NEW OBSTRUCTIONS TO THE THICKENING OF CW -COMPLEXES.

OCTAVIAN CORNEA

Abstract. In this note we use Morse theory, in particular, a result of [4], to produce new obstructions to the existence of thickenings of CW -complexes in low codimension. The obstructions are expressed as non existence of solutions x to an equation of type Σ^{k}L = Σ^{s}x with L a Ganea-Hopf type invariant.

1. Introduction

Let X be a connected space with the homotopy type of a CW -complex and let M^{n} be a manifold. A thickening of X in M is a homotopy equivalence X ≃ W where W ⊂ M is a compact submanifold with boundary of dimension n = dim(M). In this paper we shall work in the smooth category (and thus our thickenings will be smooth) but the concept exists for P L-manifolds as well. Thickenings have been introduced by C.T.C.Wall in [12]. We shall produce here some new obstructions to the existence of thickenings. This problem is closely related to that of embedding CW -complexes up to homotopy and it was brought to my attention by a discussion with the authors of the recent paper [7].

The approach of the present note is based on a Morse theoretic result from [4] and is new to this problem. For this reason, we do not attempt here for the most general results possible and rather prefer to focus on the main geometric idea. In particular, we shall only consider here the case M = S^{n} and thickenings that are regular in the sense that π_{1}(∂W) → π_{1}(W) is required to be an isomorphism.

1.1. Statement. We work in the category of pointed spaces with the homotopy type of CW -complexes. Fix a connected CW -complex X. Assume that

(1)  C_{1} : S^{q−1} →^{u} X′ → X′′ , C_{2} : S^{p−1} →^{v} X′′ → X′′′

are two cofibration sequences, p ≥ q ≥ 1, and suppose that X′′ = X^{(p−1)} and X′′′ is a subcomplex of X^{(p)}, the p-skeleton of X. Consider the composition

H : S^{p−1} →^{v} X′′ →^{∇} S^{q} ∨ X′′ ↪ S^{q} ∨ X

where ∇ is the coaction associated to C_{1} and the last map is the inclusion. Let p_{2} : S^{q} ∨ X → X be the obvious projection. Clearly, p_{2} ∘ H ≃ ∗. Recall that the homotopy fibre of p_{2} is canonically homotopy equivalent to S^{q} ∧ ((ΩX)^{+}) where A^{+} is the space A with an added disjoint point. Moreover, the fiber inclusion j : S^{q} ∧(ΩX)^{+} → S^{q} ∨X is monomorphic in homotopy. Therefore, the homotopy class [H] lifts uniquely to a homotopy class H(v, u) ∈ π_{p−1}(Σ^{q}(ΩX)^{+}) which is called the Hopf invariant of u relative to v.

Date: October 31, 2002.

2000 Mathematics Subject Classification. Primary 57R12, 55Q25; Secondary 37D15.

Theorem 1. Assume the setting and the notation above. Moreover, suppose that X and X′ are simply-connected and fix n ∈ ℕ, n > p + 2. If Σ^{n−p+1}H(v, u) is not a suspension of order q + 1, then X does not admit a regular thickening in S^{m} for any m ≤ n.

The main idea of the proof appears in a particular case and is as follows. Assume that N is a thickening of X in S^{n} and suppose that u, v have the special property that there exists a Morse-Smale function f : N → ℝ with two critical points Q and P of indexes respectively q and p such that u, v are the handle attaching maps corresponding to Q and P by classical Morse theory. Under these assumptions, a Morse theoretic result from [4] shows that H(v, u) is equal to the homotopy class of a certain Thom-Pontryaguin map associated to the framed moduli space Z_{f}(P, Q) of negative gradient flow lines of f that connect P to Q (see §2.1). But −f is also a Morse-Smale function and the moduli space Z_{−f}(Q, P) is also defined as well as the corresponding Thom-Pontryaguin map S^{n−q−1} → S^{n−p} ∧ (ΩX)^{+}. These two moduli spaces coincide as sets and we shall see that, after suspending both maps the number of times needed such as to obtain maps defined on S^{n}, the two Thom-Pontryaguin maps coincide up to sign. Therefore, in this particular case, Σ^{n−p+1}H(v, u) de-suspends q + 1-times.

We comment on the theorem below. The impatient reader may want to go directly to the proof in the next section. A relevant example is discussed in the last section of the paper.

1.2. Comments.

1.2.1. Relation to other results in the literature. There are other thickening criteria which are expressed in terms of the existence of solutions x to an equation of the type

(2)  Σ^{k}L = Σ^{s}x

for an appropriate homotopy class L. Probably, the first such condition that appears in the literature is due to Connolly and Williams in [1] (and is related to earlier work of Cooke [2]). With our notation, it shows that if

(i) X′ = ∗, X = X′′′, L = v (with v from (1)), (ii) the suspension orders verify k = n − p − 1, s = q, (iii) n > 2p − 2q, p ≥ q ≥ 2, n > p + 2,

then X admits a P L-thickening in S^{n} iff (2) has a solution. A more recent result - restated again for P L-thickenings - is that of Lambrecht, Stanley and Vandembroucq [7]. They provide a sufficient condition for thickening which applies to two-cones (these are spaces which are homotopy equivalent to the cofiber of a map between two suspensions) and not only to two-cell complexes. This reference is also the first where the role of some Hopf invariants is recognized (in the particular case of two-cell complexes). Indeed, with our notation, the authors show that if X′ = ∗, X = X′′′, L = γ^{t}(v) is the t-th James-Hopf invariant of v, k = n − p − 2, s = q, then the existence of a solution of (2) for all t’s implies that X thickens in S^{n} (again, under the additional assumption p ≥ q ≥ 2, n ≥ p + 3).

