, N) = ∂(∂_{2}(Z_{p})).

Let us denote by ∂_{1}(Z_{p}) the closure of the complement of ∂_{2}(Z_{p}) in ∂Z_{p}. It is identified with ((∂_{1}(Z) × N ⋃ Z × ∂(N)) − Int(Z × A_{N})) ⋃(∂_{1}(Z) × A_{N} × [0, 1]) ⋃ Z×∂(A_{N})×[0, 1]. Note that ∂_{1}(Z_{p}) coincides with the intersection of ∂(Z_{p}) with ∂(L(K, N)). Indeed, as ∂_{1}(Z) ⊂ ∂(A_{K}) we have (∂_{1}(Z) × N ⋃ Z × ∂(N)) ⊂ (∂(A_{K}) × N ⋃ A_{K} × ∂(N)). Also, ∂_{1}(Z) × [0, 1] ⊂ W_{K} so ∂_{1}(Z) × A_{N} × [0, 1] ⊂ ∂(K) × A_{N}, similarly Z × [0, 1] ⊂ K so, Z × ∂(A_{N}) × [0, 1] ⊂ K × ∂A_{N}.

Identification of the joins. The connection map k is given by the composition w ∘ v with

v : L(K, N) ⟶ L(K, N)/\overline{(L(K, N) − Z_{p})} ⟶ Z_{p}/∂_{2}(Z_{p})

where the first map is the collapsing map the second being the obvious homeomorphism and w : Z_{p}/∂_{2}(Z_{p}) ⟶ (K′ × N′)/C(K′, N′) is induced by the inclusions.

There is a homeomorphism of Z_{p} with Z × N which extends to a homeomorphism of L(K, N). We use it to write L(K, N) as a union of two components the first, U, homeomorphic to A_{K} × N the second ,V, homeomorphic to K × A_{N} with U ⋂ V homeomorphic to A_{K} × A_{N} and such that Z_{p} ⊂ U. We now identify via this homeomorphism U to A_{K} × N, V to K × A_{N}, Z_{p} to Z × N and ∂_{2}(Z_{p}) with the subset of ∂(Z × N) given by ∂_{2}(Z) × N ⋃ Z × A_{N}. Under this identification Z × A_{N} × {1} ⊂ Z_{p} corresponds to Z × A_{N}.

We need the following general facts. Assume A, B are two spaces U ⊂ A.

1. We have the homeomorphisms: (A × CB)/((U × CB) ⋃(A × B)) ⟶ ((A × CB)/(U ×CB))/((A×B)/(U ×B)) ⟶ ((A/U ×CB)/(∗×CB))/((A/U ×B)/(∗× B)).

2. The collapsing map (A/U)∗B = (A/U ×CB) ⋃(C(A/U)×B) ⟶ ((A/U × CB)/(∗ × CB))/((A/U × B)/(∗ × B)) gives a natural homotopy equivalence.

It follows that we have homeomorphisms: Z_{p}/∂_{2}(Z_{p}) = (Z × N)/(∂_{2}(Z) × N ⋃ Z×A_{N}) ⟶ ((Z×N)/(∂_{2}(Z)×N))/((Z×A_{N})/(∂_{2}(Z)×A_{N})) ⟶ ((Z/∂_{2}(Z)× N)/(∗ × N))/((Z/∂_{2}(Z) × A_{N})/(∗ × A_{N})). This last space is naturally homotopy equivalent with (Z/∂_{2}(Z)) ∗ A_{N}.

To end the case 0 ≠ min we need to remark that v is identified to the join of v′ : A_{K} ⟶ A_{K}/\overline{(A_{K} − Z)} ⟶ Z/∂_{2}(Z) with the identity of A_{N} and that w is identified to the join of Z/∂_{2}(Z) ⟶ K′/(A_{K′} ⋃ W_{K′}) with the flow induced homotopy equivalence A_{N} ⟶ A_{N′}.

