A_{∞}-method in Lusternik-Schnirelmann category

Norio Iwase^{∗}

Address: Graduate School of Mathematics, Kyushu University, Japan.

e-mail: iwase@math.kyushu-u.ac.jp

June 30, 2000

Abstract

To clarify the method behind [11], a generalisation of Berstein-Hilton Hopf invariants is defined as ‘higher Hopf invariants’. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea’s conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of [11] for all the cases except the case when p = 2. As an application, conditions in terms of homotopy invariants of the characteristic maps are given to determine the LS category of sphere-bundles-over-spheres. Consequently, a closed manifold M is found not to satisfy Ganea’s conjecture on LS category and another closed manifold N is found to have the same LS category as its ‘punctured submanifold’ N − {P }, P ∈ N. But all examples obtained here support the conjecture in [11].

1 Introduction

In this paper, each space is assumed to have the homotopy type of a CW complex. The LS

category of X is the least number m such that there is a covering of X by m + 1 open subsets

each of which is contractible in X, which is (by Whitehead [27]) the least number m such that

the diagonal map Δ_{m+1} : X → X^{m+1} can be compressed into the ‘fat wedge’ T^{m+1}(X) or

X^{[m+1]}. Hence cat {∗} = 0.

As is well-known, the LS category of a product space cat X×S^{n} is either cat X or cat X + 1.

A problem was posed by Ganea in [6]: Can only the latter case occur for any X and n ≥ 1?

The affirmative answer had been supposed to be true and came to be known as ‘Ganea’s

conjecture’ (see [8]) or ‘the Ganea conjecture’ (see [15]). A major advance in this subject was made by Jessup [16] and Hess [8] working in the rational category: the rational version of the conjecture is true for n ≥ 2. Also by Singhof [20] and Rudyak [18], [19], the conjecture is true for a large class of manifolds.

∗1991 Mathematics Subject Classification. Primary 55M30, Secondary 55P35, 55Q25, 55R35, 55S36.

Keywords and phrases. LS category, higher homotopy associativity, homology decomposition, sphere bundles over spheres, manifold counter example to the Ganea conjecture.

From April to June in 1997, the author studied in Aberdeen University to provide a proof of Ganea’s conjecture using higher Hopf invariants [12] on the projective spaces with the properties of Stasheff’s fibrations (see Sugawara [23] and [24], Stasheff [22] and Mimura-Iwase [13]) associated with the A_{∞}-structure of a loop space. The author knows that Stanley was aiming to find out a counter example using the ordinary James-Hopf invariants, and also a related result on a sufficient condition to Ganea’s conjecture was obtained by Vandembroucq [26] at about the same peirod.

The author’s attempt was failed on showing the stable non-triviality of non-trivial higher Hopf invariants, but it turned to suggest a straight and right way to construct counter examples, and eventually the first counter example was found (see [10] or [11]). Soon after, the higher Hopf invariants given in [12] had begun to be studied by several authors, e.g. Stanley [21].

This paper is organised as follows: In Section 2, to clarify the method behind [11], a generalisation of the Berstein-Hilton Hopf invariants is defined with its related invariants to detect the higher homotopy associativity of a Hopf space. In Section 3, under a condition between dimension and LS category, some conjectures on LS category are verified by using fibrations associated with the A_{∞}-structure of a loop space. In Section 4, a result of Boardman-Steer is generalised to give a sufficient condition to determine LS category in terms of a generalised version of the Berstein-Hilton crude Hopf invariants. In Section 5, the relation between a homology decomposition and LS category of (product) spaces is shown, by extending a result of Curjel [4]. In Section 6, generalising the main result of [11] for all the cases except the case when p = 2, some more examples are obtained by the properties of the higher Hopf invariants given in Section 2. In Section 7, we give some conditions to determine the LS category of sphere-bundles-over-spheres. Using it, we construct, in Section 8, an orientable closed manifold N_{p}, for each prime p ≥ 5, with the LS category same as its ‘punctured submanifold’ N_{p} − {P }, P ∈ N_{p}. Also another orientable closed manifold M is constructed as a counter example to Ganea’s conjecture.

The author would like to express his gratitude to Ioan James for giving an attractive lecture on LS category at the University of Aberdeen which inspired him to consider LS category again, John Hubbuck, Yuly Rudyak, Kouyemon Iriye, Donald Stanley, Daniel Tanré, Octavian Cornea and Hans Baues for valuable conversations and encouragement which helped to organise his thoughts, the University of Aberdeen for its hospitality during the author’s stay in 1997, Max-Planck-Institut für Mathematik for its hospitality during the author’s stay in 2000 and the members of the Graduate School of Mathematics Kyushu University for allowing him to be away for a long term, without which this work could not be done.

2 Projective spaces and higher Hopf invariants

In this section, we introduce a generalised version of the Berstein-Hilton Hopf invariant (see [2]), a higher Hopf invariant for short, in terms of projective spaces associated with the A_{∞}- structure of a loop space, to detect the higher homotopy associativity, or the A_{m}-structure of a Hopf space (see Example 2.7 and Conjecture 2.8). We also show that a higher Hopf invariant gives the obstruction for increasing the LS category by one by attaching a cone, as in [11].

For a given space, its loop space is an A_{∞}-space with the given space as its A_{∞}-structure. More precisely, every space X has a filtration given by the projective spaces P^{m}(ΩX) of its loop space ΩX. There is a ladder of Stasheff’s fibrations E^{m+1}(ΩX) →^{p_{m}^{ΩX}} P^{m}(ΩX) with the fibre ΩX contractible in E^{m+1}(ΩX). The total space E^{m+1}(ΩX) has the homotopy type of the m+1-fold unreduced join of ΩX which is denoted by Ē^{m+1}(ΩX) and the base space P^{m+1}(ΩX) has the homotopy type of the mapping cone of p_{m}^{ΩX}, m ≥ 0: the fibration is induced by the inclusion e_{m}^{X} : P^{m}(ΩX) → P^{∞}(ΩX) from the universal fibration E^{∞}(ΩX) →^{p_{∞}^{ΩX}} P^{∞}(ΩX) with contractible total space. (see [22] for details.)

Theorem 2.1 (Ganea) Let X be a connected CW complex. Then cat X ≤ m if and only if the inclusion e_{m}^{X} : P^{m}(ΩX) ⊂ P^{∞}(ΩX) ≃ X has a right homotopy inverse (homotopy section).

This result enables us to define local versions of cat , e.g, cat_{p} is defined in [11], for a prime p, as the least number m such that e_{m}^{X} : P^{m}(ΩX) ⊂ P^{∞}(ΩX) ≃ X has a homotopy section at p, e.g. cat_{p} S_{(p)}^{1} = 2 while cat S^{1} = 1. We remark that in some of the literature, the composition functor P^{m}^{∘}Ω is abbreviated as G_{m}, the ‘Ganea space’ functor. We use the following fact.

Fact 2.2 ([11]) Let X be a connected CW complex with cat X = m. Then cat X×S^{n} = m if and only if (e_{m}^{X}×1_{S^{n}})| : P^{m}(ΩX)×{∗} ∪ P^{m−1}(ΩX)×S^{n} ⊂ P^{∞}(ΩX)×S^{n} ≃ X×S^{n} has a homotopy section.

In [11], a conjecture was posed instead of Ganea’s conjecture (Conjecture 1.4 in [11]):

Conjecture 2.3 ([11]) For any space X, there exists an integer n(X), 1 ≤ n(X) ≤ ∞, such that cat X×S^{n} is equal to cat X + 1, if n < n(X); cat X, if n ≥ n(X).

When V is the suspension of a co-H-space, say V = ΣV_{0} with V_{0} a co-H-space, we fix a canonical structure map σ(V) = Σ ad(1_{V}) : V → ΣΩV for cat V ≤ 1, which makes V a homotopy commutative and homotopy associative co-H-space, where ad(1_{V}) : V_{0} → ΩΣV_{0} is the adjoint of the identity. In this section, we fix a non-contractible co-H-space V with right and left inversion (e.g, V is a suspension space), together with a structure map σ(V) : V → ΣΩV for cat V = 1.

Definition 2.4 Let X be a space with cat X ≤ m, m ≥ 1. For a choice of the homotopy section σ(X) : X → P^{m}(ΩX) (m ≥ 1), we define a higher Hopf invariant as

H_{m} = H_{m}^{σ(X)} : [V, X] → [V, E^{m+1}(ΩX)],

which is a homomorphism when V is homotopy associative and homotopy commutative: For a map f : V → X, the difference between σ(X)^{∘}f and P^{m}(Ωf)^{∘}σ(V), in the algebraic loop [V, P^{m}(ΩX)], is given by a map d_{m}^{σ(X)}(f) : V → P^{m}(ΩX) so that σ(X)^{∘}f + d_{m}^{σ(X)}(f) ∼ P^{m}(Ωf)^{∘}σ(V). Since d_{m}^{σ(X)}(f) vanishes in [V, P^{∞}(ΩX)] ≅ [V, X], it has a unique lift H_{m}^{σ(X)}(f) : V → E^{m+1}(ΩX) to the total space of Stasheff’s fibration

ΩX ↪ E^{m+1}(ΩX) →^{p_{m}^{ΩX}} P^{m}(ΩX) →^{e_{m}^{X}} X, m ≥ 1.  (2.1)

Remark 2.5 1. When V is a homotopy associative co-H-space, σ(V) is a co-H-map by Theorem 2.2 of Ganea [5]. Hence a simple calculation shows that d_{m}^{σ(X)} and H_{m}^{σ(X)} are homomorphisms, if V is a homotopy associative and homotopy commutative co-H-space.

2. By Berstein-Dror [1], a co-associative co-H-space admits another co-associative co-H-structure which has right and left inversions.

3. When V is a Moore space of type (A, n), H_{m} can be regarded as the Berstein-Hilton Hopf invariant H : π_{n}(X; A) → π_{n}(X^{m+1}, T^{m+1}X; A) (see [2]), since π_{n}(X^{m+1}, T^{m+1}X; A) = π_{n}(E^{m+1}(ΩX); A) by Ganea (see the proof of Theorem 1.1 in [11]).

4. When V = ΣV_{0} a suspension space, a map f : V → X factors through ΣΩX as f = e_{1}^{X}^{∘}Σ ad(f), where e_{1}^{X} denotes the evaluation map and ad(f) = Ωf^{∘} ad(1_{V}) : V_{0} → ΩX gives the adjoint of f. Then the higher Hopf invariant H_{m}^{σ(V)}(e_{1}^{X}) of e_{1}^{X} : ΣΩX → X has a kind of ‘universality’: H_{m}^{σ(V)}(f) = H_{m}^{σ(ΣΩX)}(e_{1}^{X})^{∘}Σ ad(f). So we call H_{m}^{σ(ΣΩX)}(e_{1}^{X}) the \underline{universal Hopf invariant} for X and m ≥ cat X.

5. If X is a suspension space, say X = ΣY, then we have P^{1}(ΩX) = ΣΩX = ΣJ(Y) ≃ ΣY ∨ Σ∧^{2}Y ∨ Σ∧^{3}Y ∨ .... Let us recall that H_{1}^{σ(X)} is uniquely determined by d_{1}^{σ(X)}, whose projection to Σ∧^{j}Y gives exactly the j-th James Hopf invariant h_{j}, j ≥ 2. Thus we may regard H_{1} as the collection of all James Hopf invariants h_{j}, j ≥ 2.

The above definition of a higher Hopf invariant also allows us to define a generalisation of the Berstein-Hilton crude Hopf invariant as follows.

Definition 2.6

H̄_{m} = H̄_{m}^{σ(X)} : [V, X]  [V, E^{m+1}(ΩX)]  [ΣV, ∧^{m+1}ΣΩX]  [ΣV, ∧^{m+1}X],^{H_{m}^{σ(X)}} w  w^{∘(Σh_{m+1}^{X}_{∗})∘Σ}  ^{(∧^{m+1}e_{1}^{X})_{∗}}w

where e_{1}^{X} is the evaluation map and h_{m+1}^{X} : E^{m+1}(ΩX) → Ē^{m+1}(ΩX) ≃ ΩX∧(∧^{m}ΣΩX) denotes the natural homotopy equivalence (see Stasheff [22]).