Clearly, the result of Theorem 1 is a necessary condition and, not suprinsingly, the difference s − k = p + q − n which tracks how many desuspensions are needed with respect to the number of suspensions is smaller in our case than in the sufficient criteria mentioned above. Obviously, the main interest for Theorem 1 comes from the fact that it applies to arbitrary CW -complexes and not only to two-cones. From this perspective, our result provides a bridge between the results valid for 2-cell complexes (or 2-cones) and the classical obstructions to thickening like those discovered by Thom [10] or Peterson-Stein [9] and which are based on homology operations. It is likely that all these results are particular instances of some more general statement yet to be discovered.

1.2.2. The relative Hopf invariant and the Ganea-Hopf invariant. The invariant H(v, u) is closely related to the Hopf type invariant introduced by Ganea in [6] and defined as follows.

Consider the relative attaching map δ(v, u) = p_{1}∘∇∘v (where p_{1} : S^{q}∨X → S^{q} is the projection on the first factor) and the homotopy class h′ = [∇ ∘ v − i_{2} ∘ v − i_{1} ∘ δ(v, u)] ∈ π_{p−1}(S^{q} ∨ X′′) (where i_{j} is the inclusion of the j-th term in the wedge S^{q} ∨ X′′, j = 1, 2). There exists a homotopy fibration:

ΩS^{q} ∗ ΩX′′ ⟶^{r} S^{q} ∨ X′′ ⟶^{l} S^{q} × X′′

where l is the inclusion. It has the property that π_{∗}(r) is monomorphic. Clearly, l(h′) = 0. Therefore, h′ lifts to a unique homotopy class H(v, u) ∈ π_{p−1}(ΩS^{q} ∗ ΩX′′) which is, by definition, the Ganea-Hopf invariant of (v, u).

Now let k : S^{q} ↪ Σ^{q}(ΩX)^{+} = S^{q} ∨ Σ^{q}ΩX be the obvious inclusion and let

\overline{H}(v, u) = H(v, u) − [k ∘ δ(v, u)] ∈ π_{p−1}(Σ^{q}(ΩX)^{+} .

To see the relation between H(v, u) and H(v, u) consider the commutative diagram

\xymatrix{S^{q} ∨ X′′ \ar[d]_{l} \ar[r]^{id∨incl} & S^{q} ∨ X \ar[d]^{p_{2}} \\ S^{q} × X′′ \ar[r] & X .}

The commutativity of the diagram provides an induced map s : ΩS^{q} ∗ ΩX′′ → Σ^{q}(ΩX)^{+} between the homotopy fibers of the vertical arrows.

The images of h′ and \overline{H}(v, u) inside π_{p−1}(S^{q} ∨X) coincide. Therefore, we have

\overline{H}(v, u) = s ∘ H(v, u)

and ΣH(v, u) = Σs(H(v, u)) + Σδ(v, u).

2. Proof of Theorem 1.

We start in §2.1 by recalling some well-known elements of Morse theory as well as the result from [4] which is the key of the proof which is contained in §2.2.

2.1. Some elements of Morse theory.

2.1.1. Setting. Assume that f : N → ℝ is a Morse function defined on a smooth, connected, compact manifold N of dimension n. In case ∂N ≠ ∅, we assume that f is constant and regular on each component of ∂N. Assume also that α is a riemannian metric on N and let ∇f be the corresponding gradient vector field induced by f. Denote by γ the flow induced on N by −∇f.

Recall that the stable and unstable manifolds of a critical point R of f are defined respectively by

W^{u}(R) = {x ∈ Int(N) : lim_{t→−∞} γ_{t}(x) = R} ,

W^{s}(R) = {x ∈ Int(N) : lim_{t→+∞} γ_{t}(x) = R} .

The sets W^{u}(R) and W^{s}(R) admit canonical smooth manifold structures and they are diffeomorphic to open disks of dimension respectively ind_{f}(R) and n − ind_{f}(R). We now assume that (f, α) satisfies the Morse-Smale condition. By definition this requires the transversality of the intersection between W^{u}(P) and W^{s}(Q) for all the pairs of critical points P, Q of f. For f (or α) fixed this condition is verified for generic choices of α (respectively f).

Consider two critical points P and Q of f and assume that they verify f(P) > f(Q) and no critical point R of f verifies f(R) ∈ (f(Q), f(P)). Fix also the indexes of P and Q to be respectively p and q. Let a ∈ ℝ be a regular value of f such that f(Q) < a < f(P). Let V = f^{−1}(a) and let S_{a}^{u}(P) = W^{u}(P) ∩ V, S_{a}^{s}(Q) = W^{s}(Q) ∩ V. It is easy to see that S_{a}^{u}(P), S_{a}^{s}(Q) are manifolds diffeomorphic to spheres of dimensions equal to p−1 and n−q−1 respectively. Let Z_{f}(P, Q) = S_{a}^{u}(P) ∩ S^{s}(Q). The Morse-Smale transversality assumption implies that Z_{f}(P, Q) is a compact manifold of dimension p − q − 1 and by translating Z_{f}(P, Q) along the flow γ it is immediate to see that its diffeomorphism type does not depend of a. Following [5] we call Z_{f}(P, Q) the connecting manifold of P and Q.