For v notice that it equals the composition: (A_{K} × N) ⋃(K × A_{N}) ⟶ (A_{K} ×

N)/(b × N ⋃ A_{K} × A_{N}) ⟶^{v′′} (A_{K} × N)/(\overline{(A_{K} − Z)} × N ⋃ A_{K} × A_{N}) ⟶ (Z ×

N)/(∂_{2}(Z) × N ⋃ Z × A_{N}) = Z_{p}/∂_{2}(Z_{p}) where the first two maps are the obvious collapsing maps, the first being also a homotopy equivalence, the last map is a homeomorphism (because of 1. above and Z/∂_{2}(Z) = A_{K}/\overline{A_{K} − Z}) and b is a point in \overline{(A_{K} − Z)}. In its turn v′′ is naturally identified with the map ((A_{K} × N)/(b × N))/((A_{K} × A_{N})/(b × A_{N})) ⟶ (((A_{K}/\overline{(A_{K} − Z)}) × N)/(∗ × N))/(((A_{K}/\overline{(A_{K} − Z)}) × A_{N})/(∗ × A_{N})) which is induced by the product of the collapsing A_{K} ⟶ A_{K}/\overline{(A_{K} − Z)} with the identity of A_{N}. This identifies v′′ with the join of these two maps and hence expresses v (up to homotopy) as the join of v′ and id_{A_{N}}.

For w we write C(K′, N′) = ((A_{K′} ⋃ W_{K′}) × N′) ⋃(K′ × (A_{N′} ⋃ W_{N′})) and then use the same type of identifications as in the case of v.

If 0 is a local minimum of g we have L(K, N) = A_{K}×N and the same arguments as above imply the result in this case also.

Corollary 3.2. If g is a quadratic form of index i, then k_{f⊕g}((Q, 0), (P, 0)) ≃ Σ^{i}k_{f}(Q, P)

Remark 5. 1. All the contructions in the proof above can be made in any given neighborhood of I((P, 0), (Q, 0)). In fact, the analogue of the index pair in Proposition 2.1 is obtained here by taking the union of K × N with K′ × N′ and Z_{p} × [0, 1] using the inclusions of Z_{p} in L(K, N) and in J(K′, N′). We denote such a neighborhood by T (K, N; K′, N′).

2. The space level of the join formula above is an extension of a formula of King [9] which concerns only the case when g is a nondegenerate quadratic form.

3.2. Duality. Here we compare the connection maps corresponding to the direct and inverse flows.

Spanier-Whitehead duality. We recall a few basic facts [14], [15].

Two pointed, connected CW -complexes X, X′ are Spanier-Whitehead dual 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. 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 map induced by μ. There is a good notion of duality for maps also (all maps and spaces are pointed). Assume X, X′ are duals by an m-duality, μ, and Y, Y′ are duals by an m-duality ν. Two maps f : X ⟶ Y, g : Y′ ⟶ X′ are duals if for some (possibly) large 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]

The reason why suspending is necessary is that only by allowing k and k′ to be sufficiently large this notion of duality corresponds to a duality isomorphism between (stable) classes of maps D_{m}(μ, ν) : {X, Y } ≈ {Y′, X′}.

We need now to indicate one way to obtain duality maps [14]. 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. Then there is a duality map μ : A ∧ B ⟶ S^{m}. This map is defined in the following way: we 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. There is also a way to identify dual maps in this setting. For example 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.

Duality mod u. Suppose (X, X′) and (Y, Y′) are two pairs of m-dual spaces. We say that the two maps f : X ⟶ Y and g : Y′ ⟶ X′ are Spanier-Whitehead duals mod u if there is a k ∈ N and a space H together with a self homotopy equivalence u : H ⟶ H that induces the identity in homology (with Z/2 coefficients in general, or Z coefficients in an orientable context) with the property that that Σ^{k}f factorizes as Σ^{k}X ⟶^{r} H ⟶^{l} Σ^{k}Y and g and l ∘ u ∘ r are Spanier-Whitehead duals.

Some useful homotopy equivalences. We need to fix another homotopical construction.