Example 2.7 For an A_{m}-space G in the sense of Stasheff [22], the adjoint of the inclusion ι_{1,m}^{G} : ΣG ↪ P^{m}(G) is an A_{m}-map ad(ι_{1,m}^{G}) : G ↪ ΩP^{m}(G) whose A_{m}-structure map P^{m}(ad(ι_{1,m}^{G})) : P^{m}(G) → P^{m}(ΩP^{m}(G)) is given by a splitting of e_{m}^{P^{m}(G)} : P^{m}(ΩP^{m}(G)) → P^{m}(G) (see [13]). By putting X = P^{m}(G), V = Ē^{m+1}(G) and σ(X) = P^{m}(ad(ι_{1,m}^{G})), we have

H_{m} = H_{m}^{σ(X)} : [Ē^{m+1}(G), P^{m}(G)] → [Ē^{m+1}(G), Ē^{m+1}(ΩP^{m}(G))].  (2.2)

The group [Ē^{m+1}(G), Ē^{m+1}(ΩP^{m}(G))] contains the image of 1 = 1_{Ē^{m+1}(G)} under the homomorphism ad(ι_{1,m}^{G})_{∗} : [Ē^{m+1}(G), Ē^{m+1}(G)] → [Ē^{m+1}(G), Ē^{m+1}(ΩP^{m}(G))]. As is clearly seen, if G is an A_{m+1}-space, then there is a ‘higher Hopf invariant one’ element, i.e. there is a map f : Ē^{m+1}(G) → P^{m}(G) such that H_{m}(f) is the image of 1 as mentioned above. The converse is also verified when G is a sphere: Let f : Ē^{m+1}(G) → P^{m}(G) be a ‘higher Hopf invariant one’ element. Then an easy calculation of the Serre spectral sequence shows that the fibre of f has the homotopy type of the sphere G. Thus G must be an A_{m+1}-space (see [22]).

This suggests the following conjecture.

Conjecture 2.8 An A_{m}-space G has an A_{m+1}-structure extending the given A_{m}-structure if and only if there is a map f : Ē^{m+1}(G) → P^{m}(G) with ‘higher Hopf invariant one’, where P^{m}(G) is the G-projective m-space associated with the A_{m}-structure in the sense of Stasheff.

We bring this higher Hopf invariant H_{m}^{σ(X)} into stable homotopy theory: Definition 2.9 We define a stabilised higher Hopf invariant as

H_{m} = H_{m}^{σ(X)} : [V, X] →^{H_{m}^{σ(X)}} [V, E^{m+1}(ΩX)] →^{Σ_{∗}^{∞}} {V, E^{m+1}(ΩX)}

and the stabilised crude higher Hopf invariant as

H̄_{m} = H̄_{m}^{σ(X)} : [V, X] →^{H̄_{m}^{σ(X)}} [ΣV, ∧^{m+1}X] →^{Σ_{∗}^{∞}} {ΣV, ∧^{m+1}X}.

These definitions of higher Hopf invariants depend on the choice of the structure map σ(X). So it might be useful to define the following set-valued functions. Definition 2.10

H_{m}^{S}(f) = {H_{m}^{σ_{(}X)}(f) | σ(X) is a structure map for cat X = m} ⊂ [V, E^{m+1}(ΩX)], H̄_{m}^{S}(f) = {H̄_{m}^{σ_{(}X)}(f) | σ(X) is a structure map for cat X = m} ⊂ [ΣV, ∧^{m+1}X], H_{m}^{S}(f) = {H_{m}^{σ_{(}X)}(f) | σ(X) is a structure map for cat X = m} ⊂ {V, E^{m+1}(ΩX)} H̄_{m}^{S}(f) = {H̄_{m}^{σ_{(}X)}(f) | σ(X) is a structure map for cat X = m} ⊂ {ΣV, ∧^{m+1}X}.

We show the fundamental properties of higher Hopf invariants. Proposition 2.11 Let V, X and f be as above. Then the following two statements hold.

(1) Let V′ be the suspension space of a co-H-space. If g : V′ → V is a co-H-map (or equivalently, P^{1}(Ωg)^{∘}σ(V′) ∼ σ(V)^{∘}g in P^{1}(ΩV)), then H_{m}^{σ(X)}(f^{∘}g) ∼ H_{m}^{σ(X)}(f)^{∘}g.

(2) Let X′ be a space of LS category ≤ m with a structure map σ(X′). If h : X → X′ is mprimitive in the sense of Berstein-Hilton [2] (or equivalently, σ(X′)^{∘}h ∼ P^{m}(Ωh)^{∘}σ(X)), then H_{m}^{σ(X′)}(h^{∘}f) ∼ E^{m+1}(Ωh)^{∘}H_{m}^{σ(X)}(f).

Corollary 2.11.1 For any map f : V → X, we have the following homotopy relation.

H_{m}^{σ(X)}(f) ∼ H_{m}^{σ(X)}(e_{1}^{X})^{∘}ΣΩ(f)^{∘}σ(V).  (2.3)

Remark 2.12 The statement (2) in Proposition 2.11 is a generalisation of Proposition 3.2 in Berstein-Hilton [2].

Proof of Proposition 2.11. Firstly we show (1): Let d_{m}^{σ(X)}(f) be the difference between σ(X)^{∘}f and P^{m}(Ωf)^{∘}σ(V), in the algebraic loop [V, P^{m}(ΩX)]. Since g is a co-H-map, we obtain the following equation up to homotopy:

σ(X)^{∘}(f^{∘}g) + d_{m}^{σ(X)}(f)^{∘}g ∼ {(σ(X)^{∘}f) + d_{m}^{σ(X)}(f)}^{∘}g ∼ P^{m}(Ωf)^{∘}σ(V)^{∘}g.

Again, since g is a co-H-map, P^{m}(Ωg)^{∘}σ(V′) ∼ σ(V)^{∘}g, and hence we proceed as

σ(X)^{∘}(f^{∘}g) + d_{m}^{σ(X)}(f)^{∘}g ∼ P^{m}(Ωf)^{∘}P^{m}(Ωg)^{∘}σ(V′) = P^{m}(Ω(f^{∘}g))^{∘}σ(V′).

This implies the difference between σ(X)^{∘}(f^{∘}g) and P^{m}(Ω(f^{∘}g))^{∘}σ(V′) is given by d_{m}^{σ(X)}(f^{∘}g) = d_{m}^{σ(X)}(f)^{∘}g, and hence H_{m}^{σ(X)}(f^{∘}g) = H_{m}^{σ(X)}(f)^{∘}g.

Secondly we show (2): Let d_{m}^{σ(X)}(f) be the difference between σ(X)^{∘}f and P^{m}(Ωf)^{∘}σ(V). Since h is an m-primitive map, it follows that σ(X′)^{∘}h ∼ P^{m}(Ωh)^{∘}σ(X). Hence we obtain the following equation up to homotopy:

σ(X′)^{∘}(h^{∘}f) + P^{m}(Ωh)^{∘}d_{m}^{σ(X)}(f) ∼ P^{m}(Ωh)^{∘}{σ(X)^{∘}f + d_{m}^{σ(X)}(f)}
∼ P^{m}(Ωh)^{∘}P^{m}(Ωf)^{∘}σ(V) = P^{m}(Ω(h^{∘}f))^{∘}σ(V).

Thus the difference between σ(X′)^{∘}(h^{∘}f) and P^{m}(Ω(h^{∘}f))^{∘}σ(V) is given by the map d_{m}^{σ(X′)}(h^{∘}f) = P^{m}(Ωh)^{∘}d_{m}^{σ(X)}(f), and hence H_{m}^{σ(X′)}(h^{∘}f) = P^{m}(Ωh)^{∘}H_{m}^{σ(X)}(f). QED.

3 Higher Hopf invariant and LS category

In the remainder of this paper, we always assume that V = ΣV_{0} a suspension space and we fix the structure map σ(V) = Σ ad(1_{V}), unless otherwise stated. We begin this section with the following results by James [14] and by Berstein and Hilton (see Proposition 2.5 in [2]).

Proposition 3.1 (James) Let a CW complex X be (d − 1)-connected, d ≥ 2. Then the following inequality holds: dim X ≥ d · cat X.

Proposition 3.2 (Berstein-Hilton) Let a CW complex X be (d − 1)-connected, d ≥ 2. If dim X ≤ dm + d − 2, then a structure map σ(X) : X → P^{m}(ΩX) for cat X ≤ m is determined uniquely up to homotopy. In particular, when X is simply connected and dim X ≤ 2 cat X, the structure map σ(X) : X → P^{m}(ΩX) for cat X ≤ m is determined uniquely up to homotopy.

Proposition 3.1 implies that quite a few complexes satisfy the hypothesis of Proposition 3.2. Such spaces satisfy the uniqueness of higher Hopf invariants as follows:

Corollary 3.2.1 Let CW complexes V and X be (e−1)-connected and (d−1)-connected resp., d, e ≥ 2 with dim X ≤ dm + d − 2, cat X ≤ m. Then the higher Hopf invariant H_{m} = H_{m}^{σ(X)} is uniquely determined.

Proof. By Proposition 3.2, the hypothesis on dim X and cat X implies that the structure map σ(X) for cat X ≤ m is uniquely determined. By the definition of the homomorphism H_{m} = H_{m}^{σ(X)}, H_{m} is uniquely determined. QED.

We describe the relationship between the higher Hopf invariant and the LS category:

Proposition 3.3 For a given structure map σ(X) for cat X = m and a given map f : V → X, let W be the mapping cone of f. Then the following diagram without the dotted arrows commute up to homotopy.

V  X  W  ΣV E^{m+1}(ΩX)  P^{m}(ΩX) E^{m+1}(ΩW)  P^{m}(ΩW)  P^{m+1}(ΩW)  W,^{f}  w_{_{H_{m}(f)} u}  y  w^{i}_{u_{σ(X)}}  _{u_{σ′(W)}} ^{q}  w w_{p_{m}^{ΩX}} y u_{E^{m+1}(Ωi)}  y u_{P^{m}(Ωi)} w_{p_{m}^{ΩW}}  y  w_{ι_{m}^{ΩW}}  w_{e_{m+1}^{W}}

where i : X ↪ W and ι_{m}^{ΩW} : P^{m}(ΩW) → P^{m+1}(ΩW) denote the inclusions.

Proof. Let us recall that H_{m}(f) is given by the unique lift of the difference between σ(X)^{∘}f and P^{m}(Ωf)^{∘}σ(V) in the group [V, P^{m}(ΩX)]. Hence the composition P^{m}(Ωi)^{∘}p_{m}^{ΩX}^{∘}H_{m}(f) gives the difference between P^{m}(Ωi)^{∘}σ(X)^{∘}f and P^{m}(Ωi)^{∘}P^{m}(Ωf)^{∘}σ(V) = P^{m}(Ωi^{∘}f)^{∘}σ(V) ∼ 0. Thus we obtain the commutativity of the diagram. QED.

Remark 3.4 By the homotopy commutativity of the left rectangle of the diagram, there is a map σ′(W) : W → P^{m+1}(ΩW) making the diagram commutative, since both top and bottom rows except for the map e_{m+1}^{W} are cofibration sequences.

Now we discuss about the naturality of σ′(W) which is determined as above. A suspension map g = Σg_{0} : V → V′ between suspension spaces gives the equality

σ(V)^{∘}g = ΣΩg^{∘}σ(V′).

For maps f : V → X and f′ : V′ → X, we have their cofibres i : X ↪ W = {∗}∪[0, 1]×V ∪_{f} X and i′ : X ↪ W′ = {∗} ∪ [0, 1]×V′ ∪_{f′} X with null-homotopies F : 0 ∼ i^{∘}f and F′ : 0 ∼ i′^{∘}f′:

F (t, v) = t∧v, F (t′, v′) = t′∧v′, for v ∈ V, v′ ∈ V′ and t, t′ ∈ [0, 1].

