HOMOTOPICAL DYNAMICS II: HOPF INVARIANTS, SMOOTHINGS AND THE MORSE COMPLEX.

OCTAVIAN CORNEA

Abstract. The ambient framed bordism class of the connecting manifold of two consecutive critical points of a Morse-Smale function is estimated by means of a certain Hopf invariant. Applications include new examples of non-smoothable Poincaré duality spaces as well as an extension of the Morse complex.

1. Introduction.

Let M be a smooth, compact, riemannian manifold and let f : M ⟶ R be a smooth Morse-Smale function, regular and constant on ∂M. The flow γ : M × R ⟶ M used below is induced by −∇f. Assume that P and Q are consecutive critical points of f (this means that f(P) > f(Q) and that there are no broken flow lines connecting P to Q) of indexes, respectively, p and q.

An important, classical problem in Morse theory is to use the topology of M to understand the properties of the moduli space Z(P, Q) of flow lines that connect P to Q.

This problem is the main motivation of the paper. The key new idea introduced here is that knowledge of the homotopy of the based loop space ΩM leads to significant information on these moduli spaces.

More precisely, recall that the genericity of the Morse-Smale condition implies that Z(P, Q) is a manifold of dimension p − q − 1 called connecting manifold of P and Q. It has a canonical normal framing and a classical result of John Franks [10] claims that its framed bordism class {Z(P, Q)} ∈ Ω_{∗}^{fr} is given (via the Thom-Pontryagin construction) by the relative attaching map δ_{f}(P, Q) associated to the succesive attachments of the cells corresponding to the critical points Q and P.

The closure of the space of all the points situated on some flow line joining P to Q is identified to the unreduced suspension ΣZ(P, Q). Therefore, we have an inclusion ΣZ(P, Q) ↪ M and if M is simply connected (which we will asssume) a well defined adjoint

Date: December 10, 1998.

1991 Mathematics Subject Classification. Primary 58F09, 55Q50, 55Q25; Secondary 57R70, 58F25.

l(P, Q) : Z(P, Q) ⟶ ΩM

The above mentioned normal framing of Z together with the map l(P, Q) provide, via the Thom-Pontryagin construction, a homotopy class T (P, Q) : S^{p−1} ⟶ Σ^{q}(ΩM^{+}) = S^{q} ∨ S^{q} ∧ ΩM. By Franks’ result the projection of T (P, Q) on S^{q} is δ_{f}(P, Q). Let h(P, Q) ∈ π_{p−1}(Σ^{q}ΩM) be the projection of T (P, Q) on the second factor.

The main result of the paper continues the work of Franks by giving a purely homotopical, computable description of the ambient, framed, bordism class of Z(P, Q), [Z(P, Q)]^{fr} ∈ Ω_{p−q−1}^{fr}(ΩM). In fact, we show that Σh(P, Q) equals the suspension of a certain Hopf invariant, H(P, Q), associated to the succesive cell attachments corresponding to Q and P. As T (P, Q) represents [Z(P, Q)]^{fr} we conclude that this bordism class equals the stable image of H(P, Q) + δ_{f}(P, Q).

The homotopy classes h(P, Q) turn out to be highly relevant for understanding the respective connecting manifolds and also for the topology of M itself. We study some of their properties.

Specializing to the Morse-Smale case the Spanier-Whitehead duality results of [5], [6], we show that, stably, ϵδ_{−f}(Q, P) equals δ_{f}(P, Q) + Δ(P, Q) where ϵ ∈ {−1, +1} with the twisting term Δ(P, Q) = J^{q}(Ω^{q}Σ^{q}(Ων) ∘ h^{∗}(P, Q)). Here ν : M ⟶ BSO classifies the stable normal bundle of M, h^{∗}(P, Q) is the q-th order adjoint of h(P, Q) and J^{q} : π_{k}(Ω^{q}Σ^{q}SO) ⟶ π_{k}^{S} is a factor of the classical J-homomorphism J : π_{k}(SO) ⟶ π_{k}^{S}. This can be used to deduce a measure of the embedding complexity of Z(P, Q). We also use this result to construct examples of non-smoothable Poincaré duality spaces (many of which are PL-manifolds). When the obstructions to smoothing concern only the relative attaching maps of the top cell the relevant morphism is, of course, J = J^{0} and one recovers in this case classical results (see for instance [33]) and the examples of Smith [32] which are thus seen to be part of a more general pattern.

In a different direction, if P, R are critical points such that f(P) > f(R) and all flow lines connecting P to R are broken at most once, let I(P, R) be the set of intermediate critical points. We show that the relation ∑_{Q∈I(P,R)}(−1)^{(p+r)q}[Z(P, Q)]^{fr} • [Z(Q, R)]^{fr} = 0 is satisfied inside the ring

Ω_{∗}^{fr}(ΩM) (where the product • is induced by loop composition). This can be used to define various chain complexes which, in the simplest possible case (when the index of successive critical points differs by just one), provide the Morse complex of f.

The material is organized as follows. After a second section, containing recalls and notations, comes the technical heart of the paper, in section three, very much in the spirit of the work of Franks [10]. The description of Σh(P, Q) in terms of the Hopf invariant is given at this point, as well as the proof of the relations among the bordism classes of the connecting manifolds. The fourth section contains applications and most homotopy theoretical arguments are concentrated here.

Acknowledgements. I thank Raoul Bott, Fred Cohen, Pascal Lambrechts, Chuck McGibbon and Alberto Verjovsky for useful discussions as well as John Harper whom I also thank for pointing out [32]. I am most grateful to Mark Mahowald for his valuable suggestions and for his encouragement.

2. Recalls and notations.

2.1. Hopf invariants. For further use we fix some standard facts. We denote by A ∗ B the join of A and B. For pointed spaces, A ∗ B ≃ ΣA ∧ B, ΣA = S^{1} ∧ A. Recall also that A^{+} is the pointed space obtained by the disjoint union of A and a disjoint base point; if A is pointed we have (A × B)/(∗ × B) = A ∧ B^{+}. For A and B connected we will also need the existence of the fibration ΩA ∗ ΩB ⟶ A ∨ B ⟶ A × B which is trivial after looping.

Assume that X is a CW -complex and that X′ ↪ X′′ ↪^{i} X are connected subcomplexes such that there is a cofibration sequence S^{q−1} ⟶^{f} X′ ⟶ X′′. Fix also a map S^{p−1} ⟶^{g} X′′.

The Hopf invariants that we will use are defined using this data (they are versions of invariants defined by Ganea [11][8]).

The Hopf invariant of g relative to f is obtained as follows. Consider the map

t : S^{p−2} ⟶ ΩS^{p−1} ⟶^{Ωg} ΩX′′ ⟶^{Ω∇} Ω(S^{q} ∨ X′′) ⟶^{p} Ω(ΩS^{q} ∗ ΩX′′)

Here, ∇ is the coaction X′′ ⟶ S^{q} ∨ X′′ and p is the canonical projection in the splitting Ω(S^{q} ∨ X′′) ≃ ΩS^{q} × ΩX′′ × Ω(ΩS^{q} ∗ ΩX′′); the first map in the composition is the restriction to the bottom cell.

We have a projection r : ΩS^{q}∗ΩX′′ ⟶ S^{q}∧ΩX′′ induced by the canonical evaluation ΣΩS^{q} ⟶ S^{q}.

The needed Hopf invariant is the homotopy class:

H(g, f) = (id_{S^{q}} ∧ Ωi) ∘ r ∘ t^{∗} : S^{p−1} ⟶ S^{q} ∧ ΩX

where t^{∗} : S^{p−1} ⟶ ΩS^{q} ∗ ΩX′′ is the adjoint of t.

We denote by δ = δ(f, g) : S^{p−1} ⟶ S^{q} the relative attaching map given by projecting ∇ ∘ g onto S^{q}.

Remark 1. For example, let ∗ : S^{1} ⟶ ∗ and let η : S^{3} ⟶ S^{2} be the Hopf map and X = CP^{2}. Then H(η, ∗) : S^{3} ⟶ S^{2} ∧ ΩCP^{2} is the inclusion of the bottom cell. Similarly, let ∗ : S^{q−1} ⟶ S^{t} be the trivial map and w : S^{q+t−1} ⟶ S^{q} ∨ S^{t} be the obvious Whitehead product and X = S^{q} × S^{t}. Then H(w, ∗) : S^{q+t−1} ⟶ S^{q} ∧ Ω(S^{q} × S^{t}) is the inclusion S^{q} ∧ S^{t−1} ⟶ S^{q} ∧ ΩS^{t} ⟶ S^{q} ∧ (ΩS^{q} × ΩS^{t}).

2.2. Elements of Morse theory. We follow here the fundamental paper of John Franks [10].

Let M^{n} be a smooth compact manifold and let f : M ⟶ R be a smooth function. If ∂M ≠ ∅ we assume that the function is constant and regular on ∂M. We assume also that a riemannian metric is fixed on M and we denote by γ : M × R ⟶ M the flow induced by −∇f. A critical point P of M is non-degenerate if Hess_{P} (f) is a non-degenerate matrix. The index of the induced bilinear form is called the index of P. The set W^{u}(P) = {x ∈ M : lim_{t→−∞} γ_{t}(x) = P } is called the unstable manifold of P and W^{s}(P) = {x ∈ M : lim_{t→+∞} γ_{t}(x) = P } is the stable manifold of P. If P is non-degenerate and of index p, then W^{s}(P) ≈ Int(D^{n−p}) and W^{u}(P) ≈ Int(D^{p}).

We assume from now on that f is Morse which means that all its critical points are non-degenerate, and even Morse-Smale which means that if P and Q are two critical points of f, then W^{u}(P) and W^{s}(Q) are in general position. The Morse-Smale condition is generic.