For a CW -complex Y let Eq(Y) be the group of self-homotopy equivalences

of Y. We consider the map: s_{Y} : [Y ∗ S^{k−1}, S^{k}] ⟶^{+id_{S^{k}}} [Σ(Y × S^{k−1}), S^{k}] ⟶^{ad}

[Y × S^{k−1}, ΩS^{k}] ⟶^{id_{Y} ×} [Y × S^{k−1}, Y × ΩS^{k}] ⟶^{Σ} [Σ(Y × S^{k−1}), Σ(Y × ΩS^{k})] ⟶^{p}

Eq(Y ∗ S^{k−1}). Here ad is the adjunction; for the first map we use the formula Σ(Y × Z) = ΣY ∨ Σ(Y ∧ Z) ∨ ΣZ. To define p we use the obvious retraction

r : Σ(Y ×ΩS^{k}) ⟶ Σ(Y ×S^{k−1}) of the inclusion Σ(Y ×S^{k−1}) ⟶^{Σ(id_{Y} ×e)} Σ(Y ×ΩS^{k})

(e : S^{k−1} ⟶ ΩS^{k} being the adjoint of the identity; r is induced by the retraction r′ : ΣΩS^{k} ⟶ ΣS^{k−1} of Σe). If x ∈ [Σ(Y × S^{k−1}), Σ(Y × ΩS^{k})], then p(x) is the restriction of r ∘ x to Σ(Y ∧ S^{k−1}), projected on Σ(Y ∧ S^{k−1}) (recall that Σ(Y ∧ Z) = Y ∗ Z).

Consider an inclusion A ⟶^{i} X. We will use the map s_{X,A} : [X ∗ S^{k−1}, S^{k}] ⟶ Eq((X/A) ∗ S^{k−1}) obtained by taking s_{X,A}(x) to be the self-homotopy equivalence of (X/A) ∗ S^{k−1} induced by s_{X}(x) and s_{A}(x ∘ i) from the homotopy cofibration sequence A ∗ S^{k−1} ⟶ X ∗ S^{k−1} ⟶ X/A ∗ S^{k−1}

We return now to our familiar setting f : M ⟶ R is smooth, M is compact, riemannian of dimension n and P, Q are two consecutive, reasonable critical points of f. We assume that all spaces A_{f}(P), A_{−f}(P), A_{f}(Q), A_{−f}(Q) are connected. Moreover, we assume that the dimension of M is different from four. Then all neighborhoods K(P; ϵ; δ, δ′), K(Q; ϵ; δ, δ′) are homeomorphic to disks. Both these assumptions are not too restrictive as they can be realized by adding a suitable quadratic form to the initial function. The previous section shows that the effect of such an addition is just suspending the homotopical data.

We are also going to assume that M is embedded in a high dimensional sphere S^{k+n} with k > n and that its nomal bundle is classified by χ ∈ [M, BO(k)].

Theorem 3.3. The maps k_{f}(Q, P) and k_{−f}(P, Q) are Spanier-Whitehead duals mod u with u described as follows.

There is a triple of subspaces of M, (T_{1}, T, T′) with T_{1} ≃ ΣT and a map

Q : [M, BO(k)] ⟶^{i} [ΣT, BO(k)] ⟶^{ad} [T, O(k)] ⟶^{J} [T ∗ S^{k−1}, S^{k}] ⟶^{s_{T,T′}}

⟶ Eq((T/T′) ∗ S^{k−1}) such that u = Q(χ).

Here i is restriction and J is the generalized J-homomorphism. The composition of the last three maps is a morphism of groups.

Proof. We consider cylindrical neighborhoods K′ of Q and K_{1} of P as in the proof of Proposition 2.1. Let N = K_{1} ⋃ W′ ⋃ K′ the neighborhood of I(P, Q) constructed there. As mentioned above we assume that all the spaces A_{K′}, A_{K_{1}}, B_{K′}, B_{K_{1}} are connected and that K′ and K_{1} are both homeomorphic to D^{n}. Hence, it is clear that A_{K′}, B_{K′} and A_{K_{1}}, B_{K_{1}} are pairs of (n−2)-dual spaces. This implies that A_{K′}, ΣB_{K′} and ΣA_{K_{1}}, B_{K_{1}} are pairs of (n−1)-dual spaces. It is useful to know which are the precise duality maps for each of these pairs.