Lemma 3.5 Let g : V′ → V be a suspension map between suspension spaces. If f′ ∼ f^{∘}g with a homotopy L : [0, 1]×V′ → X, there is a map L̃ : W′ → W extending i : X → W. Moreover there is a homotopy σ′(W)^{∘}L̃ ∼ P^{m+1}(ΩL̃)^{∘}σ′(W′) relative to X, where σ′(W) and σ′(W′) above are given as σ′(W) in Remark 3.4 from σ(X) using H_{m}(f) and H_{m}(f′), resp.

Proof. Let V = ΣV_{0} and V′ = ΣV′_{0} and let g = Σg_{0}, g_{0} : V′_{0} → V_{0}. We define L̃ by the following equation.

L̃(x) = x, L̃(t∧v′) = \begin{cases}(2t)∧g(v′) ∈ W, t ≤ \frac{1}{2}, \\ L(2 − 2t, v′) ∈ X ⊂ W, t ≥ \frac{1}{2} \end{cases}

for x ∈ X and (t, v′) ∈ [0, 1]×V′. Let us consider the following diagram of homotopies:

P^{m}(Ω(L̃_{∘}i′))_{∘}σ(X)_{∘}f′  P^{m}(Ωi)_{∘}σ(X)_{∘}f_{∘}g (A) P^{m}(Ω(L̃^{∘}i′))^{∘}σ(X)^{∘}f′ − 0  P^{m}(Ωi)^{∘}σ(X)^{∘}f^{∘}g − 0 (B) \begin{matrix}P^{m}(Ω(L̃^{∘}i′))^{∘}σ(X)^{∘}f′ \\ −P^{m}(Ω(L̃^{∘}i′^{∘}f′))^{∘}ι_{V′}^{∘}σ(V′) \end{matrix}  \begin{matrix}P^{m}(Ωi)^{∘}σ(X)^{∘}f^{∘}g \\ −P^{m}(Ω(i^{∘}f))^{∘}ι_{V} ^{∘}σ(V)^{∘}g \end{matrix} (C) \begin{matrix}P^{m}(Ω(L̃^{∘}i′))^{∘}(σ(X)^{∘}e_{1}^{X} \\ −ι_{X})^{∘}Σ ad(f′) \end{matrix}  \begin{matrix}P^{m}(Ωi)^{∘}(σ(X)^{∘}e_{1}^{X} \\ −ι_{X})^{∘}Σ ad(f^{∘}g) \end{matrix} (D) P^{m}(Ω(L̃^{∘}i′))^{∘}p_{m}^{ΩX}^{∘}H_{m}^{σ(X)}(e_{1}^{X})^{∘}Σ ad(f′) P^{m}(Ωi)^{∘}p_{m}^{ΩX}^{∘}H_{m}^{σ(X)}(e_{1}^{X})^{∘}Σ ad(f^{∘}g) (E) P^{m}(Ω(L̃^{∘}i′))^{∘}p_{m}^{ΩX}^{∘}H_{m}^{σ(X)}(f′)  P^{m}(Ωi)^{∘}p_{m}^{ΩX}^{∘}H_{m}^{σ(X)}(f^{∘}g)^{P^{m}(Ωi)∘σ(X)∘L}  _{w}_{u}  _{u}_{w^{P^{m}(Ωi)∘σ(X)∘L−c(0)}}_{_{\begin{matrix}c(P^{m}(Ω(L̃∘i′))∘σ(X)∘f′) \\ −P^{m}(Ω(L̃∘F′))∘ι_{V′}∘σ(V′) \end{matrix}} u}  _{u_{\begin{matrix}c(P^{m}(Ωi))∘σ(X)∘f∘g \\ −P^{m}(ΩF)∘ι_{V′}∘σ(V′) \end{matrix}}}_{w^{\begin{matrix}P^{m}(Ωi)∘σ(X)∘L \\ −P^{m}(Ω(i∘L))∘ι_{V′}∘σ(V′) \end{matrix}}}_{^{\begin{matrix}P^{m}(Ωi)∘(σ(X)∘e_{1}^{X} \\ −ι_{X})∘Σad (L) \end{matrix}} w}_{u}  _{u}_{w}_{w^{\begin{matrix}P^{m}(Ωi)∘p_{m}^{ΩX}∘ \\ H_{m}^{σ(X)}(e_{1}^{X})∘Σad (L) \end{matrix}}}  (3.1)

where c(−) denotes a constant homotopy and ι_{X} : P^{1}(ΩX) ↪ P^{m}(ΩX), ι_{V} : P^{1}(ΩV) ↪ P^{m}(ΩV) and ι_{V′} : P^{1}(ΩV′) ↪ P^{m}(ΩV′) are inclusions. The homotopy commutativities of (C) and (E) are trivial.

To show the homotopy commutativity of (A), we define a map θ_{A} : [0, 1]×[0, 1]×V′ → W.

θ_{A}(s, t, u∧v′_{0}) = \begin{cases}L(s, (\frac{2u}{2−t}) ∧v′_{0}) ∈ W, u ≤ \frac{2−t}{2} \\ ∗ ∈ X ⊂ W, u ≥ \frac{2−t}{2} \end{cases} for s, t ∈ [0, 1] and u∧v′_{0} ∈ V′.

We have θ_{A}(s, 0, u∧v′_{0}) = L(s, u∧v′_{0}), θ_{A}(s, 1, u∧v′_{0}) = (L−c(0))(s, u∧v′_{0}) and that θ_{A}(0, t, u∧v′_{0}) and θ_{A}(1, t, u∧v′_{0}) give canonical homotopies from f′(u∧v′_{0}) and f^{∘}g(u∧v′_{0}) = f(u∧g_{0}(v′_{0}) to (f − 0)(u∧v′_{0}) and (f^{∘}g − 0)(u∧v′_{0}) = (f − 0)(u∧g_{0}(v′_{0}). Thus θ_{A} gives a homotopy

f′  f_{∘}g f′ − 0  f^{∘}g − 0.^{L}  w u  u w_{L−c(0)}

where the upper and lower rows are given by t = 0 and t = 1 and left and right columns are given by s = 0 and s = 1. By applying the composition with P^{m}(Ωi)^{∘}σ(X) from the left, we get the homotopy commutativity of (A).

To show the homotopy commutativity of (B), we define a map θ_{B} : [0, 1]×[0, 1]×V′ → W.

θ_{B}(s, t, v′) = \begin{cases}(\frac{2t}{1+s}) ∧g(v′) ∈ W, t ≤ \frac{1+s}{2} \\ L(2 + s − 2t, v′) ∈ X ⊂ W, t ≥ \frac{1+s}{2} \end{cases} for s, t ∈ [0, 1] and v′ ∈ V′.

We have θ_{B}(s, 0, v′) = ∗, θ_{B}(0, t, v′) = L̃^{∘}F′(t, v′), θ_{B}(s, 1, v′) = i^{∘}L(s, v′) and θ_{B}(1, t, v′) = F (t, g(v′)). Thus θ_{B} gives a homotopy

0  0 P^{m}(Ω(L̃^{∘}i′^{∘}f′))  P^{m}(Ω(i^{∘}f^{∘}g))^{c(0)}  w_{_{P^{m}(Ω(L̃∘F′))} u}  _{u_{P^{m}(ΩF)}} w_{P^{m}(Ω(i∘L))}

where the upper and lower rows are given by t = 0 and t = 1 and left and right columns are given by s = 0 and s = 1. By applying the composition with ι_{V′}^{∘}σ(V′) from the right, we get the homotopy commutativity of (B).

To show the homotopy commutativity of (D), we fix a homotopy

σ(X)^{∘}e_{1}^{X} − ι_{X} ∼ p_{m}^{ΩX}^{∘}H_{m}^{σ(X)}(e_{1}^{X})

as a map K : [0, 1]×ΣΩX → P^{m}(ΩX) with K(0, −) = (σ(X)^{∘}e_{1}^{X} − ι_{X})(−) and K(1, −) = p_{m}^{ΩX}^{∘}H_{m}^{σ(X)}(e_{1}^{X})(−). Using it, we define a map θ_{D} : [0, 1]×[0, 1]×V′ → W.

θ_{D}(s, t, u∧v′_{0}) = K(t, u∧ ad(L)(s, v′_{0})) for s, t ∈ [0, 1] and u∧v′_{0} ∈ V′.

We have θ_{D}(s, 0, u∧v′_{0}) = (σ(X)^{∘}e_{1}^{X} −ι_{X})(u∧ ad(L)(s, v′_{0})), θ_{D}(0, t, u∧v′_{0}) = K(t, u∧ ad(f′)(v′_{0})), θ_{D}(s, 1, u∧v′_{0}) = p_{m}^{ΩX}^{∘}H_{m}^{σ(X)}(e_{1}^{X})(u∧ ad(L)(s, v′_{0})) and θ_{D}(1, t, u∧v′_{0}) = K(t, u∧ ad(f^{∘}g)(v′_{0})). Thus θ_{D} gives a homotopy

(σ(X)_{∘}e_{1}^{X} − ι_{X})_{∘}Σ ad(f′)  (σ(X)_{∘}e_{1}^{X} − ι_{X})_{∘}Σ ad(f_{∘}g) p_{m}^{ΩX}^{∘}H_{m}^{σ(X)}(e_{1}^{X})^{∘}Σ ad(f′)  p_{m}^{ΩX}^{∘}H_{m}^{σ(X)}(e_{1}^{X})^{∘}Σ ad(f^{∘}g) w^{(σ(X)∘e_{1}^{X}−ι_{X})∘Σ ad(L)} u  u w_{P^{m}(Ωi)∘σ(X)∘L−c(0)}

where the upper and lower rows are given by t = 0 and t = 1 and left and right columns are given by s = 0 and s = 1. By applying the composition with P^{m}(Ωi) from the left, we get the homotopy commutativity of (D).

The maps σ′(W)^{∘}L̃ and P^{m+1}(Ω(L̃))^{∘}σ′(W′) are given by the homotopies in the diagram (3.1) indicated by the top and right arrows and left and bottom arrows respectively, and hence the homotopy commutativity of the homotopies in the diagram (3.1) implies that σ′(W)^{∘}L̃ ∼ P^{m+1}(ΩL̃)^{∘}σ′(W′). QED.

Remark 3.6 Since e_{m+1}^{W}^{∘}σ′(W) and the identity 1_{W} coincide on X up to homotopy, the difference between them with respect to the co-action of ΣV is given by a map γ : ΣV → W. Here, we know the fibration (2.1), induces the following split short exact sequence:

0 → [ΣV, E^{m+1}(ΩW)] →^{p_{m}^{ΩW}_{∗}} [ΣV, P^{m}(ΩW)] →^{e_{m}^{W}_{∗}} [ΣV, W] → 0.  (3.2)

Thus γ can be pulled back to a map γ_{0} : ΣV → P^{m}(ΩW) ⊂ P^{m+1}(ΩW). By adding γ_{0} to σ′(W), we obtain a genuine compression σ(W) of 1_{W}. We often call this σ(W) a ‘standard’ structure map for cat W ≤ m + 1 which gives a subset of H_{m}^{S}(f) for f ∈ [V′, W] as follows:

H_{m}^{SS}(f) = {H_{m}^{σ(W)}(f) | σ(W) is a ‘standard’ structure map for cat W ≤ m + 1}

The cofibration sequence V →^{f} X →^{i} W →^{q} ΣV induces the following cofibration sequence:

V ∗S^{n−1} →^{f̂} W ∪ X×S^{n} →^{î} W ×S^{n} →^{q̂} Σ^{n+1}V;

here f̂ is given by the relative Whitehead product [χ_{f}, 1_{S^{n−1}}]^{r} : V ∗S^{n} → W ∪ X×S^{n}, where χ_{f} : (C(V), V) → (W, X) denotes the characteristic map given as a relative homeomorphism.