We say that two critical points P and Q are consecutive if f(P) > f(Q) and there are no broken flow lines connecting P to Q. In this case, let f(Q) < a < f(P) and let S^{s}(Q) = W^{s}(Q) ⋂ f^{−1}(a) and S^{u}(P) = W^{u}(P) ⋂ f^{−1}(a). If P and Q are critical and consecutive we may (and will) assume also (possibly after slightly isotoping the function f) that P and Q are the only critical points in f^{−1}([f(Q), f(P)]). In this case, it is easy to see that S^{u}(P) ≈ S^{p−1} and S^{s}(Q) ≈ S^{n−q−1}. The Morse-Smale condition insures that S^{u}(P) and S^{s}(Q) intersect transversely. Their intersection, Z(P, Q), is called the connecting manifold of P and Q (a.k.a. the moduli space of flow lines connecting P to Q). It is a p − q − 1 -dimensional manifold. Fix for each critical point R of f an orientation on the linear subspaces V^{u}(R) and V^{s}(R) of T_{R}(M) that are respectively tangent to W^{u}(R) and W^{s}(R). If the manifold M is oriented, pick these orientations such that at each point R they give on V^{u}(R)⊕V^{s}(R) the fixed orientation of T_{R}(M). It is easy to see that a choice of a basis of V^{u}(R) induces a framing of the normal bundle of W^{s}(R) and, similarly, a choice of basis for V^{s}(R) induces a normal framing of W^{u}(R). We choose these bases in a way compatible with the fixed orientations and then these framings are unique up to isomorphism. We now return to the two consecutive critical points P and Q. The normal bundle of Z(P, Q) in S^{u}(P) is induced by the normal bundle of W^{s}(Q) in M and hence inherits a standard framing (coming from the choices in V^{u}(Q)).

It is well known that the passage through a non-degenerate critical point corresponds to the attachment of a cell of dimension the index of the critical point. In other words, if M′ = f^{−1}(−∞, f(Q)−ϵ], M′′ = f^{−1}(−∞, f(P)−ϵ], M′′′ = f^{−1}(−∞, f(P)+ϵ] with ϵ small enough we have cofibration sequences:

S^{q−1} ⟶^{α(Q)} M′ ⟶ M′′ and S^{p−1} ⟶^{α(P)} M′′ ⟶ M′′′. Denote by δ_{f}(P, Q) :

S^{p−1} ⟶ S^{q} the obvious relative attaching map.

One of the key results in [10] is that δ_{f}(P, Q) corresponds to the standard framing of Z(P, Q) in S^{u}(P) via the Thom-Pontryagin construction. When p−q = 1 this comes down to just counting (with sign) the number of elements in Z(P, Q). Hence, as immediate application of this result of Franks, one obtains that if C_{i} = Z/2 < x : ∇f(x) = 0, ind(x) = i > are Z/2 vector spaces and d : C_{i} ⟶ C_{i−1} is the unique linear application defined on basis elements by d(x) = ∑_{ind(y)=i−1} #(Z(x, y))y, then (C_{i}, d) is a complex, called the Morse complex of f, and H_{∗}(C_{i}, d) ≃ H_{∗}(M; Z/2) (in the oriented case, by using some appropriate signs when counting the elements of Z(x, y) in the definition of d, one obtains the integral homology of M). There are at least a couple of other proofs of this fact. An analytical one appears in [39]. Another, that will be extended in sections 3 and 4, is based on understanding the boundaries of certain moduli spaces of connecting flow lines (see for example [29]) .

For a smooth Morse-Smale function f on M there is an associated CW - decomposition of M (we consider now only functions that are constant, regular and maximal on ∂M). If f is continuously deformed via Morse-Smale functions to a second Morse-Smale function f′, then the CW -decompositions associated to f and f′ are equivalent. When the metric is allowed to vary, the CW -decomposition corresponding to f is determined up to a contractible choice [16].

Conversely, if M and ∂M are simply-connected, n > 5 and H_{∗}(M; Z) torsion free, then all CW -complexes Y of the homotopy type of M and satisfying a certain minimality condition correspond to some Morse-Smale function on M [10] (the minimality condition is the following: Y has a unique 0-dimensional cell, at most one n- cell and if e_{1}^{k} and e_{2}^{l} are two cells of Y with k > l, then either 2k − l < n − 1 or the smallest subcomplex of Y that contains e_{1}^{k} contains also e_{2}^{l}). In particular, there are self-indexed Morse functions having a number of critical points of index k equal to the rank of H_{k}(M; Z) [31]. They are called perfect Morse functions.

2.3. Duality and Flows. Let X be a CW -complex and consider an orthogonal fiber bundle of rank n over X, μ : E ⟶ X. Let E_{S}(X) ⟶ X be the associated spherical bundle and T^{μ}(X) the respective Thom space. Assume that for a certain cell-decomposition of X we have cofibration sequences S^{q−1} ⟶ X′ ⟶ X′′ and S^{p−1} ⟶ X′′ ⟶ X′′′ with X′′′ a subcomplex of X. Denote by δ : S^{p−1} ⟶ S^{q} the respective relative attaching map. It is easy to see that the CW -decomposition of X induces one for T^{μ}(X). For example, by pulling back the cofibration sequence S^{q−1} ⟶ X′ ⟶ X′′ to the spherical fibration of μ one obtains a push-out square:

\xymatrix{S^{q−1} × S^{n−1} \ar[d] \ar[r] & E_{S}(X′) \ar[d] \\ S^{n−1} \ar[r] & E_{S}(X′′)}

by pushing down this square into the original cofibration we get the pushout square:

\xymatrix{S^{q−1} ∧ S^{n} ∨ S^{n} \ar[d] \ar[r] & T^{μ}(X′) \ar[d] \\ S^{n} \ar[r] & T^{μ}(X′′)}

As the left vertical map is just the projection onto S^{n} it is easy to transform this push-out square into a cofibration sequence S^{n+q−1} ⟶ T^{μ}(X′) ⟶ T^{μ}(X′′).

In particular we obtain a relative attaching map δ^{μ} : S^{n+p−1} ⟶ S^{n+q}. We return now to the context and notations of the previous sub-section.

Thus f : M ⟶ R is a smooth Morse-Smale function; P and Q are consecutive critical points of f of indexes respectively p and q; δ_{f}(P, Q) is the corresponding relative attaching map. The function −f is also Morse-Smale and Q and P are consecutive critical points for −f. Thus, we also have a relative attaching map δ_{−f}(Q, P) : S^{n−q−1} ⟶ S^{n−p}.

Let ν be the stable normal bundle of M. The results in [5] imply that δ_{f}(P, Q)^{ν} agrees stably up to sign with δ_{−f}(Q, P) and if ν is trivial, then δ_{f}(P, Q)^{ν} and δ_{f}(P, Q) agree stably.

Remark 2. When ν is trivial the result appears already in the paper of Franks [10]. In fact, one has much more general results valid for general flows and isolated invariant sets (in the context of Conley index theory) in which the equality up to sign is replaced by Spanier-Whitehead duality [5].

2.4. The J-homomorphism. We need to recall a few elements of classical homotopy theory. We denote by π_{k}^{S} the stable k-stem. Similarly, the stable homotopy groups of any space X are denoted by π_{k}^{S}(X).

The J-homomorphism J : π_{k}(SO) ⟶ π_{k}^{S} is defined as follows. For α ∈ π_{k}(SO) there is some m ∈ N such that α ∈ π_{k}(SO(m)). Consider the composition

α′ : S^{m−1} × S^{k} ⟶^{id×α} S^{m−1} × SO(m) ⟶^{·} S^{m−1}

Apply the Hopf construction to this map α′ (or, equivalently, suspend α′ and use the splitting of the suspension of the domain to restrict to S^{m+k}) thus getting a map α′′ : S^{m+k} ⟶ S^{m}. The image of α′′ in the stable k-stem is J(α).

One can show [38] that this construction does not depend of the different choices involved and that the resulting map is a group homomorphism.

As the image of J is stable, it is immediate to see that this homomorphism factors as π_{k}(SO) ⟶ π_{k}^{S}(SO) ⟶^{J′} π_{k}^{S}. The homomorphism J′ is called the ”very stable J” and also the ”bi-stable J-homomorphism” [18]. It is easy to see that, for each q ≥ 0, in between J and J′ there is an intermediate factor J^{q} : π_{k}(Ω^{q}Σ^{q}SO) ⟶ π_{k}^{S}. These are also homomorphisms and they commute with the morphisms induced in homotopy by the inclusions Ω^{q}Σ^{q}SO ⟶ Ω^{q+1}Σ^{q+1}SO; J^{0} = J and J′ is the limit of the J^{q}’s.

For later use we give an explicit description of J^{q}. Let α ∈ π_{k}(Ω^{q}Σ^{q}SO). There is some m ∈ N such that α ∈ π_{k}(Ω^{k}Σ^{k}(SO(m))). Then J^{q}(α) is the stable image of the composition

S^{k+m+q} ⟶^{Σ^{m}(α^{∗})} Σ^{m+q}SO(m) ⟶^{μ′} S^{m+q}

where α^{∗} is the q-th order adjoint of α and μ′ is the q-th suspension of the Hopf construction applied to the multiplication μ : S^{m−1} × SO(m) ⟶ S^{m−1}.

Remark 3. The image of J has been computed by Adams [1], Quillen [27] and Sullivan [35]. By the Kahn-Priddy theorem [13] it is known that J′ is surjective at the prime 2. However, J′ is not surjective at any odd prime [18].

3. Morse-theoretic interpretation of the Hopf invariants

As before, let M^{n} be a smooth, riemannian, compact manifold and f : M ⟶ R a smooth Morse-Smale function regular, maximal and constant on ∂M.

We assume from now on that M is simply connected and that f has a single local minimum.

Suppose that P and Q are consecutive critical points of f (in particular f(P) > f(Q)) of indexes, respectively, p and q.

For q ≥ 1, as in 2.2, consider the induced cofibration sequences: S^{q−1} ⟶^{α(Q)}

M′ ⟶ M′′ and S^{p−1} ⟶^{α(P)} M′′ ⟶ M′′′. In this case, denote by H(P, Q) the Hopf invariant H(α(P), α(Q)) : S^{p−1} ⟶ Σ^{q}ΩM. When q = 0 the relevant

cofibration sequence is S^{p−1} ⟶^{α(P)} ∗ = M′′ ⟶ S^{p} = M′′′ ↪ M and we let H(P, Q) be the adjoint of the inclusion S^{p} ↪ M.