2.1.2. Additional structure of connecting manifolds. The facts above are standard. They are carefully presented in Franks’ paper [5] where the significance of the existence of a canonical normal framing to Z_{f}(P, Q) has been first noticed. This framing will play an important role here too so we now describe it. To fix terminology recall that a framing of a fiber bundle ξ, of base B and of rank k, is a choice of k everywhere linearly independent sections of ξ modulo the equivalence relation which identifies two such choices if they are the 0 and 1 ends of a choice of k linearly independent sections for the bundle p^{∗}ξ where p : B × [0, 1] → B is the projection. Given an embedding, A ↪ N, a normal framing of A is a framing of the associated normal bundle.

Pick a normal framing of W^{s}(Q) associated to the embedding W^{s}(Q) ⊂ N (as W^{s}(Q) is contractible there is a unique such framing). By transversality this framing induces a normal framing of Z_{f}(P, Q) inside S_{a}^{u}(P) ≈ S^{p−1}, denoted by η_{f}(P, Q).

There is a second structure naturally associated to Z_{f}(P, Q). This is a map j_{f}(P, Q) : Z_{f}(P, Q) ⟶ ΩN which is defined as follows. Fix a point o ∈ Z_{f}(P, Q) and denote by ω the path in N which coincides geometrically with the flow line of γ that passes through o but is reparametrized by the interval [1/2, 1] and goes from Q to P. The map j_{f}(P, Q) associates to each point x ∈ Z_{f}(P, Q) the path in N which follows from P to Q the flow line of γ that passes through x and then returns from Q to P along the fixed path ω. Of course, this is slightly imprecise as first all the γ-flow lines joining P to Q need to be reparametrized in a uniform way by a finite interval, say [0, 1/2]. This is however possible and immediate due to the fact close to the critical points ∇f is almost linear. With this definition it is immediate that the map j_{f}(P, Q) is continuous.

We apply the Thom construction to the framed embedding

Z_{f}(P, Q) ⟶^{η_{f} (P,Q)} S^{p−1}

and the map

j_{f}(P, Q) : Z_{f}(P, Q) → ΩN

thus getting a homotopy class [Z_{f}(P, Q)] ∈ π_{p−1}Σ^{q}((ΩN)^{+})).

2.1.3. Relation between Hopf invariants and connecting manifolds. By classical Morse theory, in the setting above, the critical points P and Q correspond to cell attachments of cells of dimensions respectively p and q. In other words, we

have cofibration sequences: S^{q−1} ⟶^{α(Q)} N′ → N′′ and S^{p−1} ⟶^{α(P)} N′′ → N′′′ with

N′ = f^{−1}(−∞, f(Q) − ϵ], N′′ ≃ f^{−1}(−∞, f(P) − ϵ], N′′′ = f^{−1}(−∞, f(P) + ϵ] where the attaching maps α(P) and α(Q) are represented by the inclusions

α(P) : S_{(f(P)−ϵ)}^{u}(P) → N′′ , α(Q) : S_{(f(Q)−ϵ)}^{u}(Q) → N′

and ϵ is small. Clearly, we also have an inclusion N′′′ ↪ N and, as described in §1.1, we define the Hopf invariant H_{f}(P, Q) = H(α(P), α(Q)) ∈ π_{p−1}(Σ^{q}(ΩN)^{+}). For further use we let δ_{f}(P, Q) be the relative attaching map of the cells corresponding to P and Q, δ_{f}(P, Q) = l_{∗}(H_{f}(P, Q)) with l : Σ^{q}(ΩN)^{+} ≃ Σ^{q}N ∨ S^{q} → S^{q} the projection. The result from [4] that we shall use is that, up to sign, and when N is simply connected (such that [Z_{f}(P, Q)] does not depend of the choice of ω) we have:

(3)  [Z_{f}(P, Q)] = H_{f}(P, Q) .

Notice that this equality implies the stable equality up to sign of δ_{f}(P, Q) and the framed cobordism class of Z_{f}(P, Q) (result first proved in [5]).

2.2. The proof.

2.2.1. Consequence of time reversion on Hopf invariants. Time reversion for a flow simply replaces the flow γ by the flow −γ_{t}(x) = γ_{−t}(x). In our case, when γ is induced by −∇f for some smooth function f, this simply corresponds to replacing the function f with its opposite −f. Clearly, if the pair (f, α) satisfies the Morse-Smale condition the same is true for the pair (−f, α). Obviously, the critical points Q, P have respectively indexes n−q and n−p with respect to the function −f and the manifold Z_{−f}(Q, P) coincides as a set with Z_{f}(P, Q). The map j_{−f}(Q, P) is simply j_{f}(P, Q) composed with the loop reversion ΩN ⟶^{−1} ΩN. The framings η_{f}(P, Q) with η_{−f}(Q, P) may also be compared and, as a byproduct, it was seen in [3] (thus extending again a result from [5]) that the relative attaching maps, δ_{f}(P, Q), δ_{f}(Q, P), coincide stably up to a twisting depending on the the stable normal bundle of N.

In the present paper we shall need a slightly different result than those mentioned above.