By the proof of Proposition 5.8 in [11], the following result is obtained using Remark 3.4.

Proposition 3.7 For a given structure map σ(X) for cat X = m, the map f̂ makes the following diagram without the dotted arrows commute up to homotopy.

V ∗S^{n−1}  W ∪ X×S^{n}  W ×S^{n}  Σ^{n+1}V E^{m+1}(ΩX)∗S^{n−1} P^{m+1}(ΩW) ∪ P^{m}(ΩX)×S^{n} E^{m+1}(ΩW)∗ΩS^{n} P^{m+1}(ΩW) ∪ P^{m}(ΩW)×S^{n} P^{m+1}(ΩW)×S^{n}  W ×S^{n},^{f̂}  w_{u_{H_{m}(f)∗1_{S^{n−1}}}}  y  w^{î}_{u_{σ′(W)×1_{S^{n}}|_{W ∪X×S^{n}}}}  _{u_{σ(W ×S^{n})}}  ^{q̂}  w y u_{E^{m+1}(Ωi)∗j_{n−1}} w_{p̂_{m}^{ΩX}}  y u w_{p̂_{m}^{ΩW}}  y w  w_{e_{m+1}^{W}×1_{S^{n}}}

where p̂_{m}^{ΩX} = [χ_{p_{m}^{ΩX}}, 1_{S^{n}}]^{r}, p̂_{m}^{ΩW} = [χ_{p_{m}^{ΩW}} , e_{1}^{S^{n}}]^{r} and j_{n−1} denotes the bottom cell inclusion S^{n−1} ⊂ ΩS^{n} and σ′(W) is the extension of σ(X) by Proposition 3.3 and Remark 3.4.

We then have the following result.

Theorem 3.8 The following two statements hold for W with cat W ≤ m + 1, m = cat X.

(1) cat W ≤ cat X if the set H_{m}^{S}(f) contains a trivial element.

(2) cat W ×S^{n} ≤ cat X + 1 if the set Σ_{∗}^{n}H_{m}^{S}(f) contains a trivial element.

Corollary 3.8.1 Let W be the space constructed as in the above theorem with cat W = m + 1. Then W is a counter example to Ganea’s conjecture if the set H_{m}^{S}(f) contains trivial element.

Proof of Theorem 3.8. Firstly we show (1): If E^{m+1}(Ωi)^{∘}H_{m}^{σ(X)}(f) : V → E^{m+1}(ΩW) is trivial for some structure map σ(X), then, by Proposition 3.3 and Remark 3.4, σ(X) is extendible to W. By the argument given in Remark 3.6, we obtain a genuine compression σ(W) : W → P^{m}(ΩW) of 1_{W}. Thus cat W ≤ m.

Secondly we show (2): If (E^{m+1}(Ωi)∗j_{n−1})^{∘}Σ^{n}H_{m}^{σ(X)}(f) : V ∗S^{n−1} → E^{m+1}(ΩW)∗ΩS^{n} is trivial for some structure map σ(X), then, by Proposition 3.7, the map σ′(W)×1_{S^{n}}|_{W ∪X×S^{n}} : W ∪ X×S^{n} → P^{m+1}(ΩW) ∪ P^{m}(ΩX)×S^{n} ⊂ P^{m+1}(ΩW) ∪ P^{m}(ΩW)×S^{n} is extendible to W ×S^{n}. By the argument given in Remark 3.6 together with the fact that the natural map [Σ^{n+1}V, Z×B∪C×Y] → [Σ^{n+1}V, Z×Y] ≅ [Σ^{n+1}V, Z]×[Σ^{n+1}V, Y] is split surjective for any pointed pairs (Z, C) and (Y, B), we obtain a compression σ′′ : W ×S^{n} → P^{m+1}(ΩW) ∪ P^{m}(ΩW)×S^{n} of σ′(W)×1_{S^{n}}, as an extension of σ′(W)×1_{S^{n}}|_{W ∪X×S^{n}}. Also by Remark 3.6, the difference between e_{m+1}^{W}^{∘}σ′(W) and 1_{W} with respect to the co-action of ΣV can be pulled back to γ_{0} : ΣV → ΣΩW ⊂ P^{m}(ΩW). We define a map σ(W ×S^{n}) by

σ(W ×S^{n}) : W ×S^{n} →^{μ×1_{S^{n}}} (W ∨ ΣV)×S^{n} P^{m+1}(ΩW)×{∗} ∪ P^{m}(ΩW)×S^{n},^{σ′′∪(γ_{0}×1_{S^{n}})} ^{w}

Since σ′′ is homotopic to σ′(W)×1_{S^{n}} in P^{m+1}(ΩW)×S^{n} with the subspace {∗}×S^{n} left fixed, σ(W ×S^{n}) is homotopic to (σ′(W) + γ)×1_{S^{n}} in P^{m+1}(ΩW)×S^{n} which is a compression of 1_{W} ×1_{S^{n}}. Thus cat W ×S^{n} ≤ m + 1. QED.

Let V be a (e−1)-connected co-H-space and X a (d−1)-connected CW complex, e ≥ d ≥ 2.

Theorem 3.9 Let n ≥ 1 and X be of dim X ≤ d·m + d − 2 and of cat X = m. Then the following two statements hold for W and H_{m}(f) = H_{m}^{σ(X)}(f) where σ(X) is the unique structure map.

(1) cat W = m + 1 if E^{m+1}(Ωi)^{∘}H_{m}(f) ≠ 0.

(2) cat W ×S^{n} = m + 2 if (E^{m+1}(Ωi)∗j_{n−1})^{∘}Σ^{n}H_{m}(f) ≠ 0 and n > dim V − d(m + 1) + 2.

(3) cat W ×S^{n} = m + 2 if (E^{m+1}(Ωi)∗j_{n})^{∘}Σ^{n+1}H_{m}(f) ≠ 0.

Remark 3.10 If the condition dim V < d · cat X + e − 1 is satisfied, then for dimensional reasons, we may remove the maps ‘E^{m+1}(Ωi)’, ‘E^{m+1}(Ωi)∗j_{n−1}’ and ‘E^{m+1}(Ωi)∗j_{n}’ from the statements in Theorem 3.9.

Corollary 3.10.1 Let W be the space constructed as the mapping cone of f : V → X from simply connected spaces V and X with dim X ≤ d · cat X + d − 2 and dim V ≤ d · cat X + e − 2, where V and X are (e − 1)-connected and (d − 1)-connected, resp. If cat W ×S^{k} = cat W for some k, then cat W ×S^{n} = cat W for all n ≥ k. Thus Conjecture 2.3 is true for such W.

Corollary 3.10.2 Let W be the space constructed as the mapping cone of f : V → X from simply connected spaces V and X with dim X ≤ d · cat X + d − 2 and dim V ≤ d · cat X + e − 2, where V and X are (e − 1)-connected and (d − 1)-connected, resp. Then Ganea’s conjecture for W is true if and only if the unique stabilised higher Hopf invariant H_{m}(f) is non-trivial.

Proof of Theorem 3.9. Let dim X = k. By the assumption, cat W ≤ m + 1.

Firstly we show (1): If cat W ≤ m, then there exists a compression σ : W → P^{m}(ΩW) of the identity 1_{W}. Since the pair (P^{∞}(ΩW), P^{m}(ΩW)) is (d(m+1)−1)-connected, the inclusion map ι_{m} : P^{m}(ΩW) → P^{∞}(ΩW) induces a bijection of homotopy sets [Z, P^{m}(ΩW)] → [Z, P^{∞}(ΩW)] for any space Z of dimension up to d(m + 1) − 2. Then, for dimensional reasons, it follows that the restriction σ_{0} = σ|_{X} is unique up to homotopy in P^{m}(ΩW). Hence we may assume that σ_{0} equals to σ(X), the unique structure map for cat X = m by Proposition 3.2. Hence P^{m}(Ωi)^{∘}σ(X)^{∘}f ∼ p_{m}^{ΩW}^{∘}E^{m+1}(Ωi)^{∘}H_{m}(f) is null-homotopic. Then by the short exact sequence in Remark 3.6, it follows that E^{m+1}(Ωi)^{∘}H_{m}(f) is null-homotopic. Thus the existence of the compression σ of 1_{W} implies the triviality of E^{m+1}(Ωi)^{∘}H_{m}(f) : V → E^{m+1}(ΩW).

Secondly we show (2): Let σ(W) be a structure map for cat W ≤m + 1 obtained from σ(X) by Proposition 3.3, Remarks 3.4 and 3.6, and let ĵ_{m,n} : P^{m+1}(ΩW) ∪ P^{m}(ΩW)×S^{n} ⊂ P^{∞}(ΩW)∪P^{m}(ΩW)×S^{n} and î_{m,n} : P^{∞}(ΩW) ∪ P^{m}(ΩW)×S^{n} ⊂ P^{∞}(ΩW)×S^{n} be inclusions. If cat W ×S^{n} ≤ m + 1, then by Fact 2.2, there exists a compression σ : W ×S^{n} → P^{m+1}(ΩW) ∪ P^{m}(ΩW)×S^{n} of the identity 1_{W ×S^{n}}. Hence we have

e_{∞}^{W}×1_{S^{n}}|_{P^{∞}(ΩW)∪P^{m}(ΩW)×S^{n}}^{∘}ĵ_{m,n}^{∘}σ|_{W ∪X×S^{n}} ∼ 1_{W ×S^{n}} and
e_{∞}^{W}×1_{S^{n}}|_{P^{∞}(ΩW)∪P^{m}(ΩW)×S^{n}}^{∘}ĵ_{m,n}^{∘}(σ(W)×1_{S^{n}}|_{W ∪X×S^{n}}) ∼ 1_{W ×S^{n}}.

Here we know the pair (P^{∞}(ΩW)×S^{n}, P^{∞}(ΩW)∪P^{m}(ΩW)×S^{n}) is (dm + n + d − 1)-connected and there is a co-action μ′_{W} : W ∪X×S^{n} → W ∪X×S^{n} ∨ ΣV associated with the cofibration sequence V → X×S^{n} → W ∪ X×S^{n}. Since the subspace X×S^{n} is of dimension k + n ≤ dm + d + n − 2, the restrictions ĵ_{m,n}^{∘}σ|_{X×S^{n}} and ĵ_{m,n}^{∘}σ(W)×1_{S^{n}}|_{X×S^{n}} = σ(X)×1_{S^{n}} are homotopic and the difference between ĵ_{m,n}^{∘}σ and σ(W)×1_{S^{n}}|_{W ∪X×S^{n}} with respect to the co-action μ′_{W} is given by a map into E^{m+1}(ΩW)∗ΩS^{n} the fibre of e_{∞}^{W}×1_{S^{n}}|_{P^{∞}(ΩW)∪P^{m}(ΩW)×S^{n}}:

σ|_{W ∪X×S^{n}} ∼ σ(W)|_{W ∪X×S^{n}} + γ, γ = ĵ_{m,n}^{∘}p̂_{m}^{ΩW}^{∘}γ_{0}, γ_{0} : ΣV → E^{m+1}(ΩW)∗ΩS^{n}.

If we assume that n > dim V − d(m + 1) + 2, then we also obtain that ĵ_{m,n}^{∘}σ|_{W ∪X×S^{n}} ∼ ĵ_{m,n}^{∘}σ(W)×1_{S^{n}}|_{W ∪X×S^{n}} for dimensional reasons. In this case, we have

ĵ_{m,n}^{∘}p̂_{m}^{ΩW}^{∘}(E^{m+1}(Ωi)∗j_{n−1})^{∘}Σ^{n}H_{m}(f) ∼ ĵ_{m,n}^{∘}(σ(W)×1_{S^{n}}|_{W ∪X×S^{n}})^{∘}f̂ ∼ ĵ_{m,n}^{∘}σ|_{W ∪X×S^{n}}^{∘}f̂

which is trivial, since σ|_{W ∪X×S^{n}} is extendible on W ×S^{n}. Thus (E^{m+1}(Ωi)∗j_{n−1})^{∘}Σ^{n}H_{m}(f) is trivial provided that n > dim V − d(m + 1) + 2.