Notice also that K(P, Q) = \overline{W^{u}(P) ⋂ W^{s}(Q)} is homeomorphic to the (un)reduced suspension ΣZ(P, Q). As M is simply connected there is a canonical adjunct (up to homotopy) of the inclusion i_{Z} : ΣZ(P, Q) ↪ M that we denote by l(P, Q) : Z(P, Q) ⟶ ΩM. By making the choices described in 2.2 we have a standard framing of the normal bundle of Z(P, Q) in S^{p−1} ≈ S^{u}(P). Consider an inclusion of a tubular neighborhood U ≈ D^{q} × Z ⊂ S^{p−1} induced by this framing. We fix an orientation of Z such that this inclusion is oriented. The Thom-Pontryagin construction applied to the fixed framing together with the map l(P, Q) gives the map

T (P, Q) : S^{p−1} ⟶ S^{p−1}/\overline{S^{p−1} − U} = (D^{q} × Z)/(S^{q−1} × Z) ⟶^{Δ} (D^{q} ×

Z)/(S^{q−1} × Z) ∧ Z^{+} ⟶^{p_{1}∧l(P,Q)} S^{q} ∧ (ΩM)^{+} = S^{q} ∨ S^{q} ∧ ΩM (Δ is induced

by the diagonal and p_{1} is the projection on S^{q}). Recall that h(P, Q) is the projection of T (P, Q) on S^{q} ∧ ΩM and that the projection of T (P, Q) on S^{q} is δ_{f}(P, Q).

Theorem 3.1. We have the equality: Σh(P, Q) = ΣH(P, Q). In particular, [Z(P, Q)]^{fr} ∈ Ω_{p−q−1}^{fr}(ΩM) equals the stable image of H(P, Q) + δ_{f}(P, Q).

Proof. When q = 0 the statement is immediate as K(P, Q) = S^{p} ↪ M. Assume from now on q > 0. If p = q + 1 we have h(P, Q) = H(P, Q) = 0 hence assume also that p > q + 1. Denote by Z_{i}(P, Q) the connected components of Z(P, Q) and notice that h(P, Q) is also given by the composition

S^{p−1} ⟶^{t} ∨_{i}S^{q} ∧ Z_{i}(P, Q) ⟶^{Σ^{q}l(P,Q)} S^{q} ∧ ΩM. Here t is a degree one map

given by the sum of the maps S^{p−1} ⟶ (D^{q} ×Z_{i}(P, Q))/(S^{q−1} ×Z_{i}(P, Q)) =

S^{q} ∨ S^{q} ∧ Z_{i}(P, Q) ⟶^{p_{2}} S^{q} ∧ Z_{i}.

The proof has five steps.

Some special neighborhoods of a critical point. As above, assume that for ϵ > 0 sufficiently small, P and Q are the only critical points in f^{−1}([f(Q) − ϵ, f(P)+ϵ]). As before, let M′ = f^{−1}(−∞, f(Q)−ϵ] and M′′ = f^{−1}(−∞, f(P)− ϵ]. Recall that γ is the flow induced by −∇f and let D(x, r) be the closed disk in M of radius r and center x. Let U′_{τ,ϵ} = {x ∈ M : f(Q) − ϵ ≤ f(x) ≤ f(Q) + ϵ, ∃t ∈ R ⋃{+∞, −∞} such that γ_{t}(x) ∈ D(Q, τ) ⋂ f^{−1}(f(Q))}.

It is useful to recall at this time that, by the Morse lemma, the topology of f inside U′_{τ,ϵ} is independent of τ, for τ sufficiently small.

Let U′ = U′_{τ,ϵ} for a small, fixed τ. Then U′ ≈ D^{q} × D^{n−q}, ∂U′ = A′ ⋃ B′ ⋃ C′ with S^{q−1} × D^{n−q} ≈ A′ = U′ ⋂ f^{−1}(f(Q) − ϵ), D^{q} × S^{n−q−1} ≈ C′ = U′ ⋂ f^{−1}(f(Q) + ϵ), B′ ≈ S^{q−1} × S^{n−q−1} × [0, 1] and ∇f is tangent to B′ in all points x ∈ Int(B′) (we consider the 0-end of B′ to be contained in A′) . Of course, M′′ ≃ M′ ⋃_{A′} U′. It is obvious that there is a second neighborhood U ⊂ U′ very close to U′, homeomorphic to U′, whose boundary admits the same structure as that of U′ and such that if we denote by A, B, C the respective pieces of the boundary of U, then A = A′, C ⊂ C′, C ≈ C′, B ≈ B′ and ∇f is transverse to B and points inside U.

The attaching map α(P). We may define a deformation retract r : M′′ × [0, 1] ⟶ M′′ whose 1-end r_{1} sends each point in M′′ along the flow γ (induced by −∇f) to the point where it first reaches M′ ⋃_{A} U. Let us denote Z = Z(P, Q). Consider the inclusion S^{p−1} ⊂ M′′ that represents the relative attaching map α(P). We see that S^{p−1} ⋂ C ≈ D^{q} × Z (where we choose the framing of the respective tubular neighborhood of Z in S^{p−1} as described in 2.2). With this identification, we have ∂D^{q} × Z = S^{q−1} × Z ⊂ ∂C ≈ S^{q−1} × S^{n−q−1}. By using the deformation r we may assume that the image of α(P) lies in M′ ⋃_{A} U and that α(P)(D^{q} × Z, S^{q−1} × Z) ⊂ (C, ∂C). Moreover, we may consider a collar neighborhood V of S^{q−1} ×Z in S^{p−1} − Int(D^{q} × Z) such that V ≈ S^{q−1} × Z × [0, 1] with S^{q−1} × Z × {1} ⊂ ∂D^{q} × Z and with the property that (with the identifications described above) α(P)|_{V} : S^{q−1} × Z × [0, 1] ⟶ B equals α(P)|_{(S^{q−1}×Z)} × id_{[0,1]} and α(P)(S^{p−1} − (V ⋃ D^{q} × Z)) ⊂ M′ − A.

Denote by V′ = V ⋃ D^{q} × Z, M^{∗} = M′ ⋃_{A} U and let T = \overline{S^{p−1} − V′}. Of course, V′ ≈ D^{q} × Z.

We now intend to describe the composition

c : S^{p−1} ⟶^{α(P)} M′′ ⟶^{∇} S^{q} ∨ M′′ ⟶ S^{q} ∨ M

Here, ∇ is the coaction and the last map is induced by inclusion. By the

definition of ∇ and making use of r this map is homotopic to S^{p−1} ⟶^{α(P)}

M′′ ⟶^{r_{1}} M^{∗} ⟶^{∇′} M^{∗}/A ⟶^{w} S^{q} ∨ M^{∗} ⟶ S^{q} ∨ M. Here ∇′ is the obvious

collapsing map and the last map is, as before, the inclusion. The homotopy equivalence w is the inverse of the obvious one obtained from the fact that A ≃ S^{q−1} and using the standard flow induced nullhomotopy of A ↪ M^{∗} (this is defined by first collapsing A to its core S^{q−1} = S^{u}(Q) and then collapsing this one to Q along flow lines). In other words we have w :

M^{∗}/A ⟶^{id} U/A ∨ M′/A ⟶^{w′} S^{q} ∨ M^{∗} whith w′ respecting the wedge, its

restriction to U/A is U/A = (D^{q} ×D^{n−q})/(S^{q−1}×D^{n−q}) ⟶^{p_{1}} D^{q}/S^{q−1} = S^{q}.

In the wedge S^{q} ∨ M^{∗} the point Q ∈ M^{∗} is identified with the image of S^{q−1} in the quotient D^{q}/S^{q−1}. The restriction of w′ to M′ sends A to Q by means of a map w′′ : M′ ⟶ M′ ⋃ W^{u}(Q) ↪ M^{∗} which is the inclusion outside a neighborhood of A, is defined inside this neighborhood by using the null-homotopy mentioned above and induces a homotopy equivalence M′/A ⟶ M^{∗}.

Description of c. The map c can be described by writting S^{p−1} = V′ ⋃_{∂V′} T and giving its restrictions to each of these two pieces. On V′ the map c is induced by the map of pairs (V′ ≈ D^{q} × Z, S^{q−1} × Z) ↪ (U, A) and composition with w′; this sends ∂T = ∂V′ = S^{q−1} × Z to Q. On T it is defined by k = w′′ ∘ r_{1} ∘ α(P).

We will now see that the composition k′ : T ⟶^{k} M^{∗} ↪ M is homotopic rel(∂T) to a map k′′ defined as follows. Consider the inclusion of pairs (D^{p}, S^{p−1}) ↪ (M, M′′) whose restriction to S^{p−1} is α(P) (here D^{p} ⊂ W^{u}(P), D^{p} ⋂ M′′ = ∂D^{p}). There is a deformation d : D^{p} ⟶ M of this inclusion that is induced by the flow γ, collapses Z ⊂ S^{p−1} to Q along γ, is constant ouside a neighborhood of Z in D^{p} and factors as D^{p} ⟶ D^{p} ⋃ W^{s}(Q) ↪ M. As D^{p} ⋃(W^{s}(Q) ⋂ M′′) ≃ ΣZ that means that, up to homotopy, d factors through i_{Z}. In S^{p−1} there is a deformation l : T ⟶ S^{p−1} that is constant outside a neighborhood of ∂T and that sends the point (x, y) ∈ S^{q−1} × Z = ∂T to (0, y) ∈ Z. The map k′′ is given by i_{Z} ∘ d ∘ l. To see that k′ and k′′ are homotopic, notice that they are both homotopic (rel boundary) to the following map k′′′: transport T homeomorphically along the flow till it reaches f^{−1}(f(Q)). Let T′ be the image of T inside this singular hypersurface. Use the conical structure of f^{−1}(f(Q)) around Q to deform ∂T′ to Q without leaving f^{−1}(f(Q)) and whithout moving points that are outside a small neighborhood of ∂T′.

It follows that we may use the map k′′ instead of k′ in the description of c. We get the following commutative diagram:

\xymatrix{S^{q−1} × Z \ar[rr] \ar[dd] \ar@{-->}[dr] &  & D^{q} × Z \ar@{.>}[dd] \ar@{-->}[dr] \\  & T \ar[dd]^{l} \ar[rr] &  & S^{p−1} \ar[dd]^{α′(p)} \\ S^{q−1} × Z \ar@{.>}[rr] \ar[dd] \ar@{-->}[dr] &  & D^{q} × Z \ar@{.>}[dd] \ar@{.>}[dr] \\  & D^{p} \ar[dd]^{d} \ar[rr] &  & ∨_{i}(S^{q} ∨ S^{q} ∧ Z_{i}) \ar[dd]^{j} \\ Q \ar@{.>}[rr] \ar[dd] \ar[dr] &  & S^{q} \ar@{.>}[dd] \ar@{.>}[dr] \\  & ΣZ \ar[dd]^{i_{Z}} \ar[rr] &  & S^{q} ∨ ΣZ \ar[dd] \\ Q \ar@{.>}[rr] \ar[dr] &  & S^{q} \ar@{.>}[dr] \\  & M \ar[rr] &  & S^{q} ∨ M}

In this diagram all horizontal squares are push outs and the vertical maps in the right corner are induced by the other three; Z_{i} are the connected components of Z. We have identified V′ to D^{q} × Z. The composition originating in T is the map k′′ described above and that defined on D^{q} × Z is the restriction of c (which, as mentioned above is the projection on D^{q} followed by the collapsing to D^{q}/S^{q−1} = S^{q}). Therefore, the composition in the right corner is homotopic to c. The map α′(P) is additive with respect to the connected components of Z. Its projection onto ∨_{i}Σ^{q}Z_{i} is homotopic to the degree one map t. Thus, the Hopf invariant being additive (in the first variable), we may assume from now on Z connected.