Lemma 2.1. Assume the setting and notation above and suppose also that N is a compact, n-dimensional submanifold of S^{n}. We have

Σ^{n−p+1}H_{f}(P, Q) = ±Σ^{q+1}H_{−f}(Q, P) .

Proof. Fix the regular value a ∈ (f(Q), f(P)) and V = f^{−1}(a). To simplify notation let S^{u} = S_{a}^{u}(P) and let S^{s} = S_{a}^{s}(Q), Z = Z_{f}(P, Q). Let s_{i}, i ∈ {1, …, q} be the linearly independent sections of the normal bundle of W^{s}(Q) which induce the framing η_{f}(P, Q) of Z ↪ S^{u} by projection on T S^{u}. Similarly, let l_{j}, j ∈ {1, …, n − p} be linearly independent sections of the normal bundle of W^{u}(P) wich induce (by projection on T S^{s}) the framing η_{−f}(Q, P) of Z ↪ S^{s}. Let η be the framing of Z ⊂ N ⊂ S^{n} given by (s_{1}, …, s_{q}, l_{1}, …, l_{n−p}, −∇f). In view of the equality (3) and given the definition of the Thom map, the statement follows if we show that η is equivalent (up to orientation) to η_{f}(P, Q) ⊕ ϵ^{n−p+1} as well as to η_{−f}(Q, P) ⊕ ϵ^{q+1}. We recall that if ξ is a normal framing of M ↪ S^{k}, then ξ ⊕ ϵ^{r} is the normal framing of M ⊂ S^{k} ↪ S^{k+r} which is obtained by completing ξ by the sections coming from the trivial normal framing of the standard embedding of S^{k} ↪ S^{k+r} (addition of such a trivial framing suspends r times the corresponding Thom map). We shall only prove η = ±η_{f}(P, Q)⊕ϵ^{n−p+1} as the other equality does then follow by symmetry. First, let s′_{i} be the projections of s_{i} on T S^{u}, i = 1, …, q. By definition, (s′_{1}, …, s′_{q}) represents the framing η_{f}(P, Q). Let η′ be the framing represented by (s′_{1}, …, s′_{q}, l_{1}, …, l_{n−p}, −∇f). The first remark is that η′ = η. Indeed, consider the q-plane bundle μ associated to the normal bundle of Z in S^{n}. The planes spanned by the normal sections (s_{1}, …, s_{q}) and (s′_{1}, …, s′_{q}) give two sections s and s′ of this bundle. The fact that both s and s′ are complementary in each point to the n − q plane given by (l_{1}, …, l_{n−p}, −∇f) ⊕ T Z implies that s and s′ are homotopic via sections that remain complementary to this n − q -plane (because the space of q-planes in ℝ^{n} which are complementary to ℝ^{n−q} is contractible). It is then immediate to use precisely this homotopy to show that η and η′ coincide. Therefore, the proof of the lemma is reduced to showing η′ = ±η_{f}(P, Q)⊕ϵ^{n−p+1}. At this point is it useful to assume a particular form of α around P. Fix a Morse chart around P such that with respect to this chart f(x_{1}, x_{2}, …, x_{n}) = f(P)−x_{1}^{2} −…−x_{p}^{2} +x_{p+1}^{2} +…+x_{n}^{2} and the metric α is the flat euclidean one in this chart, α = ∑_{i} dx_{i}^{2}. This is not restrictive as we may perturb the initial metric to one with this property. Of course, f remains Morse with respect to all the metrics appearing along this deformation and thus our framing of S^{u}(P) given by (l_{1}, …, l_{n−p}, −∇f) is not changed by this modification. Let S_{ϵ}^{n−1} be a sphere around P, of radius ϵ, which is contained in our special chart and let S_{ϵ}^{p−1} ⊂ S_{ϵ}^{n−1} be the intersection W^{u}(P) ∩ S_{ϵ}^{n−1}. The embedding S_{ϵ}^{p−1} ↪ S_{ϵ}^{n−1} is the standard one. Moreover, as S_{ϵ}^{p−1} = S_{P −ϵ^{2}}^{u}(P) we may assume a = f(P) − ϵ^{2} (otherwise we transport Z along the flow −γ till it reaches f^{−1}(f(P) − ϵ^{2})) and so Z ⊂ S_{ϵ}^{p−1} = S^{u}. On S_{ϵ}^{p−1} the framing represented by (l_{1}, …, l_{n−p}) is the trivial one and so the normal framing of Z in S_{ϵ}^{n−1} is represented by (s′_{1}, …, s′_{q}, l_{1}, …, l_{n−p}) and coincides with η_{f}(P, Q) ⊕ ϵ^{n−p}. When restricted to S_{ϵ}^{p−1} the vector field −∇f coincides with the vector field normal to the sphere S_{ϵ}^{n−1}. Clearly, the embedding S_{ϵ}^{n−1} ↪ S^{n} is the trivial one so, finally, η′ coincides with (η_{f}(P, Q) ⊕ ϵ^{n−p}) ⊕ ϵ^{1}. □

2.2.2. The last step. Notice that the non-desuspension condition in Theorem 1 implies that q + p > n, p > q + 1 and as p + 2 < n we have p − 1 > q > 3, n ≥ 9. Clearly, if X admits a regular thickening in S^{m}, then it also admits a regular thickening in S^{m+1}. Therefore, the proof of Theorem 1 is now immediate - by contradiction - from the duality Lemma 2.1 and the result below.