Thirdly we show (3): In this case, the difference γ might not be trivial. Since the composition

pr_{1} ^{∘}ĵ_{m,n}^{∘}σ : W ×S^{n} →^{ĵ_{m,n}∘σ} P^{∞}(ΩW)∪P^{m}(ΩW)×S^{n} →^{pr_{1}} P^{∞}(ΩW) factors through P^{∞}(ΩW)×S^{n}

as pr_{1} ^{∘}ĵ_{m,n}^{∘}σ = pr_{1} ^{∘}î_{m,n}^{∘}ĵ_{m,n}^{∘}σ, we have e_{∞}^{W}^{∘} pr_{1} ^{∘}ĵ_{m,n}^{∘}σ = pr_{1} ^{∘}(e_{∞}^{W}×1_{S^{n}})^{∘}î_{m,n}^{∘}ĵ_{m,n}^{∘}σ ∼ pr_{1}, which yields the following commutative diagram up to homotopy.

W  P^{∞}(ΩW) W ×S^{n}  P^{∞}(ΩW)∪P^{m}(ΩW)×S^{n} W  P^{∞}(ΩW).^{(e_{∞}^{W})^{−1}}  w u_{pr_{1}}  _{^{ĵ_{m,n}∘σ}  w}_{_{pr_{1}} u}  u_{pr_{1}}_{u_{pr_{1}}}_{^{(e_{∞}^{W})^{−1}}  w}

Taking push-outs of both right and left columns of the above diagram, we have a map

σ_{1} : W ×S^{n+1} → P^{∞}(ΩW)∪P^{m}(ΩW)×S^{n+1} with σ_{1}^{∘}(1_{W} ×E_{n}) = E_{m,n}^{∘}ĵ_{m,n}^{∘}σ, where E_{n} denotes the inclusion S^{n} = S^{n}×{0} ↪ S^{n+1} = S^{n}×[−1, 1]/∼, (x, −1) ∼ (x, 1) ∼ (∗, t) for x ∈ S^{n} and t ∈ [−1, 1] and E_{m,n} = 1_{P^{∞}(ΩW)}×E|_{P^{∞}(ΩW)∪P^{m}(ΩW)×S^{n}}. By the definition of σ_{1}, we have σ_{1}|_{X×S^{n+1}} ∼ ĵ_{m,n+1}^{∘}(σ(X)×1_{S^{n+1}}) and the difference between σ_{1}|_{W ∪X×S^{n+1}} and ĵ_{m,n+1}^{∘}(σ(W)×1_{S^{n+1}})|_{W ∪X×S^{n+1}} are given by E_{m,n}^{∘}γ. Since P^{∞}(ΩW)∪P^{m}(ΩW)×(S^{n}×0) is compressible into P^{∞}(ΩW), so is E_{m,n}^{∘}γ. Thus E_{m,n}^{∘}γ ∼ in_{1} ^{∘} pr_{1} ^{∘}E_{m,n}^{∘}γ = in_{1} ^{∘} pr_{1} ^{∘}γ ∼ in_{1} ^{∘} pr_{1} ^{∘}î_{m,n}^{∘}ĵ_{m,n}^{∘}γ ∼ 0, where in_{1} or pr_{1} denotes an appropriate inclusion or projection. Thus by the same argument as in (2), we obtain (E^{m+1}(Ωi)∗j_{n})^{∘}Σ^{n+1}H_{m}(f) ∼ 0. QED.

4 Higher Hopf invariants and the reduced diagonal

We state here another property of the higher Hopf invariants, which is a generalisation of Theorem 5.14 of Boardman-Steer [3]. Theorem 4.1 Let V →^{f} X →^{i} W →^{q} ΣV be a cofibration sequence as above with cat X = m. Then the reduced diagonal Δ̂_{m+1} : W → W^{m+1} → ∧^{m+1}W is given by the composition ∧^{m+1}e_{1}^{W}^{∘}Σh_{m+1}^{W}^{∘}Σ(E^{m+1}(Ωi)^{∘}H_{m}(f))^{∘}q. Corollary 4.1.1 Let V, X, f and W be as above with cat X = m and cat W = m + 1. Then the n-fold suspension of the reduced diagonal Δ̂_{m+1}∧1_{S^{n}} : W ×S^{n} → W^{m+1}×S^{n} → ∧^{m+1}W ∧S^{n} is given by the composition Σ^{n}(∧^{m+1}e_{1}^{W}^{∘}Σh_{m+1}^{W}^{∘}Σ(E^{m+1}(Ωi)^{∘}H_{m}(f)))^{∘}q̂, where q̂ denotes the collapsing map W ×S^{n} → W ×S^{n}/(W ∪ X×S^{n}) = (W/X)∧S^{n} = Σ^{n+1}V.

For any space Z, Ganea showed that there is a commutative ladder of fibrations up to homotopy (see [5] or the proof of Theorem 1.1 of [11]):

E^{m+1}(ΩZ)  P^{m}(ΩZ)  Z Ē^{m+1}(ΩZ)  Z^{[m+1]}  Z^{m+1},^{p_{m}^{Z}} _{w}_{u_{h_{m+1}^{Z}}}  ^{e_{m}^{Z}}  _{w}_{u_{Δ′_{m+1}}}  _{u_{Δ_{m+1}}}_{^{q_{m+1}^{Z}} w}  _{y  w}  (4.1)

where the fibration q_{m+1}^{Z} : Ē^{m+1}(ΩZ) → Z^{[m+1]} is given by q_{m+1}^{Z}(Σt_{i}ℓ_{i}) = {ℓ_{i}(t_{i}/t_{M})}_{0≤i≤m}, t_{M} = max(t_{0}, ..., t_{m}) and Δ′_{m+1} denotes a map which makes the right hand square of (4.1) a homotopy pull-back diagram.

Using the precise description of the fibration q_{m+1}^{Z}, we obtain the following proposition. Proposition 4.2 There exists a map e_{m+1} : Z^{[m+1]} ∪ Ĉ(Ē^{m+1}(ΩZ)) → Z^{m+1} which makes the following diagram commute up to homotopy, where Ĉ denotes the functor taking unreduced cones with base point 0∧∗:

Ē^{m+1}(ΩZ)  Z^{[m+1]}  Z^{[m+1]} ∪ Ĉ(Ē^{m+1}(ΩZ))  ∧^{m+1}ΣΩZ Ē^{m+1}(ΩZ)  Z^{[m+1]}  Z^{m+1}  ∧^{m+1}Z,^{q_{m+1}^{Z}}  w  y  w  w_{u_{e_{m+1}}}  _{u_{∧^{m+1}e_{1}^{Z}}}_{^{q_{m+1}^{Z}}  w}  y  w  w

where the upper row is a cofibration sequence. Proof. We define e_{m+1} : Z^{[m+1]} ∪ Ĉ(Ē^{m+1}(ΩZ)) → Z^{m+1} by

e_{m+1}(t, Σt_{i}ℓ_{i}) = (ℓ_{1}(t·t_{1}/t_{M}), ⋯, ℓ_{m}(t·t_{m}/t_{M})),

where ∑_{0≤i≤m} t_{i} = 1 and t_{M} = max(t_{0}, ..., t_{m}) > \frac{1}{m+1}. By collapsing Z^{[m+1]}, we obtain a map ê_{m+1} : Σ̂Ē^{m+1}(ΩZ) → ∧^{m+1}Z from e_{m+1}, where Σ̂ denotes the functor taking unreduced suspensions with base point 0∧∗. ê_{m+1} is given by ê_{m+1}(t∧Σt_{i}ℓ_{i}) = ℓ_{0}(tt_{0}/t_{M})∧...∧ℓ_{m}(tt_{m}/t_{M}).

Let us recall the homotopy equivalence s_{m+1} : Σ̂Ē^{m+1}(ΩZ) → ∧^{m+1}Σ̂ΩZ given by

s_{m+1}(t, Σt_{i}ℓ_{i}) = (u_{i} ∧ ℓ_{i}), u_{i} = t_{i} \frac{t}{t_{M}} .

Then we have the following relation of maps: ∧^{m+1}e_{1}^{Z}^{∘}h^{∘}s_{m+1} = ê_{m+1}, where h : ∧^{m+1}Σ̂ΩZ → ∧^{m+1}ΣΩZ is the canonical homotopy equivalence. Thus the proposition follows. QED.

Proof of Theorem 4.1. By Remarks 3.4 and 3.6 and Propositions 3.3 and 4.2, we have the following commutative diagram up to homotopy.

V  X  W  ΣV E^{m+1}(ΩW)  P^{m}(ΩW)  P^{m+1}(ΩW)  ΣE^{m+1}(ΩW) Ē^{m+1}(ΩW)  W^{[m+1]}  W^{m+1}  ∧^{m+1}W.^{f}  w_{_{E^{m+1}(Ωi)∘H_{m}(f)} u}  y  w^{i}_{u_{σ(X)}}  ^{q}  w_{u_{σ(W)}}  _{u_{Σ(E^{m+1}(Ωi)∘H_{m}(f))}} w_{_{h_{m+1}^{W}} u}  y  w u  w_{u_{Δ_{m+1}∘e_{m+1}^{W}}}  _{u_{∧^{m+1}e_{1}^{W} ∘Σh_{m+1}^{W}}} w_{q_{m}^{W}}  y  w  w_{∧}

QED.

Remark 4.3 For any structure map σ(X), by naturality, we have

Δ̂_{m+1} = (∧^{m+1}e_{1}^{W})^{∘}Σh_{m+1}^{W}^{∘}Σ(E^{m+1}(Ωi)^{∘}H_{m}^{σ(X)}(f))^{∘}q
= (∧^{m+1}e_{1}^{W})^{∘}Σh_{m+1}^{W}^{∘}ΣE^{m+1}(Ωi)^{∘}ΣH_{m}^{σ(X)}(f)^{∘}q
= (∧^{m+1}e_{1}^{X})^{∘}(∧^{m+1}ΣΩi)^{∘}Σh_{m+1}^{X}^{∘}Σ(H_{m}^{σ(X)}(f))^{∘}q
= (∧^{m+1}i)^{∘}(∧^{m+1}e_{1}^{X})^{∘}Σh_{m+1}^{X}^{∘}Σ(H_{m}^{σ(X)}(f))^{∘}q = (∧^{m+1}i)^{∘}H̄_{m}^{σ(X)}(f)^{∘}q,

where the composition H̄_{m}^{σ(X)}(f) = (∧^{m+1}e_{1}^{X})^{∘}Σh_{m+1}^{X}^{∘}Σ(H_{m}^{σ(X)}(f)) is the generalised version of the Berstein-Hilton crude Hopf invariant (see Definition 2.6). Thus we have

(∧^{m+1}i)_{∗}^{∘}q^{∗}H̄_{m}^{S}(f) = {Δ̂_{m+1}}.

5 Homology decomposition and product spaces

In this section, we always assume that X is simply connected finite complex. Then there is a homology decomposition {X_{t}, f_{t} : S_{t}(X) → X_{t}}_{t≥1} of X, where S_{t}(X) is the Moore space of type (H_{t+1}(X), t) for t ≥ 1. By modifying the arguments given in Curjel [4], we obtain the following result.

Theorem 5.1 The homology decomposition {X_{t}, f_{t} : S_{t}(X) → X_{t}}_{t≥1} of X satisfies 0 = cat X_{1} ≤ cat X_{2} ≤ ... ≤ cat X_{t−1} ≤ cat X_{t} ≤ ... ≤ cat X. If, in addition, cat X_{k−1} = cat X_{k} = m, then we can choose structure maps σ_{k−1}(X) : X_{k−1} → P^{m}(ΩX_{k−1}) and σ_{k}(X) : X_{k} → P^{m}(ΩX_{k}) for cat X_{k−1} = m and cat X_{k} = m to be compatible with each other, i.e. σ_{k}(X)|_{X_{k}} ≃ σ_{k−1}(X) in P^{m}(ΩX_{k}).