Identification of a Whitehead product. With this assumption, the next step is to consider the map j : S^{q} ∨ S^{q} ∧ Z ⟶ S^{q} ∨ ΣZ of the diagram above and show that its restriction to S^{q} ∧ Z is homotopic to the (generalized) Whitehead product of the inclusions S^{q} ↪ S^{q} ∨ ΣZ and ΣZ ↪ S^{q} ∨ ΣZ.

For this consider the next commutative diagram.

\xymatrix{S^{q−1} × Z \ar[rr] \ar[dd] \ar@{-->}[dr] &  & D^{q} × Z \ar@{.>}[dd] \ar@{-->}[dr] \\  & S^{q−1} × CZ \ar[dd] \ar[rr] &  & S^{q} ∧ Z \ar[dd] \\ S^{q−1} × Z \ar@{.>}[rr] \ar[dd] \ar@{-->}[dr] &  & D^{q} × Z \ar@{.>}[dd] \ar@{.>}[dr] \\  & D^{p} \ar[dd] \ar[rr] &  & S^{q} ∨ S^{q} ∧ Z \ar[dd] \\ Q \ar@{.>}[rr] \ar[dr] &  & S^{q} \ar@{.>}[dr] \\  & ΣZ \ar[rr] &  & S^{q} ∨ ΣZ}

Again, the horizontal squares are push outs and the vertical maps in the right corner are induced by the respective three others, CZ is the cone on Z. The map S^{q−1} × Z ⟶ D^{p} which is the restriction of l to ∂T factors as S^{q−1} × Z ⊂ S^{q−1} × CZ ⟶^{p_{2}} CZ ↪ D^{p} (the last map being the inclusion of CZ in W^{u}(P), the vertex of this cone being identified to P) and it is this factorization that is used in the upper, vertical, left square. It is clear that the top, vertical map in the right corner is, up to homotopy, the inclusion on the first factor. Now, the composition S^{q−1} × CZ ⟶ D^{p} ⟶ ΣZ is given by projection onto CZ and collapsing onto ΣZ. Similarly, D^{q} × Z ⟶ S^{q} is projection onto D^{q} and then collapsing onto S^{q}. This shows, by the definition of the Whitehead product [38], that the composition in the right corner is the wanted Whitehead product.

We therefore obtain that c is homotopic to the composition

c′ : S^{p−1} ⟶^{α′(P)} S^{q} ∨ S^{q} ∧ Z ⟶^{h} S^{q} ∨ ΣZ ⟶^{id∨i_{Z}} S^{q} ∨ M

with h = id_{S^{q}} ∨ [i_{S^{q}}, i_{ΣZ}].

Identification of the Hopf invariant. The last step of the proof is to use the factorization of c′ to evaluate the relevant Hopf invariant. We now look to Ω(h) and use the standard splitting of the loop space of a wedge to write the Hopf invariant H(P, Q) as the adjoint of the composition h′ :

S^{p−2} ↪^{i} ΩS^{p−1} ⟶^{u} Ω(S^{q} ∧ Z) × Ω(ΩS^{q} ∗ Ω(S^{q} ∧ Z)) ⟶^{v} Ω(ΩS^{q} ∗ ΩΣZ) ⟶^{y}

Ω(S^{q} ∧ ΩΣZ) ⟶ Ω(S^{q} ∧ ΩM).

By the basic properties of the Whitehead product we obtain that the restriction of h′′ = y∘v to Ω(S^{q} ∧Z) is just the looping of the map S^{q} ∧Z ⟶ S^{q} ∧ ΩΣZ induced by the inclusion Z ↪ ΩΣZ. This shows that the Hopf invariant verifies H(P, Q) = ϕ + ϕ′. Here, ϕ : S^{p−1} ⟶^{t} S^{q} ∧ Z ⟶ S^{q} ∧ ΩΣZ ⟶ S^{q} ∧ ΩM. As the composition S^{q} ∧ Z ⟶ S^{q} ∧ ΩΣZ ⟶ S^{q} ∧ ΩM is the q-th suspension of the adjunction of the inclusion i_{Z} we obtain that ϕ = h(P, Q).

The second homotopy class, ϕ′, is the composition S^{p−1} ⟶^{s} ΩS^{q} ∗Ω(S^{q} ∧ Z) ⟶^{z} ΩS^{q} ∗ ΩΣZ ⟶ S^{q} ∧ ΩM with s being the adjunction of p_{2} ∘ u ∘ i and z the top composition in the next diagram.

\xymatrix{ΩS^{q} ∗ Ω(S^{q} ∧ Z) \ar[r]^{x} \ar[d] & Ω(S_{1}^{q} ∨ S_{2}^{q}) ∗ ΩΣZ \ar[r] \ar[d] & ΩS^{q} ∗ ΩΣZ \ar[d] \\ S_{1}^{q} ∨ S^{q} ∧ Z \ar[d] \ar[r]^{id∨[,]} & S_{1}^{q} ∨ S_{2}^{q} ∨ ΣZ \ar[d] \ar[r]^{m∨id} & S^{q} ∨ ΣZ \ar[d] \\ S_{1}^{q} × S^{q} ∧ Z \ar[r] & (S_{1}^{q} ∨ S_{2}^{q}) × ΣZ \ar[r] & S^{q} × ΣZ}

Here, the columns are fibrations and the top row is induced by the bottom two; m is the folding map of the two spheres S_{1}^{q} and S_{2}^{q}; the bottom, left, horizontal map is trivial when projected on ΣZ.

We now compose z with the evaluation ev : ΣΩS^{q} ∧ ΩΣZ ⟶ S^{q} ∧ ΩΣZ. The composition ev∘z factors via the evaluation ev′ : ΣΩ(S_{1}^{q}∨S_{2}^{q})∧ΩΣZ ⟶ (S_{1}^{q} ∨ S_{2}^{q}) ∧ ΩΣZ = (S_{1}^{q} ∧ ΩΣZ) ∨ (S_{2}^{q} ∧ ΩΣZ). Notice that the projection of ev′ ∘ x on each of the factors of the wedge is null. This implies that Σϕ′ = 0 and concludes the proof.

Remark 4. It is instructive to consider the case of the Morse-Smale function f : S^{2} × S^{2} ⟶ R with precisely four critical points. Denote by P the maximum and let Q be one of critical points of index two. The equality ΣH(P, Q) = Σh(P, Q) comes down to the fact that the Thom-Pontryagin construction applied disjointly to two circles embedded with linking number one and trivially framed in S^{3} produces the Whitehead product S^{3} ⟶ S^{2} ∨ S^{2}.

Here is a first context in which h(P, Q) is relevant. Suppose f, P, Q are as in the theorem and fix a fiber bundle on M that is classified by a map μ : M ⟶ BSO(m) with p−2q < m. For x ∈ Ω_{∗}^{fr}(ΩM) let \overline{μ}(x) ∈ Ω_{∗}^{fr}(ΩM)

be defined as follows. Consider X ⟶^{g} ΩM together with a framing X ↪^{i} S^{k} representing x. The class \overline{μ}(x) is represented by X ⟶^{g} ΩM together with a

framing X ↪^{i} S^{k} ↪^{j} S^{k+m} given at a point a ∈ X by ((Ωμ ∘ g)(a)(i^{∗}(j)_{a}), i) (here j is the standard framing of S^{k} in S^{k+m}).

In a tubular neighborhood Y of M in the total space of μ we may consider a function g : Y ⟶ R giving the square of the distance from M. If Y is sufficiently small this function is non-degenerate in the direction of the fibre and, in particular, the difference f′ = f ∘ p − g is Morse (here p : Y ⟶ M is the restriction of the projection of the bundle). The critical points P and Q are again consecutive nondegenerate critical points of f′. Their indexes are respectively m + p and m + q. Let Z^{μ}(P, Q) be the (framed) connecting manifold of f′ (as a space it coincides with Z(P, Q) but its framing might be different). Recall that δ_{f}(P, Q)^{μ} is the relative attaching map induced on the Thom space of μ by δ_{f}(P, Q). It is easy to see [5] that δ_{f′}(P, Q) = δ_{f}(P, Q)^{μ}.

For x ∈ π_{k+n}(S^{n}) we denote by {x} ∈ π_{k}^{S} its stable image.

Proposition 3.2. In Ω_{p−q−1}^{fr}(ΩM) we have [Z^{μ}(P, Q)]^{fr} = \overline{μ}([Z(P, Q)]^{fr}). Moreover, if h^{∗}(P, Q) ∈ π_{p−q−1}Ω^{q}Σ^{q}ΩM denotes the q-th order adjoint of h(P, Q), then the stable difference {δ_{f}(P, Q)^{μ}} − {δ_{f}(P, Q)} equals

J^{q}(Ω^{q}Σ^{q}(Ωμ) ∘ h^{∗}(P, Q))

Proof. We use the notations fixed in the proof of the theorem. As seen above, the bordism class of Z(P, Q) in ΩM is given by a map S^{p−1} ⟶^{α}

S^{q} ∧ Z^{+} ⟶^{id∧l(P,Q)} S^{q} ∧ (ΩM)^{+}. Of course, the bordism class of Z^{μ}(P, Q) is

given by a similar map S^{m+p−1} ⟶^{α′} S^{m+q} ∧ Z^{+} ⟶^{id∧l(P,Q)} S^{m+q} ∧ (ΩM)^{+}. In

general, the map α′ is not the suspension of α. It is immediate that, as in [6], α′ = e∘Σ^{m}α where e : S^{m+q} ∧Z^{+} ⟶ S^{m+q} ∧Z^{+} is a homotopy equivalence that appears at the passage (along the flow γ_{1} induced by −∇f′ ) from a neighborhood of P to one of Q. More precisely e is induced by the map of pairs e′ : (D^{m} × D^{q} × Z, S^{m−1} × D^{q} × Z) ⟶ (D^{m} × D^{q} × Z, S^{m−1} × D^{q} × Z)

that takes (x, y, z) to (μ^{∗}(z)(x), y, z) where μ^{∗} : Z ⟶^{l(P,Q)} ΩM ⟶^{Ωμ} SO(m). The map e′ induces e by collapsing S^{m+q−1} × Z to a point. The first part of the statement is now clear. Moreover, notice that p_{1} ∘ e restricted to the q + m skeleton (which is a wedge of q + m-dimensional spheres in number equal to the number of connected components of Z) is the identity because the bundle is oriented. Its restriction to S^{q+m} ∧ Z is Σ^{q−1}J(μ^{∗}). Here, J(μ^{∗}) : S^{m} ∧ Z ⟶ S^{m} is defined by the Hopf construction on the map S^{m−1} × Z ⟶ S^{m−1} given by (x, y) ⟶ μ^{∗}(y)(x). Because p − 2q < m, δ_{f′}(P, Q) = Σ^{m}δ_{f}(P, Q) + Σ^{q−1}J(μ^{∗}) ∘ t. The statement follows from the definition of J^{q}.