Proposition 2.2. Suppose that X, u, v are as in Theorem 1. Assume that N is a regular n-dimensional thickening of X inside S^{n}, n ≥ 6. Suppose also that p − 1 > q > 1, p + 2 < n. There exists a Morse-Smale pair (f, α) on N with the property that there are critical points P_{i}, Q_{i} of f with ind(P_{i}) = p, ind(Q_{i}) = q and there also exist coefficients a_{i} ∈ ℤ and a homotopy equivalence ϕ : N → X such that

ΣH(v, u) = ∑_{i} a_{i}(Σ(Σ^{q}Ωϕ ∘ H_{f}(P_{i}, Q_{i}))) .

Proof. Because n ≥ 6 and N as well as ∂N are simply-connected it follows from classical Morse theory that there exists a Morse function f : N → ℝ such that f is maximal, constant and regular on ∂N, f is self indexed (meaning that if R is a critical point of f of index r, then f(R) = r) and, moreover, f has no critical point of index i ∈ (q, p) ∪ {1}. For this last statement we use q ≥ 2 and p < n − 2 such that the cancellation technique appearing in the h-cobordism theorem (and based on the Whitney trick) works for cells of dimension ≤ p (see also [8] page 69). The attachments of the q-cells that correspond to the critical points of index q provide a cofibration sequence ∨_{i_{q}}S^{q−1} ⟶^{w} N′ → N′′ and the attachment of the cells corresponding to the index-p critical points provides yet another cofibration sequence ∨_{i_{p}}S^{p−1} ⟶^{z} N′′ → N′′′ where i_{k} is the number of critical points of index k of f. Notice that N′, N′′, N′′′ are all simply connected. We fix homotopy equivalences ϕ : N → X, ϕ^{−1} : N → X. Clearly, the whole problem comes from the fact that the cellular decomposition induced by f may be different from the decomposition given by the CW -structure of X (as 2p ≥ n even by modifying f we might not recover this last cell decomposition [5]). To compare these two cell decompositions we shall make use of the following lemma.

Lemma 2.3. Suppose that we have two cofibration sequences and maps given by the solid arrows in the next diagram

(4)  \xymatrix{V \ar@{.>}[d]^{t_{0}} \ar[r]^{a_{1}} & A_{1} \ar[d]^{t_{1}} \ar[r]^{a_{2}} & A_{2} \ar[d]^{t_{2}} \\ V′ \ar[r]_{b_{1}} & B_{1} \ar[r]_{b_{2}} & B_{2}}

such that V and V′ are wedges of q − 1-dimensional spheres, the A_{i}, B_{i}’s are all simply connected and the solid-arrow diagram commutes. There exists a map t_{0} represented by the dotted arrow in the diagram and there exists a homotopy h : V × [0, 1] → B_{1} between t_{1} ∘ a_{1} and b_{1} ∘ t_{0} which induces by passage to cofibers a map homotopic to t_{2}.

Proof. The proof of the lemma is rather standard but, as it is not completely trivial, we shall give it here in detail. We may assume q > 2 because, otherwise, the maps are both null-homotopic and the existence of t_{0} follows immediately. First we write A_{2} = A_{1} ∪ CV (CV = (V × [0, 1])/(V × {0} ∪ ∗ × [0, 1])), B_{2} = B_{1} ∪CV′. Let i : F → A_{1} be the fibration obtained by pulling back the path loop fibration ΩA_{2} → P A_{2} → A_{2} (clearly, F is the homotopy fiber of a_{1}). Notice that by pulling back this same path-loop fibration over the cofibration V → A_{1} → A_{2} we obtain the existence of a push out diagram

\xymatrix{V × ΩA_{2} \ar[d]_{p_{2}} \ar[r] & F \ar[d] \\ ΩA_{2} \ar[r] & P A_{2} .}

This shows that ΣF ≃ ΣV ∧ (ΩA_{2})^{+}. By the defining property of pull-backs the null-homotopy of the composition V → A_{1} → A_{2} given by deforming V to a point inside CV ⊂ A_{2} produces a map a′_{1} : V → F which satisfies i ∘ a′_{1} = a_{1} (and which equals the composition V × ∗ → V × ΩA_{2} → F). Let r : ΣF → ΣV be the map induced by collapsing to a point F inside P A_{2} and A_{1} inside A_{2}. The definition of a′_{1} shows that r ∘ Σa′_{1} ≃ id. Let i′ : F′ → B_{1}, r′ : ΣF′ → ΣV′, b′_{1} : V′ → F′ be the same objects constructed from the second cofibration. Let c′ : ΣV → ΣV′ be the map induced by (t_{1}, t_{2}) by collapsing to a point A_{1} and B_{1} inside, respectively, A_{2} and B_{2}. Let c : F → F′ be the comparison map also induced by (t_{1}, t_{2}). Therefore, in the diagram