A basic duality diagram. We represent B_{K_{1}} and ΣA_{K_{1}} as complemetary subspaces of S^{n}. In fact, we prefer to use instead of A_{K_{1}} the homeomorphic space A′_{K_{1}} = A_{K_{1}} ⋃ W_{K_{1}}. We embed first K_{1} ⊂ S^{n} then (as K_{1} is a disk) the complement of K_{1} in S^{n} is the cone over the boundary of K_{1}. We consider the union U of K_{1} with the subset of this cone that gives the cone over A′_{K_{1}}. It is easy to see that S^{n} − U retracts to Int(B_{K_{1}}) and that U represents ΣA_{K_{1}}. The duality map that we need is induced by this embedding. We denote this duality map by ν. Moreover, let Z ↪ B_{K_{1}} be the incidence set appearing in 2.1 (we identify here Z and Z ×{0}). Recall that ∂Z = ∂_{2}(Z) ⋃ ∂_{1}(Z) with ∂_{2}(Z) ⊂ ∂B_{K_{1}} ⊂ W_{K_{1}}. Hence, we may consider the inclusion i : Z/∂_{2}(Z) ⟶ ΣA′_{K_{1}} = K_{1}/A′_{K_{1}}. It is again useful to think about it in S^{n}. It is the inclusion of the union, T, of Z and the part of the cone on ∂K_{1} that gives the cone over ∂_{2}(Z). The complement of T in S^{n} has as a deformation retract the union of K_{1} and a copy of Z that is pasted to K_{1} along ∂_{1}(Z) (this type of argument is very classical in nature see [1]). Clearly, there is a (nontrivial) inclusion j of a copy of B_{K_{1}} in this complement. The conclusion is that Z/∂_{2}(Z), Z/∂_{1}(Z) are also n − 1-dual by a duality map η such that with respect to ν and η the two inclusions i and j are dual.

We can proceed in a similar fashion for the pair A_{K′}, ΣB_{K′}. We denote by ν′ the duality map obtained for this pair. Clearly Z ⊂ A_{K′} and ∂_{1}(Z) ⊂ W_{K′}. As above we obtain a duality map η′ for the pair Z/∂_{2}(Z), Z/∂_{1}(Z). We denote by i′ the inclusion Z/∂_{1}(Z) ⟶ ΣB_{K′}. It is dual (with respect to η′ and ν′) to the inclusion j′ : A_{K′} ⟶ Z/∂_{2}(Z).

We use the notations k = k(K′, K_{1}) and k′ = k(K_{1}, K′). The definitions of the connection maps imply that k and respectively k′ factor as

A_{K′} ⟶^{p} Z/∂_{2}(Z) ⟶^{i} ΣA_{K_{1}} and B_{K_{1}} ⟶^{p′} Z/∂_{1}(Z) ⟶^{i′} ΣB_{K′}.

We obtain that the following diagram (∗∗) commutes up to homotopy:

[figure: A_{K′} ∧ ΣB_{K′}; S^{n−1}; ν′; A_{K′} ∧ Z/∂_{1}(Z); Z/∂_{2}(Z) ∧ Z/∂_{1}(Z); η′; id; A_{K′} ∧ B_{K}; id∧k′; id∧p′; k∧id; p∧id; Z/∂_{2}(Z) ∧ B_{K_{1}}; i∧id; Z/∂_{2}(Z) ∧ Z/∂_{1}(Z); η; id∧j; ΣA_{K_{1}} ∧ B_{K_{1}}; S^{n−1}; ν]

If η and η′ are the same we obtain that the two connection maps are Spanier-Whitehead duals. In turn, this can be assumed if all the constructions above are done with respect to a neighborhood of type N that is embedded in a sphere of dimension n (= dim(M)).

The stably parallelizable case. Assume now that i(χ) = 0. Consider a nondegenerate quadratic form g : D^{k} ⟶ R. On a tubular neighborhood of N, U = N × D^{k} we can define the function f ⊕ g : N × D^{k} ⟶ R. For the invariant set I((P, 0), (Q, 0)) there is a neighborhood of type T (K, N; K′′, N′′) (as in the last remark of subsection 3.1). All the above argument can be done with respect to this neighborhood by replacing K′ with K × N, K_{1} with K′′ × N′′, Z with Z_{p}, ∂_{1}(Z) with ∂_{1}(Z_{p}) and ∂_{2}(Z) with ∂_{2}(Z_{p}).

By the product formula we know that the connection maps constructed in this way (for f ⊕ g) are iterated suspensions of the corresponding maps for f. Hence, in the stably parallelizable case we obtain that the two connection maps are dual.