Remark 5. a. It is clear that, as Σh(P, Q) = ΣH(P, Q), we may replace in the formula above h(P, Q) by H(P, Q). When p > q + 1 notice also that [h(P, Q)] = [H(P, Q] when viewed in H_{p−q−1}(ΩM) . Moreover, in this case, [h(P, Q)] is the fundamental class of Z(P, Q) in ΩM.

b. The second part of the above proposition, with H(P, Q) in the place of h(P, Q), is also a consequence of purely homotopical results of Dula [8].

c. The proof of the proposition is in fact the direct specialization to the Morse-Smale case of a result established in [6] for reasonable critical points (a class that contains all isolated, analytic singularities).

For the next result we consider two critical points P and R of f (which is a function as before) such that P and R are not necessarily consecutive but f(P) > f(R) and if Q is a critical point such that Q ∈ \overline{W^{u}(P)} ⋂ \overline{W^{s}(R)}, then P and Q are consecutive and so are Q and R (in other words all broken conecting flow lines between P and R are broken in just one point). We denote by I(P, R) the set of all such intermediate critical points Q associated to P and R.

For P and Q consecutive of indexes respectively p and q recall that we denote by [Z(P, Q)]^{fr} ∈ Ω_{p−q−1}^{fr}(ΩM) the ambient bordism class of Z(P, Q). We assume M oriented and make the choice of orientations described in 2.2. As before, assume the connecting manifolds Z(P, Q) oriented such that the standard framing inside S^{p−1} summed with this orientation gives the standard orientation of S^{p−1}.

Theorem 3.3. For the choice of orientations described above we have:

∑_{Q∈I(P,R)}(−1)^{(p+r)q}[Z(P, Q)]^{fr} • [Z(Q, R)]^{fr} = 0

where • is the product in Ω_{∗}^{fr}(ΩM).

Proof. We may assume whithout loss of generality that I(P, R) is contained is the same critical level f^{−1}(c). Let

K(P, R) = {x ∈ M : lim_{t→+∞}(γ_{t}(x)) = R, lim_{t→−∞}(γ_{t}(x)) = P } ⋃{P, R}

The proof is based on understanding how the ends of K(P, R) are embedded in M.

Identification of the ends of K(P, R). Let p, r, q be repectively the indexes of P, R and Q ∈ I(P, R). Consider a ∈ R such that f(R) < a < f(P) and let Z(P, R) = f^{−1}(a) ⋂ K(P, R). Then Z(P, R) is a manifold of dimension p − r − 1 (of course, Z(P, R) is not closed) and its homeomorphism type does not depend on the choice of a. Clearly, K(P, R) is homeomorphic to the unreduced suspension of Z(P, R). We also recall the notation K(P, Q) = \overline{W^{u}(P) ⋂ W^{s}(Q)}.

Around each critical point in I(P, R) we assume fixed a Morse chart inside which the metric is the canonical one. Let K_{τ} (P, R) be the set of all points x ∈ K(P, R) such that if for some t ∈ R we have γ_{t}(x) ∈ f^{−1}(c), then d(γ_{t}(x), Q) ≥ τ for all Q ∈ I(P, R) (d(,) being the distance in M). For τ sufficiently small this set is the (unreduced) suspension over Z_{τ} (P, R) = (K(P, R) ⋂ f^{−1}(c)) − ⋃_{Q∈I(P,R)} D(Q, τ) where D(Q, τ) is the disk in M of center Q and of radius τ (the intersection of S(Q, τ) = ∂D(Q, τ) and f^{−1}(c) is certainly transverse for τ small enough).

Notice that Z_{τ} (P, R) is a manifold with boundary whose homeomorphism type does not depend on the choice of τ, if this constant is smaller than some fixed τ′ > 0, and that its interior is homeomorphic to K(P, R). These statements follow from the Morse-Smale condition. Indeed, for Q ∈ I(P, R) let G(Q, τ) = f^{−1}(c) ⋂ S(Q, τ). Clearly, G(Q, τ) ≈ S^{q−1}×S^{n−q−1}. Consider a neighborhood of Q as described at the beginning of the proof of 3.1, U′ = U′_{τ,ϵ} and recall that for small ϵ and τ, C′ = ∂U′ ⋂ f^{−1}(c + ϵ) = D^{q} × S^{n−q−1} and with this identification {0} × S^{n−q−1} = S^{s}(Q). The intersection of W^{u}(P) with f^{−1}(c + ϵ) is identified with S^{u}(P) and therefore it intersects transversely S^{s}(Q) as well as ∂C′. This implies immediately that W^{u}(P) intersects transversely G(Q, τ). By the same method we obtain that W^{s}(R) intersects transversely G(Q, τ). This shows that for all small enough τ the intersection of K(P, Q) and G(Q, τ) is transverse. This implies all the claimed properties of Z_{τ} (P, R).

Fix Q and some τ as above and let ∂_{Q}Z = ∂Z_{τ} (P, R) ⋂ S(Q, τ), ∂_{Q}K = {x ∈ K_{τ} (P, R) : ∃t, γ_{t}(x) ∈ S(Q, τ) ⋂ f^{−1}(c)} ⋃{P, R}. Clearly, ∂_{Q}K is the suspension over ∂_{Q}Z and, in particular, if for a ∈ R such that f(R) < a < f(P) we denote ∂_{Q}^{a} = ∂_{Q}K ⋂ f^{−1}(a), then ∂_{Q}^{a} is homeomorphic to ∂_{Q}Z.

Embedding of ∂_{Q}K in M. Let W (P, Q) = {x ∈ W^{u}(P) : f(x) ≥ c, ∃t γ_{t}(x) ∈ f^{−1}(c), d(γ_{t}(x), Q) ≤ τ} ⋃ K(P, Q) and W (R, Q) = {x ∈ W^{s}(R) : f(x) ≤ c, ∃t γ_{t}(x) ∈ f^{−1}(c), d(γ_{t}(x), Q) ≤ τ} ⋃ K(Q, R). Let a and b be such that f(R) < b < c < a < f(P). Clearly, the interiors of W (P, Q) and W (R, Q) are (open) cones over their intersection with f^{−1}(a) and respectively f^{−1}(b). In particular, we have W_{a}(P) = W (P, Q) ⋂ f^{−1}(a) ≈ D^{q} × Z(P, Q) and W_{b}(R) = W (R, Q) ⋂ f^{−1}(b) ≈ Z(Q, R) × D^{n−q}. The set H(Q, τ) = {x ∈ f^{−1}(c) : d(x, Q) ≤ τ} is clearly a cone over its boundary which is G(Q, τ). Also ∂_{Q}^{a} ⊂ W_{a}(P) and ∂_{Q}^{b} ⊂ W_{b}(R).

Notice, that the union W (P, Q) ⋃ W (R, Q) ⋃ H(Q, τ) has the homotopy type of the wedge ΣZ(P, Q) ∨ ΣZ(Q, R) and H(Q, τ) ⋃ ∂_{Q}K ≃ Σ∂_{Q}Z ∨ Σ∂_{Q}Z. Therefore the inclusion ∂_{Q}K ⟶ H(Q, τ) ⋃ ∂_{Q}K ⟶ W (P, Q) ⋃ W (R, Q) ⋃ H(Q, τ) ⟶ M is seen to be homotopic to

∂_{Q}K = Σ∂_{Q}Z ⟶ Σ∂_{Q}Z ∨ Σ∂_{Q}Z ⟶ ΣZ(P, Q) ∨ ΣZ(Q, R) ⟶ M

where the first map is the pinch map, the second is given by wedging the

suspensions of t_{1} : ∂_{Q}^{a} ⊂ W_{a}(P) = D^{q} × Z(P, Q) ⟶^{p_{2}} Z(P, Q) and t_{2} : ∂_{Q}^{a′} ⊂

W_{a′}(R) = Z(Q, R) × D^{n−q} ⟶^{p_{1}} Z(Q, R) and the third map is the inclusion of K(P, Q) ∨ K(Q, R).

Because M is simply connected we have a well defined map l(P, R) : ∂_{Q}Z ⟶ ΩM which is the adjunct of the inclusion ∂_{Q}K ⊂ M. From the factorization above, by adjunction, we obtain that l(P, R) factors as

∂_{Q}Z ⟶ Z(P, Q) × Z(Q, R) ⟶ ΩM × ΩM ⟶ ΩM (1)

where the first map is the product t_{1} × t_{2}, the second map is the product of the adjoints of the respective inclusions and the third is the loop product.