Proof. To prove the former part of the theorem, it is sufficient to show cat X_{k} ≤ cat X_{k+1} for k ≥ 1. So we may assume that X = X_{k+1} and cat X = m. If m = 0, then Theorem 5.1 is clearly true, and hence we may assume that m ≥ 1. Then there is a homotopy section σ(X) : X → P^{m}(ΩX). By induction on k, we show the existence of a compression σ_{k}(X) : X_{k} → P^{m}(ΩX_{k}) of σ(X)|_{X_{k}} : X_{k} → P^{m}(ΩX). In the case k = 1, we have X_{1} = {∗}, and hence the existence of σ_{1}(X) is clear. In the general case k > 1, by the induction hypothesis, we have a compression σ_{k−1}(X) : X_{k−1} → P^{m}(ΩX_{k−1}) of σ(X)^{∘}i_{k−1} : X_{k−1} → P^{m}(ΩX), where i_{t} : X_{t} ↪ X denotes the canonical inclusion, t ≥ 1; in addition, e_{m}^{X_{k−1}}^{∘}σ_{k−1}(X) ∼ 1_{X_{k−1}}. Let us consider the following commutative diagram:

E^{m+1}(ΩX_{k})  E^{m+1}(ΩX) X_{k−1}  X_{k}  X P^{m}(ΩX_{k−1})  P^{m}(ΩX_{k})  P^{m}(ΩX) X_{k−1}  X_{k}  X,_{_{p_{m}^{ΩX_{k}}} u}  _{y  w}^{E^{m+1}(Ωi_{k})}  _{u_{p_{m}^{ΩX}}}_{_{``}_{ˆ σ_{k−1}(X)}}_{y  w^{i_{k−1,k}}} _{y  w^{i_{k}}}_{´´´}_{_{ˆ```}_{σ̂(X)}}_{_{e_{m}^{X_{k−1}}} u}  _{y  w_{P^{m}(Ωi_{k−1,k})}}_{_{e_{m}^{X_{k}}} u}  _{y  w_{P^{m}(Ωi_{k})}} _{u_{e_{m}^{X}}}_{y  w^{i_{k−1,k}}} _{y  w^{i_{k}}}  (5.1)

where i_{k−1,k} : X_{k−1} ↪ X_{k} denotes the canonical inclusion. Also the k′-invariant f_{k−1} induces the following cofibre sequence:

S_{k−1} →^{f_{k−1}} X_{k−1} ↪^{i_{k−1,k}} X_{k} →^{g_{k}} ΣS_{k−1},

The obstruction to extend σ_{k−1}(X) to X_{k} is given by a map γ = P^{m}(Ωi_{k−1,k})^{∘}σ_{k−1}(X)^{∘}f_{k−1} : S_{k−1} → P^{m}(ΩX_{k}). But the commutativity of the diagram (5.1) implies that p_{m}^{ΩX_{k}}^{∘}γ = i_{k−1,k}^{∘}e_{m}^{X_{k−1}}^{∘}σ_{k−1}(X)^{∘}f_{k−1} ∼ i_{k−1,k}^{∘}f_{k−1} ∼ 0. Hence γ has a unique lift γ̂ : S_{k−1} → E^{m+1}(ΩX_{k}), which vanishes in E^{m+1}(ΩX), since P^{m}(Ωi_{k})^{∘}γ = P^{m}(Ωi_{k−1})^{∘}σ_{k−1}(X)^{∘}f_{k−1} ∼ σ(X)^{∘}i_{k−1}^{∘}f_{k−1} ∼ 0. Since X and X_{k} are simply connected and (X, X_{k}) is k-connected, (E^{m+1}(ΩX), E^{m+1}(ΩX_{k})) is (k + 2m − 1)-connected, m ≥ 1. Hence γ vanishes since the dimension of S_{k−1} is at most k ≤ k+2m−2. Thus there is a map σ′_{k} : X_{k} → P^{m}(ΩX_{k}) such that σ′_{k}^{∘}i_{k−1,k} ∼ P^{m}(Ωi_{k−1,k})^{∘}σ_{k−1}(X).

The difference between e_{m}^{X_{k}}^{∘}σ′_{k} and the identity 1_{X_{k}} with respect to the co-action of ΣS_{k−1} is given by a map δ : ΣS_{k−1} → X_{k}. By (3.2), with V = S_{k−1}, X = X_{k−1} and W = X_{k}, we have a surjection [ΣS_{k−1}, ΣΩX_{k}] → [ΣS_{k−1}, X_{k}] and hence δ can be pulled back to δ_{0} : ΣS_{k−1} → ΣΩX_{k} ⊂ P^{m}(ΩX_{k}). By adding δ_{0} to σ′_{k}, we have a compression σ_{k} of 1_{X_{k}} which is an extension of σ_{k−1}(X). Thus cat X_{k} ≤ m.

The difference between P^{m}(Ωi_{k})^{∘}σ_{k} and σ(X)^{∘}i_{k} with respect to the co-action of ΣS_{k−1} is given by a map ε : ΣS_{k−1} → P^{m}(ΩX) which vanishes in P^{∞}(ΩX) ≃ X. Thus ε can be lifted to ε_{0} : ΣS_{k−1} → E^{m+1}(ΩX). Since (E^{m+1}(ΩX), E^{m+1}(ΩX_{k})) is (k + 2m − 1)-connected and the dimension of ΣS_{k−1} is at most k + 1 ≤ k + 2m − 1, ε_{0} can be compressed into E^{m+1}(ΩX_{k}); ε_{0} : ΣS_{k−1} →^{ε′_{0}} E^{m+1}(ΩX_{k}) ⊂ E^{m+1}(ΩX). Again by adding p_{m}^{ΩX_{k}}^{∘}ε′_{0} to σ_{k}, we obtain new structure map σ_{k}(X) : X_{k} → P^{m}(ΩX_{k}) for cat X_{k} ≤ m as a compression of σ(X)^{∘}i_{k}, which is also an extension of σ_{k−1}(X). Thus we obtain the compression σ_{k}(X) of σ(X)^{∘}i_{k} for all k. The latter part of the theorem is clear by the definition of σ_{k}(X). QED.

Now we apply the results in Section 3 for homology decompositions. Then we can show Theorems 3.8 (1) and 3.9 (1) in a slightly stronger form. Let cat X_{k} = m for some k ≥ 1, and let f_{k}(X) : S_{k}(X) → X_{k} be the k′-invariant of the k-th stage. We show that the obstruction for X_{k+1} to satisfy cat X_{k+1} = m is the set of higher Hopf invariants H_{m}^{S}(f_{k}(X)).

Theorem 5.2 cat X_{k+1} = m if and only if E^{m+1}(Ωi_{k,k+1})_{∗}H_{m}^{S}(f_{k}(X)) contains 0. Moreover, if (i) m ≥ 2 or (ii) X is 2-connected or more generally Ext(H_{k+1}(X), H_{2}(X)⊗H_{k+1}(X)) = 0, then cat X_{k+1} = m if and only if H_{m}^{S}(f_{k}(X)) contains 0.

Proof. If cat X_{k+1} = m, then by Theorem 5.1, there exists a structure map σ_{k}(X) for cat X_{k} = m such that P^{m}(Ωi_{k,k+1})^{∘}σ_{k}(X) is extendible to X_{k+1}. For this particular choice of σ_{k}(X), we obtain, by Proposition 3.3, the following diagram except for the dotted arrows commutative up to homotopy.

S_{k}(X)  X_{k}  X_{k+1} ΩX_{k}  E^{m+1}(ΩX_{k})  P^{m}(ΩX_{k}) ΩX_{k+1}  E^{m+1}(ΩX_{k+1})  P^{m}(ΩX_{k+1})  P^{m+1}(ΩX_{k+1}).^{f_{k}(X)}  w_{_{H_{m}^{σ_{k}(X)}(f_{k}(X))} u}  y  w^{i_{k,k+1}}_{u_{σ_{k}(X)}}  _{u_{σ_{k+1}}} y  w y u_{Ωi_{k,k+1}}  _{^{p_{m}^{ΩX_{k}}} w} y u_{E^{m+1}(Ωi_{k,k+1})}  y u_{P^{m}(Ωi_{k,k+1})} y  w  w_{p_{m}^{ΩX_{k+1}}}  y  w_{ι_{m}^{ΩX_{k+1}}}

Thus the extendibility of P^{m}(Ωi_{k,k+1})^{∘}σ_{k}(X) implies that p_{m}^{ΩX_{k+1}}^{∘}E^{m+1}(Ωi_{k,k+1})^{∘}H_{m}^{σ_{k}(X)}(f_{k}(X)) ∼ P^{m}(Ωi_{k,k+1})^{∘}σ_{k}(X)^{∘}f_{k}(X) is trivial. Since p_{m}^{ΩX_{k+1}} induces a split monomorphism of homotopy groups with any coefficient groups, it follows that E^{m+1}(Ωi_{k,k+1})^{∘}H_{m}^{σ_{k}(X)}(f_{k}(X)) is trivial. We know the pair (E^{m+1}(ΩX_{k+1}), E^{m+1}(ΩX_{k})) is (k + 2m − 1)-connected, and hence, by Universal Coefficient Theorem for homotopy groups, we obtain

π_{k}(E^{m+1}(ΩX_{k+1}), E^{m+1}(ΩX_{k}); H_{k+1}(X)) = 0 and
π_{k+1}(E^{m+1}(ΩX_{k+1}), E^{m+1}(ΩX_{k}); H_{k+1}(X))
= Ext(H_{k+1}(X), H_{k+2}(E^{m+1}(ΩX_{k+1}), E^{m+1}(ΩX_{k})))

is trivial unless m = 1. When m = 1, for dimensional reasons, we have

H_{k+2}(E^{2}(ΩX_{k+1}), E^{2}(ΩX_{k})) ≅ H_{k+2}(S_{1}(X)∗S_{k}(X)∨S_{k}(X)∗S_{1}(X))
≅ H_{2}(X)⊗H_{k+1}(X)⊕H_{k+1}(X)⊗H_{2}(X),

and hence

π_{k+1}(E^{m+1}(ΩX_{k+1}), E^{m+1}(ΩX_{k}); H_{k+1}(X))
≅ Ext(H_{k+1}(X), H_{2}(X)⊗H_{k+1}(X))⊕ Ext(H_{k+1}(X), H_{k+1}(X)⊗H_{2}(X))

is trivial if the condition (ii) is satisfied. Hence E^{m+1}(Ωi_{k,k+1})_{∗} : [S_{k}(X), E^{m+1}(ΩX_{k})] = π_{k}(E^{m+1}(ΩX_{k})) → [S_{k}(X), E^{m+1}(ΩX_{k+1})] = π_{k}(E^{m+1}(ΩX_{k+1})) is an isomorphism. Thus we obtain that the set H_{m}^{S}(f_{k}(X)) contains 0 = H_{m}^{σ_{k}(X)}(f_{k}(X)). The converse is an immediate consequence of Theorem 3.8 (1). QED.

Let cat X_{k+1} = m + 1, in other words, the set H_{m}^{S}(f_{k}(X)) does not contain 0. Then the following theorem is an immediate consequence of Theorem 3.8 (2). Theorem 5.3 If the set Σ_{∗}^{n}H_{m}^{S}(f_{k}(X)) contains 0, then cat X_{k+1}×S^{n} = cat X_{k+1} = m + 1. Corollary 5.3.1 If the set H_{m}^{S}(f_{k}(X)) contains 0, then X_{k+1} is a counter example to Ganea’s conjecture.