The general case. We come now to the general case. Denote again by U a tubular neighborhood of N in S^{n+k}. As N = K_{1} ⋃ W′ ⋃ K′ with K_{1} and K′ contractible and W′ = Z × [0, 1], U is homeomorphic to the push out of the two inclusions Z × D^{k} ⟶ K_{1} × D^{k}, Z × D^{k} ⟶ K′ × D^{k}. The identification between Z ×D^{k} ⊂ K′ ×D^{k} and Z ×D^{k} ⊂ K_{1} ×D^{k} is given by a map t : Z ×D^{k} ⟶ Z ×D^{k} with t(x, a) = (x, t_{x}a), t_{x} ∈ O(k), clearly we have t_{x} ≃ (ad ∘ i)(χ)(x). We denote χ′ = (ad ∘ i)(χ). We then have t(x, a) = (x, χ′(x)(a))

Let g : U ⟶ R be the negative of the square of a function measuring the distance from M. Under the identifications above we may assume that the restriction of g at each of the sets K_{1} × D^{k} and respectively K′ × D^{k} coincides with the composition of g′(a) = −a_{1}^{2} − a_{2}^{2} − ... − a_{k}^{2} with the projection on D^{k}.

Consider the function f + g. It has two reasonable consecutive critical points (P, 0) and (Q, 0). The stably parallelizable case discussed before shows that k_{f+g}((Q, 0), (P, 0)) and k_{−(f+g)}((P, 0), (Q, 0)) are Spanier-Whitehead duals. We need to compare these two connection maps with k_{f}(Q, P) and k_{−f}(P, Q) respectively.

On each of the two sets K_{1} × D^{k} and K′ × D^{k} the behaviour of k_{f+g}((Q, 0), (P, 0)) is described in Theorem 3.1. However, because of the twisting given by χ′ the connection map k_{f+g}((Q, 0), (P, 0)) cannot be evaluated by directly applying that theorem. Instead, using the notations of the proof of the theorem, one sees that this connection map is given by the composition w ∘ χ′′ ∘ v (we recall v : L(K, N) ⟶ L(K, N)/\overline{(L(K, N) − Z_{p})} ⟶ Z_{p}/∂_{2}(Z_{p}), w : Z_{p}/∂_{2}(Z_{p}) ⟶ (K′ × N′)/C(K′, N′)).

Here χ′′ : Z_{p}/∂_{2}(Z_{p}) ⟶ Z_{p}/∂_{2}(Z_{p}) is derived from χ′ in the following way: recall the identification of Z_{p} with Z × N and that of ∂_{2}(Z_{p}) with ∂_{2}(Z) × N ⋃ Z × A_{N}. In our case A_{N} = S^{k−1} and N = D^{k}. The map χ′′ is induced by t and by the restriction of t to ∂_{2}(Z_{p}).

Of course the same argument applies also to the function −(f + g). In that case, one needs the function χ_{2} : Z_{p}/∂_{1}(Z_{p}) ⟶ Z_{p}/∂_{1}(Z_{p}). Here ∂_{1}(Z_{p}) is the complement of ∂_{2}(Z_{p}) in ∂Z_{p} and χ_{2} is induced by t′(x, a) = (x, (χ′(x))^{−1}(a)). In this case we have B_{N} = ∅ and we immediately obtain that ∂_{1}(Z_{p}) = ∂_{1}(Z) × D^{k}. Hence, χ_{2} ≃ id. Thus, k_{−(f+g)}((P, 0), (Q, 0)) ≃ k_{−f}(P, Q).

Identification of u. To end the proof we take T_{1} = N, T = Z_{p}, T′ = ∂_{2}(Z_{p}) and we still have to understand better the map χ′′. In particular, we want to show χ′′ ≃ Q(χ). By identifying Z_{p} to Z × D^{k} and ∂_{2}(Z_{p}) to ∂_{2}(Z) × D^{k} ⋃ Z × S^{k−1} we write Z_{p}/∂_{2}(Z_{p}) as ((Z × D^{k})/(Z × S^{k−1}))/((∂_{2}(Z) × D^{k})/(∂_{2}(Z) × S^{k−1})). The map χ′′ coincides with the map induced to the cofibre of the inclusion (∂_{2}(Z)×D^{k})/(∂_{2}(Z)× S^{k−1}) ⟶ (Z × D^{k})/(Z × S^{k−1}) by the maps t′ : (∂_{2}(Z) × D^{k})/(∂_{2}(Z) × S^{k−1}) ⟶ (∂_{2}(Z)×D^{k})/(∂_{2}(Z)×S^{k−1}) and t′′ : (Z×D^{k})/(Z×S^{k−1}) ⟶ (Z×D^{k})/(Z×S^{k−1}) induced by t. We recall that t : Z × D^{k} ⟶ Z × D^{k}, t(x, a) = (x, χ′(x)(a)).