Description of ∂_{Q}Z. We want to observe that t_{1} × t_{2} is a homeomorphism. Fix a and b such that f(R) < b ≤ c ≤ a < f(P) and c − b < ϵ, a − c < ϵ. We have the homeomorphsims ∂_{Q}^{b} ≈ ∂_{Q}Z ≈ ∂_{Q}^{a} induced by transporting the respective sets along the flow γ. Consider the inclusion ∂_{Q}^{a} ↪ S^{q−1} × Z(P, Q) (here S^{q−1} ×Z(P, Q) ⊂ D^{q} ×Z(P, Q) = W^{u}(P) ⋂ f^{−1}(a) ⋂ U′). By transporting it along the flow γ till we reach the level set f^{−1}(b) we obtain

an inclusion ∂_{Q}^{b} ↪ S^{q−1} ×Z(P, Q) ↪^{j} S^{q−1} ×S^{n−q−1} = ∂U′ ⋂ f^{−1}(b) (notice

that now S^{q−1} is identified here to S^{u}(Q)). The map j is the identity on the first factor and the usual inclusion on the second. At the same time we also have an inclusion ∂_{Q}^{b} ↪ Z(Q, R) × S^{n−q−1}. It follows that the product t_{2} × t_{1} is identified to the inclusion v : ∂_{Q}^{b} ↪ Z(Q, R) × Z(P, Q) = (Z(Q, R) × S^{n−q−1}) ⋂(S^{q−1} × Z(P, Q)). On the other hand each point of this last intersection belongs to ∂_{Q}^{b}. Indeed, with the given parametrizations all the points in S^{q−1} × Z(P, Q) belong to W^{u}(P) and all the points in Z(Q, R) × S^{n−q−1} belong to W^{s}(R). Therefore v is also surjective.

Consequences in Ω_{∗}^{fr}(ΩM). Consider the framing of ∂_{Q}Z = Z(P, Q) × Z(Q, R) inside S^{n−q−1} × S^{q−1} obtained as the restriction of the normal framing of K(P, R) inside f^{−1}(a). This framing is the product of the framing induced by the standard framing of S^{p−1} with that induced by the standard framing of S^{n−r−1}. Therefore, the framing of Z(P, Q) × Z(Q, R) inside S^{n−q−1} ×S^{q−1} coincides with the product of the standard framings of Z(P, Q) in S^{n−q−1} (induced by the framing of S^{p−1}) and the standard framing of Z(Q, R) in S^{q−1}. Let \overline{Z}(P, Q) be the framed bordism representative given by l(P, Q) and the framing of Z(P, Q) inside S^{n−q−1}. The orientation of \overline{Z}(P, Q) is such that the framing summed with this orientation gives the standard orientation on S^{n−q−1}.

We orient Z(P, R) such that the ordered sum of the framings induced first from S^{p−1}, from S^{n−r−1} and the orientation of Z(P, R) gives the orientation of f^{−1}(a).

Assume now that M has trivial stable normal bundle. In this case, after embedding M in a high dimensional sphere we see that because of (1) and as

∐_{Q∈I(P,R)} ∂_{Q}Z = ∂(Z_{τ} (P, R))

we have Σ_{Q∈I(P,R)}ϵ′(Q)[\overline{Z}(P, Q)]^{fr} • [Z(Q, R)]^{fr} = 0.

The sign ϵ′(Q) is +1 if the orientation induced from that of Z(P, R) on \overline{Z}(P, Q) × Z(Q, R) coincides with the product orientation and is −1 otherwise. With our conventions ϵ′(Q) = (−1)^{(n−r)q}ϵ_{0} where ϵ_{0} depends only on p, r, n.

We now return to the case when the stable bundle of M is general. Similarly to the proof of Proposition 3.2 we consider a function f′ = f ∘ p + g defined on a neighborhood U of the 0-section of the normal bundle ν of an embedding of M in a high dimensional sphere S^{n+m}. Here, g is the square of a distance function measuring the distance from M. We use the notation Z^{−ν}(P, Q) for the connecting manifold of P and Q for the function f′. By the formula above we have Σ_{Q∈I(P,R)}(−1)^{(m+n−r)q}[\overline{Z^{−ν}}(P, Q)]^{fr} • [Z^{−ν}(Q, R)]^{fr} = 0. We have [Z^{−ν}(Q, R)]^{fr} = [Z(Q, R)]^{fr}, [Z^{−ν}(P, Q)]^{fr} = [Z(P, Q)]^{fr}. Moreover, because the stable normal bundle of U is trivial, the framings of \overline{Z^{−ν}}(P, Q) and Z^{−ν}(P, Q) are the same up to a sign coming from a possible difference in orientations. With our conventions the sign is (−1)^{(m+n−p)q}.

Remark 6. In [3],[4] Cohen, Jones and Segal pursue a systematic analysis of the ends of the moduli spaces of connecting flow lines (even for non consecutive critical points) together with the relevant framings (the definition they use for these framings is somewhat different from ours, though). In particular, the fact that t_{1} × t_{2} above is a homeomorphism is a direct consequence of a more general result of Betz as described in [3]. For completeness we have included a direct justification in the proof above.

4. Applications and Examples.

As in the last section M^{n} is a compact, smooth, riemannian, simply connected manifold. Suppose f : M ⟶ R is a Morse-Smale function and P and Q are consecutive critical points of index, respectively, p and q. Recall that h^{∗}(P, Q) ∈ π_{p−q−1}Ω^{q}Σ^{q}ΩM is the q-th order adjoint of h(P, Q). Let h^{S}(P, Q) ∈ π_{p−q−1}^{S}(ΩM) be the class of h^{∗}(P, Q). Let k(P, Q) = min{j : h^{S}(P, Q) ∈ Im(Ω^{j}Σ^{j}ΩM)}. This gives a measure of the embedding complexity of Z(P, Q). Indeed, if the framed embedding Z(P, Q) ↪ S^{p−1} = S^{u}(P) factors as a composition of framed embeddings Z(P, Q) ↪ S^{t−1} ↪ S^{p−1}, then k(P, Q) ≤ t − p + q. For example, if k(P, Q) > 0, then Z(P, Q) is not a sphere.

Of course, as k(P, Q) is defined homotopically, it is invariant to deformations of f via Morse-Smale functions.

We intend here to give a method to estimate k(P, Q). In particular, we construct examples when k(P, Q) = ind(Q) even if δ_{f}(P, Q) = 0. Of course, we always have k(P, Q) ≤ ind(Q). Along the way, we also detect certain Poincaré complexes that are not smoothable.

4.1. Non-smoothable Poincaré duality complexes and embedding complexity of connecting manifolds. We start with two simple consequences of 3.2.

Corollary 4.1. Assume that f : M ⟶ R is a smooth Morse-Smale function and that P and Q are consecutive critical points of f. Then, for some ϵ ∈ {−1, 1}, we have ϵ{δ_{−f}(Q, P)} = {δ_{f}(P, Q)} + Δ(P, Q). Where {x} is the stable image of x and Δ(P, Q) = J^{q}(Ω^{q}Σ^{q}(Ων) ∘ h^{∗}(P, Q)) (ν is the stable normal bundle of M).

Proof. We apply 3.2 to the stable normal bundle of M together with the duality results (mentioned in 2.3) implying δ_{−f}(Q, P) = ϵδ_{f}(P, Q)^{ν}.

Suppose now that H_{∗}(M; Z) is torsion free and that for some q < p < n/2 we have H_{q}(M; Z) ≈ H_{p}(M; Z) ≈ Z, H_{∗}(M; Z) = 0 for q < ∗ < p, ∗ = q − 1, ∗ = p+1. Then, in a minimal cell decomposition of M, there are two pairs of dual cells e^{p}, e^{n−p} and e^{q} and e^{n−q} representing Poincaré dual generators in H_{p}(M), H_{n−p}(M) and, respectively, H_{q}(M) , H_{n−q}(M). As e^{p}, e^{q} are attached in succession there is a relative attaching map δ : S^{p−1} ⟶ S^{q} and similarly e^{n−q} and e^{n−p} being attached successively we have another relative attaching map δ′ : S^{n−q−1} ⟶ S^{n−p}.

In all this section ϵ ∈ {−1, +1}. Its presence in the formula below reflects an indeterminancy caused by the fact that if δ is the relative attaching map above, then there is a different minimal cell decomposition having as corresponding relative attaching map −δ.

Corollary 4.2. In the setting above δ and δ′ are (up to sign) independent of the minimal CW -decomposition used in their definition and {δ} = ϵ{δ′} mod (Im(J^{q})).

Proof. The first part is immediate.We apply the corollary above to a perfect Morse function and its consecutive critical points P, Q of indexes p and q. Because of the first part we have δ_{f}(P, Q) = δ and δ_{−f}(Q, P) = δ′.

Remark 7. a. Corollary 4.2 can also be proven by purely homotopical methods. Surprisingly, it appears not to have been known before.

b. For spaces more general than those appearing in 4.2 the relative attaching maps of the type of δ and δ′ depend on the specific minimal celldecomposition used. Here is a relevant example (well known in the study of the Mislin genus [24]). Let X = S^{9} ⋃_{5ν} e^{13} and X′ = S^{9} ⋃_{13ν} e^{13} here ν ∈ π_{12}(S^{9}) ≈ π_{3}^{S} ≈ Z/24 is a generator. These two complexes are not homotopy equivalent (even if they are so when localized at each prime p). However, it is easy to show that X′ ∨ S^{9} ≃ X ∨ S^{9}. Hence, we have two very different minimal cell decompositions for the space Y = X ∨ S^{9}. In particular, if N is a manifold with simply connected boundary having the homotopy type of Y and of dimension greater than 26, each of these two cell decompositions is induced by a perfect Morse-Smale function. These two Morse-Smale functions are in different connected components of the space of perfect Morse-Smale functions of N. However, they are in the same component of the space of perfect Morse functions. Indeed, by a result of Matsumoto [23] the space of perfect Morse functions on a simply connected manifold with simply connected boundary, of dimension greater than 5 and with torsion free homology is connected (see also [2], for more general results [30]).

It is immediate to see that if Δ(P, Q) ∉ Im(J^{t}), then k(P, Q) > t. The obvious question that we consider now is what values can take the twisting Δ(P, Q).

Remark 8. Certainly, there are examples when ΣH(P, Q) is not vanishing, but Δ(P, Q) is null. An instructive example is that of CP^{n} together with a perfect Morse-Smale function. If P and Q are two arbitrary consecutive critical points of f, then p = ind(P) = ind(Q) + 2 and H(P, Q) ∈ π_{p−1}(Σ^{p−2}ΩCP^{n}) ≈ Z is a generator. On the other hand the difference δ_{f}(P, Q) − δ_{−f}(Q, P) is non zero iff n is even (because in that case w_{2} ≠ 0).

In the constructions below the strategy will be the following:

• Construct a certain Poincaré duality CW -complex X with an explicit cell decomposition.

• Assuming X smooth, consider a perfect Morse-Smale function f on X which induces the fixed cell-decomposition at least below the middle dimension.

• Evaluate the restriction of the stable normal bundle of X to the middle dimensional skeleton.

• Identify two consecutive critical points P and Q and evaluate using 4.1, 4.2 or 3.2 the twisting Δ(P, Q).