(5)  \xymatrix{ΣV \ar[r]^{Σa′_{1}} \ar@{.>}[d]_{c′} & ΣF \ar[d]^{Σc} \ar[r]^{r} & ΣV \ar[d]^{c′} \\ ΣV′ \ar[r]_{Σb′_{1}} & ΣF′ \ar[r]_{r′} & ΣV′}

the right square commutes up to homotopy and the horizontal compositions are homotopic to the respective identities. Given the fact that V and V′ are wedges of q − 1-spheres and as we have ΣF ≃ ΣV ∧ (ΩA_{2})^{+}, ΣF′ ≃ ΣV′ ∧ (ΩB_{2})^{+} with F, F′, A_{2}, B_{2} simply connected we obtain that c′ makes homotopy commutative the left square of the diagram and moreover c′ desuspends to a map t_{0} which has the property b′_{1} ∘ t_{0} ≃ c ∘ a′_{1}. Clearly, this last equality shows that the map t_{0} makes homotopy commutative the left square in (4). We now need to show the existence of the homotopy h. We first recall that the map j_{g} : A_{2} → B_{2} induced by a homotopy g : V × [0, 1] → B_{2}, with g_{0} = b_{1} ∘ t_{0}, g_{1} = t_{1} ∘ a_{1} is defined by j_{g}(x) = t_{1}(x) if x ∈ A_{1}, j_{g}(y, t) = (t_{0}(y), 2t) if (y, t) ∈ CV and t ≤ 1/2 and j_{g}(y, t) = g(y, 2t − 1) if (y, t) ∈ CV, t > 1/2. Clearly, by passage to cofibers the pair (t_{1}, j_{g}) induces a map homotopic to Σt_{0}. Moreover, as j_{g}|_{A_{1}} = t_{2}|_{A_{1}} there is a map a_{g} : ΣV → B_{2} such that t_{2} ≃ (a_{g} ∨ j_{g}) ∘ ∇ where ∇ : A_{2} → ΣV ∨ A_{2} is the coaction associated to the top cofibration in (4). By collapsing to a point A_{1} and B_{1} inside A_{2} and B_{2}, respectively, this equation becomes Σt_{0} ≃ δ ∘ a_{g} + Σt_{0} with δ : B_{2} → ΣV′ the connectant of the bottom cofibration sequence in (6). This means δ ∘ a_{g} ≃ ∗. As B_{2}, B_{1} are simply connected we have an isomorphism π_{q}(B_{2}, B_{1}) ≈ π_{q}(ΣV′) induced by the collapsing of B_{1}, so we obtain that there exists a map a′_{g} : ΣV′ → B_{1} such that b_{2} ∘ a′_{g} ≃ a_{g}. It is this element a′_{g} that we shall use to modify the homotopy g. At this point it is simpler to assume that V contains a single sphere S^{q−1} (this is not restrictive as we can repeat our construction for each term of the wedge). We define a new homotopy h : V × [0, 1] → B_{2} as follows. First let k : V × [0, 1/2] → ΣV ∨ (V × [0, 1]) be the collapsing map obtained by contracting to a point a sphere S^{q−1} embedded in the interior of V × [0, 1/2] = S^{q−1} × [0, 1/2]. We now let h(x, t) = g(x, 2t − 1)) when t ∈ [1/2, 1], and h(x, t) = (a′_{g} ∨ (b_{1} ∘ t_{0})) ∘ k if t ∈ [0, 1/2]. We consider the induced map j_{h} and notice that, by construction, j_{h} ≃ ((b_{2} ∘ a′_{g}) ∨ j_{g}) ∘ ∇. Therefore, j_{h} ≃ t_{2}. □

We now return to the proof of the proposition. By applying the lemma in our setting we obtain diagrams

(6)  \xymatrix{∨S^{q−1} \ar[d]_{j_{0}} \ar[r]^{w} & N′ \ar[d]_{j_{1}} \ar[r]^{s_{1}} & N′′ \ar[d]_{j_{2}} \\ S^{q−1} \ar[r]_{u} & X′ \ar[r]_{l_{1}} & X′′}

and

(7)  \xymatrix{S^{p−1} \ar[d]_{t} \ar[r]^{v} & X′′ \ar[r]^{l_{2}} \ar[d]_{j_{2}^{−1}} & X′′′ \ar[d]_{j_{3}} \\ ∨S^{p−1} \ar[r]^{z} & N′′ \ar[r]^{s_{2}} & N′′′ .}

which are constructed as follows. We start with maps j_{k}, k = 1, 2, 3 obtained from ϕ, ϕ^{−1} by cellular approximation and we let the s_{i}’s, l_{i}’s be inclusions. We notice that j_{2} is a homotopy equivalence because both X′′ and N′′ are p − 1skeleta of dimension q and p − 1 > q. We then use the lemma to construct the map j_{0} and a homotopy h′ making commutative the left square in (6) and such that j_{h′} ≃ j_{2} as well as the map t and a a homotopy h′′ which makes commutative the left square in (7) and with j_{h′′} ≃ j_{3}.

An immediate consequence of the fact that j_{2} ≃ j_{h′} is that the coactions associated to the two cofibration sequences in (6) commute (up to homotopy):

(8)  \xymatrix{N′′ \ar[d]_{j_{2}} \ar[r]^{∇} & (∨S^{q}) ∨ N′′ \ar[d]^{Σj_{0}∨j_{2}} \\ X′′ \ar[r]_{∇} & S^{q} ∨ X′′ .}

By combining the left square in (7) with (8) and after composition with the respective inclusions into X and N we obtain the next homotopy commutative diagram

(9)  \xymatrix{S^{p−1} \ar[d]_{t} \ar[r] & X′′ \ar[d] \ar[r] & S^{q} ∨ X \\ ∨S^{p−1} \ar[r] & N′′ \ar[r] \ar[u] & (∨S^{q}) ∨ N \ar[u]_{Σj_{0}∨ϕ}}

and this immediately implies the claim by the definition of the respective Hopf invariants.