Also let n ≥ 1, cat Y ≥ n + 1. Then, by Theorem 5.1, one can find an integer h ≥ 1 such that cat Y_{h+1} = n + 1 and cat Y_{h} = n. Let f_{h}(Y) : S_{h}(Y) → Y_{h} be the k′-invariant of the h-th stage. We know, by Theorem 5.2, that the obstruction for Y_{h+1} to satisfy cat Y_{h+1} ≤ n is the set of Hopf invariants H_{h}^{S}(f_{h}(Y)). We define another set Definition 5.4

H_{m,n}^{S}(f_{k}(X), f_{h}(Y)) = {g_{X}∗g_{Y} | g_{X} ∈ H_{m}^{S}(f_{k}(X)) and g_{Y} ∈ H_{n}^{S}(f_{h}(Y))}

Then we have the following theorem. Theorem 5.5 If the set H_{k,h}^{S}(X, Y) contains 0, then cat X_{k+1}×Y_{h+1} < m + n + 2 = cat X_{k+1} + cat Y_{h+1}. Proof. The proof is obtained by a similar argument given in the proof of Theorem 3.8 (2) using the following diagram instead of the diagram in Proposition 3.7:

S_{k+h+1}(X_{k+1}×Y_{h+1})  _{X_{k+1}×Y_{h}∪X_{k}×Y_{h+1}}  _{X_{k+1}×Y_{h+1}}_{E^{m+1}(ΩX_{k})∗E^{n+1}(ΩY_{h})}  _{\begin{matrix}P^{m+1}(ΩX_{k+1})×P^{n}(ΩY_{h}) \\ ∪P^{m}(ΩX_{k})×P^{n+1}(ΩY_{h+1}) \end{matrix}}_{E^{m+1}(ΩX_{k+1})∗E^{n+1}(ΩY_{h+1})}  _{\begin{matrix}P^{m+1}(ΩX_{k+1})×P^{n}(ΩY_{h+1}) \\ ∪P^{m}(ΩX_{k+1})×P^{n+1}(ΩY_{h+1}) \end{matrix}}  _{P^{m+1}(ΩX_{k+1})×P^{n+1}(ΩY_{h+1}),}_{u_{H_{m+1}^{σ_{k}(X)}(f_{k}(X))∗H_{n+1}^{σ_{h}(Y )}(f_{h}(Y))}}^{[f_{k}(X),f_{h}(Y)]^{r}}  w  _{u_{(σ_{k+1}(X)×σ_{h+1}(Y))|}} y  w u z u_{E^{m+1}(Ωi_{k,k+1})∗E^{n+1}(Ωi_{h,h+1})}_{^{[p_{m}^{ΩX_{k}} ,p_{n}^{ΩY_{h}}]^{r}} w}  z u_{w^{[p_{m}^{ΩX_{k+1}},p_{n}^{ΩY_{h+1}}]^{r}}}  y w

The details are left to the reader. QED.

6 Higher Hopf invariants for some examples

In this section, we compute the higher Hopf invariants for well-known examples, which yields a generalisation of the main result of [11]: We denote by ℂ ≈ ℝ^{2} the field of complex numbers, by ℍ ≈ ℝ^{4} the algebra of quaternion numbers and by 𝕆 ≈ ℝ^{8} the Cayley algebra:

Example 6.1 We know that cat ℂP^{m} = m and dim ℂP^{m} = 2m. Hence dim ℂP^{m} = 2m ≤ 2m + 2 − 2. Thus ℂP^{m} has a unique structure for cat ℂP^{m} = m. The higher Hopf invariant H_{m} : [V, ℂP^{m}] → [V, ΩℂP^{m}∗⋯∗ΩℂP^{m}] gives a (unique) homomorphism

H_{m} : π_{2m+1}(ℂP^{m}) → π_{2m+1}(ΩℂP^{m}∗⋯∗ΩℂP^{m}) ≅ ℤ

with the canonical projection p_{m}^{S^{1}} : S^{2m+1} → ℂP^{m} a ‘higher Hopf invariant one’ element.

Example 6.2 We know that cat ℍP^{m} = m and dim ℍP^{m} = 4m. Hence dim ℍP^{m} = 4m ≤ 4m + 4 − 2. Thus ℍP^{m} has a unique structure for cat ℍP^{m} = m. The higher Hopf invariant H_{m} : [V, ℍP^{m}] → [V, ΩℍP^{m}∗⋯∗ΩℍP^{m}] gives a (unique) homomorphism

H_{m} : π_{4m+3}(ℍP^{m}) → π_{4m+3}(ΩℍP^{m}∗⋯∗ΩℍP^{m}) ≅ ℤ

with the canonical projection p_{m}^{S^{3}} : S^{4m+3} → ℍP^{m} a ‘higher Hopf invariant one’ element.

Example 6.3 We know that cat 𝕆P^{2} = 2 and dim 𝕆P^{2} = 16. Hence dim 𝕆P^{2} = 16 ≤ 8 × 2 + 8 − 2. Thus 𝕆P^{2} has a unique structure for cat 𝕆P^{2} = 2. The higher Hopf invariant H_{2} : [V, 𝕆P^{2}] → [V, Ω𝕆P^{2}∗Ω𝕆P^{2}∗Ω𝕆P^{2}] gives a (unique) homomorphism

H_{2} : π_{23}(𝕆P^{2}) → π_{23}(Ω𝕆P^{2}∗Ω𝕆P^{2}∗Ω𝕆P^{2}) ≅ ℤ.

But there are no elements of ‘higher Hopf invariant one’: The existence of such a higher Hopf invariant one element implies that the Hopf space S^{7} is homotopy associative. As is well-known, the p-local Hopf space S_{(p)}^{7} is homotopy associative for p ≥ 5 (in view of [24], [22] and [13], it actually is an A_{p−1}-space). However, by using primary cohomology operations, one can easily see that any Hopf structure of S_{(3)}^{7} is not homotopy associative. Hopf space theorists were, however, much more interested in the case p = 2. And it was known by Goncalves [7], using higher order cohomology operations, and by Hubbuck [9], using K-theory Adams operations, that any Hopf structure of S_{(2)}^{7} is not homotopy associative (but the result itself had already been known by James). Hence, the image of the higher Hopf invariant homomorphism is in 6ℤ ⊂ ℤ.

Example 6.4 For m, n ≥ 1 and p ≥ m + 2, let f_{m,p} = p_{m}^{S^{1}}^{∘}g_{m,p} : S^{2mp+2(p−1)−1} → ℂP^{m}, where g_{m,p} : S^{2mp+2(p−1)−1} → S^{2m+1} denotes the generator of π_{2mp+2(p−1)−1}(S^{2m+1}) ≅ ℤ/pℤ and p_{m}^{S^{1}} : S^{2m+1} → ℂP^{m} denotes the projection which gives a ‘higher Hopf invariant one’ element as in Example 6.1. For the dimensional reasons, the map g_{m,p} is a co-H-map, and hence H_{m}(f_{m,p}) = g_{m,p} ≠ 0, and hence Σ^{n}H_{m}(f_{m,p}) is trivial if and only if n ≥ 2, by Theorem 13.4 in Toda [25].

Let Q_{m,p} be the mapping cone of f_{m,p}: Q_{m,p} = ℂP^{m}∪_{f_{m,p}} e^{2mp+2(p−1)}. Then, by Theorems 3.8 and 3.9 and Remark 3.10, it follows that cat Q_{m,p} = m + 1 and m + 1 ≤ cat(Q_{m,p}×S^{1}) ≤ m + 2 but cat(Q_{m,p}×S^{n}) = m + 1 for n ≥ 2 by Theorem 5.3.

Remark 6.5 (1) Every example in this section supports Conjecture 2.3.

(2) The space Q_{m,p} in Example 6.4 is a generalisation of Q_{p} in [11] except Q_{2}. Actually, Ganea’s conjecture for Q_{m,p} is true if we consider cat_{q} , for q ≠ p, instead of cat or cat_{p} .

7 LS category of sphere bundles over spheres

Let r ≥ 1, t ≥ 1 and E be a fibre bundle over S^{t+1} with fibre S^{r}. Then E has a CW decomposition S^{r} ∪_{α} e^{t+1} ∪_{ψ} e^{t+r+1} with α : S^{t} → S^{r} and ψ : S^{t+r} → Q, Q = S^{r} ∪_{α} e^{t+1}. We identify H_{1}^{S}(α) with its unique element, say H_{1}(α), since a sphere S^{k} has the unique structure map for cat S^{k} = 1.

Fact 7.1 Let α = ±1_{S^{r}} the identity. Then clearly cat Q = 0 and cat E = 1. In addition, cat Q×S^{n} = 1 and cat E×S^{n} = 2 for n ≥ 1.

The following fact is an immediate consequence of Berstein-Hilton [2] and a cup length consideration.

Fact 7.2 Let α ≠ ±1_{S^{r}}. Hence 1 ≤ cat Q ≤ 2. Then cat Q = 2 if and only if H_{1}(α) ≠ 0. In particular if H_{1}(α) = 0, we can easily obtain that cat Q = 1 and cat E = 2. In this case, it also follows that cat Q×S^{n} = 2 and cat E×S^{n} = 3 for n ≥ 1.

By Theorem 3.9 and Remark 3.10, we can extend the main result of [11].

Theorem 7.3 Let H_{1}(α) ≠ 0. Hence cat Q = 2. Then for n ≥ 1, cat Q×S^{n} = 3 if Σ^{n}H_{1}(α) ≠ 0 with n ≥ t − 2r + 2 or Σ^{n+1}H_{1}(α) ≠ 0.

We give a partial answer to Ganea’s Problem 4 (see [6]) for the total space of sphere-bundles-over-spheres. To show this, we need the following lemma.

Lemma 7.4 The collapsing map q : E → E/Q = S^{t+r+1} induces a map with trivial kernel

(Σ^{n}q)^{∗} : [S^{n+r+t+1}, S^{n+kr}] → [Σ^{n}E, S^{n+kr}] for all k ≥ 3 and n ≥ 0.

Proof. The cofibration sequences S^{r+t} →^{ψ} Q →^{j} E and S^{t} →^{α} S^{r} →^{i} Q together with the bundle projection p : E → S^{t+1} induce the following commutative diagram:

[S^{n+t+2}, S^{n+kr}]

[Σ^{n+1}E, S^{n+kr}] [Σ^{n+1}Q, S^{n+kr}] [S^{n+r+t+1}, S^{n+kr}] [Σ^{n}E, S^{n+kr}]

[S^{n+r+1}, S^{n+kr}],

A A A A A A A A D

Σ^{n+1}p^{∗}

u_{(Σ^{n+1}p|_{Σ^{n+1}Q})^{∗}}

w_{Σ^{n+1}j^{∗}} w_{Σ^{n+1}ψ^{∗}} u_{Σ^{n+1}i^{∗}} w_{(Σ^{n}q)^{∗}}

where the column and row are exact sequences. Since k ≥ 3, we have n + r + 1 < n + kr and π_{n+r+1}(S^{n+kr}) = 0. Hence (Σ^{n+1}p|_{Σ^{n+1}Q})^{∗} is surjective, and so is (Σ^{n+1}j)^{∗}. Thus (Σ^{n+1}ψ)^{∗} is trivial and the map (Σ^{n}q)^{∗} has trivial kernel for k ≥ 3 and n ≥ 0. QED.

Theorem 7.5 Let H_{1}(α) ≠ 0. Hence 2 ≤ cat E ≤ 3. Then cat E = 3 if Σ^{r+1}h_{2}(α) ≠ 0. Also cat E = 2 if H_{2}^{S}(ψ) or its subset H_{2}^{SS}(ψ) (see Remark 3.6 for its definition) contains 0.

Remark 7.6 In the latter case of cat E = 2, it is known that cat E×S^{n} = 3 for n ≥ 1 by using cup-length argument on the cohomology ring (see Singhof [20]).