Some examples of Larry Smith. Let X = (S^{p} ∨ S^{n−p}) ⋃_{h} e^{n} where h = [i_{1}, i_{2}]+i_{2}∘x with p < n/2, i_{1} : S^{p} ⟶ S^{p}∨S^{n−p} and i_{2} : S^{n−p} ⟶ S^{p}∨S^{n−p} the inclusions, [−, −] the Whitehead product, x ∈ π_{n−1}(S^{n−p}) such that x ∉ Im(J). Then, obviously X is a Poincaré duality space, but it is not smoothable by 4.2 as the stable difference of the two relative attaching maps associated the first to the p and 0 cells, and the second to the n and n − p cells, is equal to x.

Smith has proved this result by purely homotopical methods [32]. We indicate below a different, very short, purely homotopical proof of this same fact. It pinpoints the homotopical content of our Morse theoretical techniques.

Assume that X is smoothable. Thus, its stable normal bundle ν is orthogonal. Fix m = rank(ν) and let T^{ν}(X) be the associated Thom space. It is Spanier-Whitehead dual to X_{+}. In particular, the first notrivial relative attaching map in T^{ν}(X), δ : S^{m+p−1} ⟶ S^{m} equals stably (up to sign) the top relative attaching map of X, δ′ : S^{n−1} ⟶ S^{n−p}. Of course, δ′ = x hence δ = ϵx. Now, δ is already present in the Thom space, T^{ν}(S^{p}), of the restriction of ν to the bottom sphere S^{p}. By classical results (appearing already in [25] or [1]) we have that T^{ν}(S^{p}) ≃ S^{m} ⋃_{τ} e^{m+p} and τ ∈ Im(J). As τ = δ we are led to a contradiction.

Remark 9. a. Of course, if x ∈ Im(J_{P L}), then X has the homotopy type of a P L manifold [32].

b. Clearly, one may extend the examples above by producing k − 1- connected Poincaré duality complexes of dimension n, 2 ≤ k < n/2 with a relative attaching map of the top cell with respect to some n−k cell that does not belong to Im(J). Of course, this is the content of Spivak’s ”first smoothing obstruction” [33]( see also [15]) which is thus recovered from 4.1. Notice, on the other hand, that Kervaire’s original non-smoothable PL-manifold [14], as well as the examples of Eells and Kuiper [9], are not detected by these means. Indeed, the non-smoothable Poincaré spaces obtained by our methods are all stable in the sense that they remain non-smoothable after crossing with a sphere.

Non-smoothable Poincaré spaces with vanishing Spivak first smoothing obstruction. We construct here an example of a non-smoothable Poincaré complex whose non-smoothability can not be detected by the relative attaching maps of the top cell.

Take q > 2 and Z = (S^{2} ∨ (S^{q} ⋃_{η} e^{q+2})) ⋃_{[i,j]} e^{q+4} with i : S^{2} ⟶ S^{2} ∨ (S^{q} ⋃_{η} e^{q+2}) the inclusion, η ∈ π_{q+1}(S^{q}) ≈ Z/2 a generator and j ∈ π_{q+2}(S^{q} ⋃_{η} e^{q+2}) such that the image of j via the pinching map p : S^{q} ⋃_{η} e^{q+2} ⟶ S^{q+2} is equal to twice a generator of π_{q+2}(S^{q+2}). Consider y ∈ π_{q+1}^{S} such that 2y ∉ Im(J^{2}). Let BF be the classifying space of stable spherical fibrations. As π_{k}(BF) = π_{k−1}^{S} there is a spherical fibration given

by the composition μ : Z ⟶^{t} S^{q+2} ⟶^{u} BF_{m} ↪^{v} BF with y ≃ (v ∘ u)^{∗}, BF_{m}

is the classifying space of spherical fibrations of fibre S^{m−1} (m big enough) and t is induced by the collapsing S^{2} ∨ (S^{q} ⋃_{η} e^{q+2}) ⟶ S^{q+2}. Let N be a trivial, smooth thickening of Z - this is a smooth manifold with boundary of dimension n > 2(q + 4) + 1 having the homotopy type of Z and which embedds in S^{n} (by [37] it is unique up to isotopy). Let N′ be the double of N. As N′ is the union of two copies of N pasted over ∂N, two copies of the spherical fibration μ can be also pasted together to give a new spherical fibration μ′ : N′ ⟶ BF_{m}. The total space K of μ′ is our example.

Remark 10. There is another more elegant way to produce μ′ out of μ. It was indicated to me by Mark Mahowald.

First a general remark of some independent interest (it appears to be folklore, for extensions see [17]). Because N′ is the double of a trivial thickening, the suspension ΣN′ splits as ΣN ∨ Σ^{2}N^{#} ∨ S^{n+1} with N^{#} homotopy equivalent to the complement of N in S^{n}. Indeed, N is embeddable in S^{n}, therefore we have S^{n} = N ⋃_{∂N} N^{#}. The inclusions of N and N^{#} in S^{n} being homotopically trivial, we obtain a cofibration sequence S^{n} ⟶ Σ∂N ⟶ ΣN ∨ ΣN^{#}. This cofibration sequence splits and thus Σ∂N ≃ ΣN ∨ΣN^{#} ∨S^{n}. Now N′ = N ⋃_{∂N} N hence ΣN′ = ΣN ⋃_{Σ∂N} ΣN. The description of Σ∂N implies the claimed splitting.

Consider μ^{∗} : ΣN ⟶ B^{2}F whose adjoint is μ. It extends trivially to (μ^{∗})′ : ΣN′ = ΣN ∨ Σ^{2}N^{#} ∨ S^{n+1} ⟶ B^{2}F. Its adjoint gives μ′.

Lemma 4.3. The space K does not have the homotopy type of a smooth manifold.

Proof. Assume K smooth. Its stable normal bundle ν : K ⟶ BSO(m′), m′ > m + n, has the property that if ν′ : K ⟶ BSO(m′) ⟶ BF_{m′} is the associated spherical fibration, then ν′|_{N} + μ = 0 in [N, BF] (because this sum gives the Spivak spherical fibration of N which is trivial). Therefore ν′|_{N} = −μ. Let f : K ⟶ R be a perfect Morse-Smale function inducing the standard cell decomposition given for Z on the q + 4 dimensional skeleton of K, K^{(q+4)} ≃ Z. We may assume also that f is such that the critical points P and Q of indexes, repectively, q + 4 and 2 are consecutive. Of course, δ_{f}(P, Q) = 0. We intend to evaluate Δ(P, Q). Consider the following diagram.

\xymatrix{S^{q+3} \ar[r] \ar[dr]_{l} \ar[ddr]_{s} & S^{2} ∧ Ω(S^{2} ∨ (S^{q} ⋃ e^{q+2})) \ar[d]^{S^{2}∧Ων} \ar[r] & S^{2} ∧ ΩS^{q+2} \ar[ddl]^{−Σ^{2}Ωu} \\  & S^{2} ∧ SO(m′) \ar[d] \\  & S^{2} ∧ ΩBF_{m′}}

By the basic properties of the Whitehead product, the top horizontal composition is twice the inclusion of the bottom cell. This implies that s = −2Σ^{2}y. Using 4.1 and the diagram below we see that δ_{−f}(Q, P) agrees stably (up to sign) with the composition k ∘ Σ^{m′}l = k′ ∘ Σ^{m′}s .

\xymatrix{S^{m′} ∧ S^{q+3} \ar[r]^{Σ^{m′}l} \ar[dr]_{Σ^{m′}s} & S^{m′} ∧ S^{2} ∧ SO(m′) \ar[r]^{k} \ar[d] & S^{m′+2} \ar[d]^{id} \\  & S^{m′} ∧ S^{2} ∧ ΩBF_{m′} \ar[r]_{k′} & S^{m′+2}}

Here k and k′ are the double suspensions of the maps induced by the Hopf construction applied respectively to the actions S^{m′−1} × SO(m′) ⟶ S^{m′−1} and S^{m′−1} × ΩBF_{m′} ⟶ S^{m′−1}. As s is itself a double suspension we obtain that k′ ∘ Σ^{m′}s is stably equal to the class of s in π_{q+1}(ΩBF) = π_{q+1}^{S}. Therefore, Δ(P, Q) = −2y which leads to a contradiction.

Clearly the relative attaching maps of the top cell S^{n+m−1} ⟶ S^{n+m−2} and S^{n+m−1} ⟶ S^{n+m−q} vanish.

Remark 11. a. It is easy to see that if y lifts to an element in π_{∗}(ΩBP L), then μ and μ′ are PL-bundles and therefore K is a PL-manifold.

b. Mark Mahowald has described another interesting non-smoothable Poincaré space that we now present (see also [22] p.408).

Let U be the total space of the spherical fibration classified by η_{4}^{∗} ∈ π_{17}(BF_{14}) where η_{4}^{∗} is the adjoint of the fourth element of Mahowald’s family [21]. Recall that η_{4} ∉ Im(J).

A cell decomposition of this space is as follows. Let i_{n} be a generator of π_{n}(S^{n}) and ν_{n} a generator of π_{n+3}(S^{n}). Let T = S^{13} ⋃_{ν_{13}} e^{17}. The Whitehead product [ν_{13}, i_{13}] being null ([36], [20]) the relative Whitehead product [i_{17}, i_{13}] ∈ π_{29}(T, S^{13}) pulls back to an element z ∈ π_{29}(T) and U = T ⋃_{z} e^{30}.

Notice that Σ^{n}U splits as S^{13+n} ⋃_{ν_{13+n}} e^{17+n} ⋃_{i_{13+n}∘η_{4}} e^{n+30}.

The space U is not smoothable. This is detectable by a top cell argument as follows.

We have T = ΣT′. Thus, we may interpret T as the cofibre of a trivial attaching map T′ ⟶ ∗. Therefore, U × S^{n} has a cone decomposition that starts with T ∨ S^{n} and has two other stages given by cofibration sequences T′ ∗ S^{n−1} ⟶ T ∨ S^{n} ⟶ U′ and S^{29} ∗ S^{n−1} ⟶ U′ ⟶ U × S^{n}. By a Spanier-Whitehead duality argument, one sees that if U and hence U × S^{n} are smoothable, then the relative attaching map S^{n+29} ⟶ Σ^{n}T should not contain η_{4}.

Obviously, this example does not fit our Morse theoretic setting. However, a critical point approach is possible by using degenerate critical points as in [6].