□

3. An example.

It is not too difficult to build examples where Theorem 1 applies and previous results do not. Before describing the construction of such an example notice that the criterion in the theorem is only relevant if n − p + 1 < q + 1 ⇔ n < p + q. Our example will be one whith q = 8, p = 30 and no thickening exists whenever n < 37.

3.1. A construction. Let α_{1} be an element of order three in π_{6}(S^{3}) ≈ ℤ/12. Consider the space T = S^{3} ∪_{α_{1}} e^{7} and let δ be the map δ : T → S^{7} obtained by collapsing S^{3} to a point in T. Let i ∈ π_{7}(T) be an element of infinite order such that the image of i by δ is homotopic to three times a generator of π_{7}(S^{7}). The existence of such an element follows from the exact sequence π_{7}(T) → π_{7}(S^{7}) → π_{6}(L) → π_{6}(T) ≈ ℤ/4 → 0 with L the homotopy fiber of T → S^{7} because

(10)  π_{6}(L) ≈ π_{6}(S^{3}) .

Before proceeding with our construction we justify the isomorphism (10). We use δ to pull-back the fibration ΩS^{7} → P S^{7} → S^{7} over the cofibration sequence S^{6} → S^{3} → T. We thus obtain a push-out square of the form

\xymatrix{S^{6} × ΩS^{7} \ar[r]^{α_{1}×id} \ar[d] & S^{3} × ΩS^{7} \ar[d] \\ ΩS^{7} \ar[r] & L}

where the vertical map at the left restricted on S^{6} × ∗ equals the inclusion of the bottom cell S^{6} ↪ ΩS^{7}. We only need to consider the 7-skeleta involved and we deduce the push out square

\xymatrix{S^{6} ∨ S^{6} \ar[r] \ar[d] & S^{3} ∨ S^{6} \ar[d] \\ S^{6} \ar[r] & L^{(7)}}

which shows that L^{(7)} ≃ S^{3} because the vertical map on the left is the folding map.

We return to our construction and let w : S^{14} → S^{8} ∨ T be given by w = [i_{1}, i] where i_{1} : S^{8} → S^{8} ∨ T is the inclusion of the first factor. Recall, [11] page 188, that there exists an element ρ_{13} ∈ π_{28}(S^{13}) ≈ ℤ/480 ⊕ ℤ/2 which is of order 32 and which does not desuspend. On the other hand the stabilization of ρ_{13}, ρ ∈ π_{15}^{S} is still of order 32 and, in fact, π_{15}^{S} ≈ π_{28}(S^{13}). Let ρ_{14} : S^{29} → S^{14} be the suspension of ρ_{13}.

We now let X be the cofibre of the map v = w ∘ ρ_{14} : S^{29} → S^{14} → S^{8} ∨ T.

3.2. Absence of thickenings. The purpose now is to show the following consequence of Theorem 1.

Corollary 3.1. The space X defined above does not admit a regular thickening in S^{n} for any n < 37.

Proof. We consider the two cofibration sequences

(11)  S^{7} ⟶^{u} T ⟶ S^{8} ∨ T , S^{29} ⟶^{v} S^{8} ∨ T ⟶^{t} X

with u ≃ ∗ and the associated Hopf invariant H(v, u). We consider the composition

ρ′ : S^{29} →^{ρ} S^{14} →^{l} S^{8} ∧ (ΩX)^{+} →^{q} S^{8} ∧ (ΩS^{7})^{+} →^{e} S^{14}

where:

- l is the unique lift of the composition S^{14} →^{w} S^{8} ∨ T ⟶^{id_{S^{8}}∨incl} S^{8} ∨ X to

the homotopy fibre of S^{8} ∨ X →^{p_{2}} X.

- q = Σ^{8}(Ωq′)^{+} with q′ : X → S^{7} the composition of the map q′′ from

the cofibration sequence S^{8} ∨ S^{3} → X →^{q′′} S^{30} ∨ S^{7} with the projection S^{30} ∨ S^{7} → S^{7}.

- e is the composition Σ^{8}(ΩS^{7})^{+} ≃ Σ^{8}ΩS^{7} ∨ S^{8} →^{p_{1}} Σ^{8}ΩS^{7} ⟶^{Σ^{7}(ev)} S^{14} with ev : ΣΩS^{7} → S^{7} the adjoint of the identity.

These maps satisfy the following two properties:

(i) H(u, v) ≃ l ∘ ρ.

(ii) e ∘ q ∘ l is a map of degree ±3.

We postpone the justification of (i) and (ii) and first describe how they imply the claim of the Corollary. Indeed, if H(v, u) would de-suspend 9 times after being suspended n−30+1 > 1 times, then the same would be true for ρ′ = e∘q∘l∘ρ but, after inverting 3, we see that e ∘ q ∘ l induces a homotopy equivalence S_{(3)}^{14} → S_{(3)}^{14} and, in particular, ρ′ remains of order 32. As there are no elements of order 32 in π_{27}(S^{12}) it follows that n − 29 ≥ 8 and, by using Theorem 1, we obtain the statement.