Notice that (Z ×D^{k})/(Z ×S^{k−1}) ≃ (Z ∗S^{k−1})∨S^{k} and (∂_{2}(Z)×D^{k})/(∂_{2}(Z)× S^{k−1}) ≃ (∂_{2}(Z) ∗ S^{k−1}) ∨ S^{k}.

The restriction of both t′ and t′′ to S^{k} is just the inclusion of S^{k} into the wedge.

Let t′′_{1} be the composition Z ∗ S^{k−1} ⟶ (Z ∗ S^{k−1}) ∨ S^{k} ⟶^{t′′} (Z ∗ S^{k−1}) ∨ S^{k} ⟶

Z ∗ S^{k−1} and define similarly t′_{1} (here the first map is the inclusion and the last one collapses S^{k} to a point).

Clearly, χ′′ is induced on the cofibre of (∂_{2}(Z) ∗ S^{k−1}) ∨ S^{k} ⟶ (Z ∗ S^{k−1}) ∨ S^{k} by t′′_{1} and t′_{1}.

We now show that t′′_{1} = (s_{Z} ∘J)(χ′). Let b : Z ×S^{k−1} ⟶ Z ×S^{k−1} be given by the restriction of t. Notice that t′′_{1} is, up to homotopy, the composition Z ∗ S^{k−1} ↪ Σ(Z × S^{k−1}) ⟶^{Σb} Σ(Z × S^{k−1}) ⟶ Z ∗ S^{k−1} (the last map being, as before, the collapsing and the first the inclusion). On the other hand J(χ′) is, by definition,

the composition Z ∗ S^{k−1} ↪ Σ(Z × S^{k−1}) ⟶^{Σ(p_{2}∘b)} S^{k} with p_{2} : Z × S^{k−1} ⟶ S^{k−1}

being the projection. Because dim(Z) < k the restriction of Σb to ΣZ coincides up to homotopy with the inclusion ΣZ ↪ Σ(Z × S^{k−1}). Hence, we can write Σ(p_{2} ∘ b) = J(χ′) + id_{S^{k}}. We take the adjoint, and get ad(J(χ′) + id_{S^{k}}) = e ∘ p_{2} ∘ b where e : S^{k−1} ⟶ ΩS^{k} is the adjoint of the identity. We take the product of the map ad(J(χ′) + id_{S^{k}}) : Z × S^{k−1} ⟶ ΩS^{k} with the identity of Z and then suspend. We precompose the resulting map with Z ∗ S^{k−1} ↪ Σ(Z × S^{k−1}) and compose it with the retraction Σ(Z × ΩS^{k−1}) ⟶ Z ∗ S^{k−1}.

It is easy to see that the map obtained in this way coincides with t′′_{1} and with s_{Z}(J(χ′)).

The same argument applies to t′_{1} and gives the wanted description for χ′′. Notice also that t′′_{1} (as well as t′_{1}) induces the identity in homology with Z/2

coefficients in general and with integral coefficients if χ is orientable.

The fact that the composition of the last three maps in the statement is a homomorphism follows immediately from the description of t′_{1} = (s_{Z} ∘ J)(χ′) and of t′′_{1} in terms of t.

Remark 6. As pointed out to me by Mark Mahowald examples where the two connection maps are not dual are easy to produce. A simple one is provided by the Morse function with three critical points on RP^{2}. Indeed, if one considers the critical points of index 0 and 1 the corresponding attaching map is null. The reverse attaching map however is S^{1} ⟶^{2} S^{1}. Our theorem implies that for an orientable manifold this type of example when the two critical points are of indexes of difference one and the direct and reverse attaching maps are not Spanier-Whitehead duals can not occur.

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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