Morse-Smale functions with twisting realizing elements of Im(J^{q}). We consider here the question of what elements in Im(J^{q}) can be realized as differences, Δ(P, Q), of attaching maps like in Corollary 4.1.

Lemma 4.4. For any x ∈ Im(J^{q}), q ≥ 2, there is a smooth manifold M and a Morse-Smale function f : M ⟶ R with consecutive critical points P and Q such that Q has index q and δ_{f}(P, Q) = 0, δ_{−f}(Q, P) = ϵx.

Proof. As x ∈ Im(J^{q}) there is x^{1} : S^{p−1} ⟶ Σ^{q}SO(m) such that x is given by J^{q}(x^{∗}) where x^{∗} is the q-th order adjoint of x^{1} and m is large enough. Denote by BS′ the p−q-dimensional skeleton of BSO(m) and let S′ = ΩBS′. Of course, x^{1} lifts to an element x′ ∈ π_{p−1}(Σ^{q}S′).

There is a fibration ΩS^{q} ∗ S′ ⟶^{j} S^{q} ∨ BS′ ⟶ S^{q} × BS′ and a natural inclusion Σ^{q}S′ = S^{q−1} ∗ S′ ↪^{i} ΩS^{q} ∗ S′. Let x′′ = j ∘ i ∘ x′. Denote by L the cofibre of x′′. Fix a smooth, trivial, n-dimensional thickening of L denoted by H and let H′ be its (smooth) double. We need to define a certain fiber bundle over H′. Denote by μ : H ≃ L ⟶ BSO the bundle defined by the trivial extension of the map S^{q} ∨ BS′ ↪ S^{q} ∨ BSO(m) ⟶^{c} BSO(m) ↪ BSO (c being the obvious collapsing). As discussed above one can extend μ to a bundle μ′ : H′ ⟶ BSO and it lifts to a bundle τ : H′ ⟶ BSO(m′) with m′ > n, m. Let G be the total space of the associated sphere bundle. As H′ is smooth we see, by a transversality argument, that we may assume τ smooth and therefore G is also smooth. There is a perfect Morse-Smale function f : G ⟶ R such that the critical point P of index p and the critical point Q of index q corresponding to S^{q} ↪ L ↪ H′ are consecutive, and such that the attaching map of the p-dimensional cell is given by x′′. We have δ_{f}(P, Q) = 0. We now intend to compute δ_{−f}(Q, P). Let ν be the stable normal bundle of G. Obviously, its restriction to L ⊂ H′ coincides with −μ. Notice also that, as μ is null on S^{q}, Δ(P, Q) can be estimated by using the composition

S^{p−1} ⟶^{x′} S^{q} ∧ S′ ↪ S^{q} ∧ ΩG instead of H(P, Q). Indeed, the image of this composition by the map S^{q} ∧ ΩG ⟶ S^{q} ∧ SO(m′) coincides with the image of H(P, Q). The composition S^{q} ∧ S′ ↪ S^{q} ∧ ΩG ⟶^{Σ^{q}ν} S^{q} ∧ SO(m′) is homotopic to the negative of the inclusion S^{q} ∧ S′ ↪ S^{q} ∧ SO(m′).

Therefore, by applying 3.2, we see that δ_{−f}(Q, P) is (up to sign) equal to J^{q}(x^{∗}) = x.

Remark 12. It is likely that the Corollaries 4.1 and 4.2 as well as Proposition 3.2 have analogues in the P L and T op categories.

4.2. An extension of the Morse complex and detection of connecting flow lines. The results of the second section can be used for the detection of critical points and connecting flow lines. Indeed, assume that P, Q are consecutive critical points of the Morse-Smale function f : M ⟶ R. If the suspension of the Hopf invariant H(P, Q) is not zero, then there are flow lines connecting P to Q. Of course, such connecting flow lines are already known to exist if δ_{f}(P, Q) ≠ 0 but, as seen before, this relative attaching map can vanish without ΣH(P, Q) being also null.

The energy functional on ΩS^{3}. This example has been suggested to me by Raoul Bott. Consider the energy functional E on Ω(S^{3}; u, v), the space of piecewise smooth curves connecting the points u, v ∈ S^{3}, (with u and v different and not antipodals). It is well known that E is a perfect Morse function [26] whose critical points are the geodesics connecting u and v. Fix two such geodesics P and Q. Approximate Ω(S^{3}; u, v) by a finite dimensional manifold that contains P and Q, on which the restriction of E, E′, is still Morse, has critical points of the same index and the sets Ω^{a} = (E′)^{−1}(−∞, a] are compact for a ≤ b = max{E(P), E(Q)} + τ (τ > 0 small) and have the same homotopy type as E^{−1}(−∞, a] [26]. Fix a metric on Ω^{b} such that E′ is Morse-Smale. We want to remark that P and Q are connected by some flow lines of the flow induced by −∇E′. Notice that Ω(S^{3}; u, v) is homotopy equivalent to ΩS^{3}. Assume R is another critical point of E in Ω^{b} such that ind(P) = ind(R) + 2, then δ_{f}(P, R) = 0. However, ΣH(P, R) ≠ 0. This implies, by transitivity, that P and Q are also connected by some flow line.

Extension of the Morse complex. The ”detection” arguments above can be pursued further. For example, if P, Q and Q, R are two pairs of consecutive critical points for which the homology classes [h(P, Q)], [h(Q, R)] ∈ H_{∗}(ΩM; Z/2) are known and if [h(P, Q)] • [h(Q, R)] ≠ 0 it follows from 3.3 that there is at least another critical point Q′ that is connected via possibly broken flow lines to both P and R (• being here the Pontryagin product).

Here is a way to encode in a somewhat global fashion the type of information given by 3.3.

As before f : M ⟶ R is a Morse-Smale function, constant, regular and maximal on ∂M. Let a_{1} < a_{2} < ... < a_{r} be a set A of real numbers such that the critical values of f all appear among the a_{i}’s as well as f(∂M). Let C_{k} = Ω_{∗}^{fr}(ΩM) < X ∈ f^{−1}(a_{k}) : ∇f(X) = 0 > (where R < X_{1}, ..., X_{n} > is the free R-module generated by X_{1}, ..., X_{n}). If X is a critical point let x be its index. Let d : C_{i} ⟶ C_{i−1} be the unique Ω_{∗}^{fr}(ΩM)-module morphism given by

d(P) = ∑_{X∈f^{−1}(a_{i−1}),∇f(X)=0}(−1)^{px}[Z(P, X)]X

Corollary 4.5. The graded Ω_{∗}^{fr}(ΩM)-module (C_{∗}, d) is a chain complex.

Proof. Notice that any pair of critical points y ∈ f^{−1}(a_{i}), x ∈ f^{−1}(a_{i−1}) are consecutive and apply 3.3.

Remark 13. a. A somewhat easier to handle complex is obtained by replacing Ω_{∗}^{fr}(ΩM) with the Pontryagin ring H_{∗}(ΩM; Z/2) and using instead of d, d′(P) = ∑_{X∈f^{−1}(a_{i−1}),∇f(X)=0} H′(P, X)X where, for two consecutive critical points P and Q, we denote by H′(P, Q) the number of elements in Z(P, Q) if p = q+1 and the homology class of H(P, Q) in H_{p−q−1}(ΩM; Z/2) if p > q + 1. If f is a self indexed function and A = N, then this complex is the Morse complex of f tensored with H_{∗}(ΩM; Z/2) (recall that M is simply connected).

b. It would be interesting to know whether one can deduce the existence of the discussed chain complex by analytic methods.

Fusion of critical points. One initial motivation for this work was the problem of constructing functions with the least possible number of (possibly degenerate) critical points on a given smooth manifold M. It is well-known that a strict lower bound for this number is the Lusternik-Schnirelmann category cat(M) of M [19]. It was shown in [7] that when M is 2-connected, and k ≥ dim(M) there is a function on M × D^{k} regular, maximal and constant on ∂(M × D^{k}) which realizes the lower bound given by the category up to one unit. The next step is to understand when two consecutive critical points of a given function can be ”fused” together. More precisely, let f : M ⟶ R be a smooth function and let P, Q be consecutive critical points of f. The question is whether there is a function f′ equal to f in the exterior of some neighborhood U of the closure of the points situated on flow lines connecting P to Q and having just one critical point in U. It is natural to first assume that P and Q are consecutive, non-degenerate critical points of indexes, respectively, p and q, and that, with respect to some fixed metric on M, f is Morse-Smale. One can also weaken the question by asking P and Q to be fused to a ”reasonable” critical point [7], a class that contains all critical points of locally analytic functions.

Lemma 4.6. Assume that f^{−1}(−∞, f(Q)) has the homotopy type of a kdimensional CW -complex with k < q − 1. If H(P, Q) ≠ 0, then P and Q cannot be fused to a reasonable critical point.

Proof. This is a simple consequence of some results in [7]. First, as P and Q are consecutive and f is Morse-Smale we may assume that f(P) > f(Q) and that P and Q are the only critical points in f^{−1}[f(Q), f(P)]. If P and Q can be fused to a reasonable critical point, one deduces [7] a cofibration sequence Z ⟶ M′ ⟶ M′′′ where M′ = f^{−1}(−∞, f(Q) − τ], M′′′ = f^{−1}(−∞, f(P) + τ] with τ > 0 small. Let M′′ = f^{−1}(−∞, f(Q) + τ]. The

composition S^{p−1} ⟶ M′′ ⟶^{∇} S^{q} ∨ M′′ is homotopic to S^{p−1} ⟶^{∇′} S^{q} ∨

S^{p−1} ⟶^{id∨α(P)} S^{q} ∨ M′′ where ∇′ is the coaction associated to the cofibration sequence S^{q−1} ⟶ Z ⟶ S^{p−1}. As π_{p−1}(S^{q}∨S^{p−1}) = π_{p−1}(S^{q})⊕π_{p−1}(S^{p−1}) it follows that H(P, Q) vanishes.

Remark 14. Controlling the behavior of Hopf invariants has recently become a key tool in the homotopical study of the Lusternik-Schnirelmann category. Results based on this technique are the negative solution of the Ganea conjecture by Iwase [12], the examples of Roitberg showing that the L.S.-category is not generic in the sense of the Mislin genus [28] and the examples of Stanley of spaces of category n but cone-length n + 1 [34]. In all these examples the non-vanishing of certain Hopf invariants associated to the attachment of a cell is used to deduce that the L.S.-category is increased by the attachment of that cell. This is clearly also the homotopical content of the lemma above.

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