We now discuss (i) and (ii). We start with (ii). This is immediate from the

fact that q ∘ l is the unique lift of the Whitehead product S^{14} ⟶^{[i_{1},3i_{2}]} S^{8} ∨ S^{7} to the homotopy fibre of the map p_{2} : S^{8} ∨ S^{7} → S^{7} which in turn follows from the commutativity of the diagram

\xymatrix{S^{14} \ar[r]^{[i_{1},i_{2}]} & S^{8} ∨ S^{7} \ar[r]^{id∨i} \ar[dr]_{id∨3} & S^{8} ∨ T \ar[d]^{id∨δ} \ar[r]^{incl} & S^{8} ∨ X \ar[dl]^{id∨q′} \\  &  & S^{8} ∨ S^{7} & .}

So, to conclude the proof we only have to show (i). For this we make use of the commutative diagram:

\xymatrix{S^{29} \ar[d]_{ρ} \ar@{-->}[drrr]^{H′} \\ S^{14} \ar[dr]_{[i_{1},i]} \ar[r] & S^{8} ∧ (ΩT)^{+} \ar[r] \ar[d] & S^{8} ∧ (Ω(S^{8} ∨ T))^{+} \ar[d] \ar[r] & S^{8} ∧ (ΩX)^{+} \ar[d]^{j} \\  & S^{8} ∨ T \ar[r]^{id_{S^{8}}∨t′} \ar[d]_{p_{2}} & S^{8} ∨ (S^{8} ∨ T) \ar[d]^{p_{2}} \ar[r]^{id_{S^{8}}∨t} & S^{8} ∨ X \ar[d]^{p_{2}} \\  & T \ar[r]_{t′} & S^{8} ∨ T \ar[r] & X}

where t′ : T → S^{8} ∨ T is the obvious inclusion and t appears in the second cofibration in (11). We see from this diagram that H′ ≃ l ∘ ρ. The proof of H(v, u) ≃ l ∘ ρ now follows from the commutativity of the diagram

(12) \xymatrix{S^{29} \ar[d]^{∇} \ar[r]^{ρ} & S^{14} \ar[d]^{∇} \ar[r]^{[i_{1},i]} & S^{8} ∨ T \ar[d]^{∇∨id_{T}} \\ S^{29} ∨ S^{29} \ar[r]_{ρ∨ρ} & S^{14} ∨ S^{14} \ar[r]_{[i_{1},i]∨[i_{2},i]} & S^{8} ∨ S^{8} ∨ T \ar[r]_{id_{S^{8}}∨t} & S^{8} ∨ X .}

Indeed, H(v, u) is uniquely defined as the lift of h′′ = (id_{S^{8}} ∨ t) ∘ (∇ ∨ id_{T}) ∘ v to the fiber of S^{8} ∨ X → X and the commutativity of (12) implies that [h′′] = [t ∘ v] + [j ∘ H′] = [j ∘ H′] and so H(u, v) ≃ H′ ( we have used here the obvious identity t ∘ v ≃ ∗ and we recall that j : S^{8} ∧ (ΩX)^{+} → S^{8} ∨ X is the inclusion of the fiber). Finally, in diagram (12), the left square is commutative because ρ = ρ_{14} is a suspension and the right square is commutative due to the following immediate commutativity of universal Whitehead products

\xymatrix{S^{14} \ar[r]^{[i_{1},i_{2}]} \ar[d]_{∇} & S^{8} ∨ S^{7} \ar[d]^{∇∨id_{S^{7}}} \\ S^{14} ∨ S^{14} \ar[r]_{[i_{1},i_{3}]∨[i_{2},i_{3}]} & S^{8} ∨ S^{8} ∨ S^{7} .}

□

References

1. F.X.Connolly, B.Williams, Embeddings up to homotopy type and geometric suspensions of manifolds, Quart. J. Math. Oxford 29 (1978), 385–401

2. G.Cooke, Embedding certain complexes up to homotopy type in Euclidean space, Ann. of Math. 90 (1969), 144–156

3. O.Cornea, Homotopical Dynamics II: Hopf invariants, smoothings and the Morse complex, Annales Scient. de l’E.N.S. (2002)

4. O.Cornea, Homotopical Dynamics IV: Hopf invariants and Hamiltonian flows, Commun. Pure and Appl. Math. LV (2002), 1033–1088

5. J.Franks, Morse-Smale flows and homotopy theory, Topology 18 (1979), 199–215

6. T.Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295–322

7. P.Lambrechts, D.Stanley, L.Vandembroucq, Embeddings up to homotopy of two-cones in Euclidean space, Trans. A.M.S. (2002)

8. J.Milnor, Lectures on the h-cobordism theorem, Princeton Math. Notes Princeton Univ. Press (1965)

9. F.P.Peterson, N.Stein, The dual of a secondary cohomology operation, Ill. J. of Math. 4 (1960), 397–404

10. N.Steenrod, D.Epstein, Cohomology operations, Annals of Math. Studies 50 Princeton Univ. Press, 1962

11. H.Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies 49 Princeton Univ. Press, 1962

12. C.T.C.Wall, Classification problems in differential topology IV , Topology 5 (1966), 73–94

Université de Lille 1 UFR de Mathématiques 8524 59655 Villeneuve D’Ascq France

E-mail address: cornea@agat.univ-lille1.fr