Proof of Theorem 7.5. Let q′ : E → E/S^{r}, q′′ : Q → Q/S^{r} be respectively the collapsing maps. The reduced diagonal map Δ̂_{2} : E → E∧E factors as E →^{q′} S^{t+1}∪e^{r+t+1} →^{Δ̂_{2}} (S^{r} ∪ e^{t+1})∧(S^{r} ∪ e^{t+1}) ⊂ E∧E, which is an extension of the map Q →^{q} S^{t+1} →^{Σh_{2}(α)} S^{r}∧S^{r} ⊂ E∧E by Theorem 5.14 of Boardman-Steer [3]. In this case, the generator in H^{r+t+1}(E; ℤ) is a cup product of generators in H^{r}(E; ℤ) and H^{t+1}(E; ℤ). Then it follows that the mapping degrees of Δ̂_{2} on e^{r}∧e^{t+1} and e^{t+1}∧e^{r} are 1. Hence the reduced diagonal map Δ̂_{3} = (Δ̂_{2}∧1_{E})^{∘}Δ̂_{2} : E → E∧E∧E

factors as E →^{q} S^{r+t+1} →^{≃} S^{t+1}∧S^{r} →^{Σ^{r+1}h_{2}(α)} S^{r}∧S^{r}∧S^{r} ⊂ E∧E∧E. Hence Δ̂_{3} is the composition

of Σ^{r+1}h_{2}(α)^{∘}q with a suitable inclusion, which does not depend on the choice of σ(Q). By Lemma 7.4, q^{∗} : [S^{r+t+1}, S^{3r}] → [E, S^{3r}] has trivial kernel, and hence the non-triviality of Σ^{r+1}h_{2}(α) implies the non-triviality of Σ^{r+1}h_{2}(α)^{∘}q. Then, for dimensional reasons, it follows that Δ̂_{3} is also non-trivial. Then by Theorem 4.1, it follows that H_{2}^{S}(ψ) does not contain 0, and hence we see that cat E = 3 by Theorem 5.2. The latter part is clear by Theorem 5.2. QED.

We next study the LS category of E×S^{n}. To do this, we need the following lemma.

Lemma 7.7 The map q̂: E×S^{n} → S^{n+r+t+1} induces a map with trivial kernel

q̂^{∗} : [S^{n+r+t+1}, S^{n+kr}] → [E×S^{n}, S^{n+kr}] for all k ≥ 3 and n ≥ 1.

Proof. Let us recall that the space E ∨ S^{n} is a retractile subspace (see Zabrodsky [28]) of

both E×S^{n} and E ∪ Q×S^{n}. The cofibration sequences S^{n+r+t} →^{ψ′} E ∪ Q×S^{n} ↪^{j′} E×S^{n},

E ∨ S^{n} ↪^{i_{0}} E×S^{n} →^{q_{0}} E∧S^{n} and E ∨ S^{n} ↪^{i′_{0}} E ∪ Q×S^{n} →^{q′_{0}} Q∧S^{n} induce the following com-

mutative diagram:

[Σ(E∧S^{n}), S^{n+kr}]  [Σ(E×S^{n}), S^{n+kr}]  [ΣE∨ΣS^{n}, S^{n+kr}] [Σ(Q∧S^{n}), S^{n+kr}]  [Σ(E ∪ Q×S^{n}), S^{n+kr}]  [ΣE∨ΣS^{n}, S^{n+kr}] [S^{n+r+t+1}, S^{n+kr}]  [S^{n+r+t+1}, S^{n+kr}] [E×S^{n}, S^{n+kr}],_{_{(Σ^{n+1}j)^{∗}} u}  ^{(Σ^{n+1}q_{0})^{∗}}w  _{u_{(Σj′)^{∗}}}  ^{(Σi_{0})^{∗}} w_{_{(Σ^{n+1}ψ)^{∗}} u}  _{w^{(Σ^{n+1}q′_{0})^{∗}}}  _{u_{(Σψ′)^{∗}}}  _{^{(Σi′_{0})^{∗}} w}_{u_{q̂^{∗}}}

where the columns are exact sequences and the rows are split short exact sequences with natural splittings (see Zabrodsky [28]). By the proof of Lemma 7.4, (Σ^{n+1}j)^{∗} is surjective, and hence so is (Σj′)^{∗}. Thus (Σψ′)^{∗} is trivial, and hence the map q̂^{∗} has trivial kernel for k ≥ 3 and n ≥ 0. QED.

Theorem 7.8 Let Σ^{r+1}h_{2}(α) ≠ 0. Hence cat E = 3. Then for n ≥ 1, cat E×S^{n} = 4 if Σ^{n+r+1}h_{2}(α) ≠ 0. Also for n ≥ 1, cat E×S^{n} = 3 if H_{3}^{S}(ψ′), Σ_{∗}^{n}H_{2}^{S}(ψ) or Σ_{∗}^{n}H_{2}^{SS}(ψ) contains 0.

Proof. By the proof of Theorem 7.5, the n-fold suspension of the reduced diagonal map Δ̂_{3}∧1_{S^{n}} : E∧S^{n} → E∧E∧E∧S^{n} is the composition of Σ^{n}H̄_{2}^{σ}(ψ)^{∘}q̂ = Σ^{n+r+1}h_{2}(α)^{∘}q̂ with a suitable inclusion, which does not depend on the choice of σ = σ(E∪Q×S^{n}). By Lemma 7.7, q̂^{∗} : [S^{n+r+t+1}, S^{n+3r}] → [E×S^{n}, S^{n+3r}] has trivial kernel, and hence the non-triviality of Σ^{n+r+1}h_{2}(α) implies the non-triviality of Σ^{n+r+1}h_{2}(α)^{∘}q̂. Then, for dimensional reasons, it follows that Δ̂_{3}∧1_{S^{n}} is also non-trivial, and hence the four-fold reduced diagonal Δ̂′_{4} of E×S^{n} is non-trivial. Thus by Theorem 4.1, H_{3}^{S}(ψ′) does not contain 0, and hence we see that cat E×S^{n} = 4 by Theorem 5.2. The latter part is clear by Theorem 5.3. QED.

8 Manifold examples

The Hopf fibration σ_{4} : S^{7} → S^{4} is given as a principal Sp(1)-bundle. Taking orbits of the action of U(1) ⊂ Sp(1) on S^{7}, we obtain a fibre bundle ℂP^{3} → S^{4} with fibre Sp(1)/U(1) ≈ S^{2}; the structure group Sp(1) acts on the fibre S^{2} via a map, say μ_{0} : S^{3}×S^{2} → S^{2}. Here the CW decomposition of ℂP^{3} is known as ℂP^{3} = ℂP^{2}∪_{p_{2}^{S^{1}}}e^{6} = S^{2} ∪_{η} e^{4}∪_{p_{2}^{S^{1}}}e^{6}. Hence the attaching map p_{2}^{S^{1}} of the top cell of ℂP^{3} is given by the composition

S^{3}∗S^{1}  C(S^{3})∪S^{3}×S^{2}  ℂP^{2},^{[C(1_{S^{3}}),Σ1_{S^{1}}]^{r}}w  ^{χ̂_{4}}  w

where χ̂_{4} denotes the map defined by μ̂|_{S^{3}×S^{2}} = μ_{0} and μ̂|_{C(S^{3})} = χ_{4} the characteristic map of the top cell of ℂP^{2}. Definition 8.1 For any map β : S^{q} → S^{3}, we may assume that the suspension Σβ : S^{q+1} → S^{4} is a C^{∞}-map by suitably deforming it up to homotopy, since S^{q+1} and S^{4} are closed C^{∞}- manifolds. We define E(β) to be the total space of the Sp(1)-bundle E(β) → S^{q+1} induced by the C^{∞}-map Σβ from the Sp(1)-bundle ℂP^{3} → S^{4}. Hence E(β) is an orientable, closed C^{∞}-manifold with CW decomposition E(β) = S^{2} ∪_{η∘β} e^{q+1}∪_{ψ(β)}e^{q+3}.

For a map β : S^{q} → S^{3} and a suspension map γ : S^{q′} → S^{q} with 3 ≤ q ≤ q′, we denote β′ = β^{∘}γ, and then we have E(β′) = S^{2} ∪_{η∘β′} e^{q′+1}∪_{ψ(β′)}e^{q′+3}. By putting Q(β) = S^{2} ∪_{η∘β} e^{q+1} ⊂ E(β) and Q(β′) = S^{2} ∪_{η∘β′} e^{q′+1} ⊂ E(β′), we have the following commutative ladder of cofibre sequences:

S^{q′}  S_{2}  Q(β′)  S^{q′+1}  S^{3}, S^{q}  S^{2}  Q(β)  S^{q+1}  S^{3},^{η∘β′}  _{w}_{_{γ} u}  _{y  w}  _{w}_{_{γ̂_{0}} u}  _{_{Σγ} u}  _{w}_{^{η∘β} w} _{y  w}  _{w}  _{w}  (8.1)

where γ̂_{0} = γ̂|_{Q(β′)} and γ̂ : E(β′) → E(β) is the bundle map induced from Σγ : S^{q′+1} → S^{q+1}. To compare the higher Hopf invariant of ψ(β) with that of ψ(β′), we show the following proposition.

Proposition 8.2 With respect to the ‘standard’ structure maps (see Remark 3.6), γ̂_{0} is ‘primitive’ in the sense of Berstein and Hilton.

Remark 8.3 ℂP^{2} has a unique structure map σ_{2}(ℂP^{2}) for cat ℂP^{2} ≤ 2.

Proof of Proposition 8.2. By Proposition 3.3 and Remark 3.4, there are maps σ′_{2}(Q(β)) : Q(β) → P^{2}(ΩQ(β)) and σ′_{2}(Q(β′)) : Q(β′) → P^{2}(ΩQ(β′)). Since γ is a suspension map, the ‘naturality’ of Lemma 3.5 imply a homotopy (relative to S^{2}):

P^{2}(Ωγ̂_{0})^{∘}σ′_{2}(Q(β′)) ∼ σ′_{2}(Q(β))^{∘}γ̂_{0}.  (8.2)

The difference between e_{2}^{Q(β)}^{∘}σ′_{2}(Q(β)) and 1_{Q(β)} with respect to the co-action of S^{q+1} is given by a map δ : S^{q+1} → Q(β) which can be pulled back to a map δ_{0} : S^{q+1} → ΣΩQ(β). Also the difference between e_{2}^{Q(β′)}^{∘}σ′_{2}(Q(β′)) and 1_{Q(β′)} with respect to the co-action of S^{q′+1} is given by a map δ′ : S^{q′+1} → Q(β′) which can be pulled back to a map δ′_{0} : S^{q′+1} → ΣΩQ(β′). Let σ_{2}(Q(β)) = σ′_{2}(Q(β)) + ι_{1}^{ΩQ(β)}^{∘}δ_{0} and σ_{2}(Q(β′)) = σ′_{2}(Q(β′)) + ι_{1}^{ΩQ(β′)}^{∘}δ′_{0}, where the addition is induced from the co-actions of S^{q+1} = Q(β)/S^{2} on Q(β) and of S^{q′+1} = Q(β′)/S^{2} on Q(β′), respectively. Then σ_{2}(Q(β)) and σ_{2}(Q(β′)) are genuine compressions of 1_{Q(β)} and 1_{Q(β′)}.

Using the homotopy (8.2), we obtain a homotopy (relative to S^{2})

γ̂_{0}^{∘}e_{2}^{Q(β′)}^{∘}σ′_{2}(Q(β′)) ∼ e_{2}^{Q(β)}^{∘}P^{2}(Ωγ̂_{0})^{∘}σ′_{2}(Q(β′)) ∼ e_{2}^{Q(β)}^{∘}σ′_{2}(Q(β))^{∘}γ̂_{0},

and hence a homotopy (relative to S^{2})

e_{2}^{Q(β)}^{∘}σ′_{2}(Q(β))^{∘}γ̂_{0} + δ^{∘}Σγ ∼ (e_{2}^{Q(β)}^{∘}σ′_{2}(Q(β)) + δ)^{∘}γ̂_{0} ∼ γ̂_{0} ∼ γ̂_{0}^{∘}e_{2}^{Q(β′)}^{∘}σ_{2}(Q(β′))
∼ γ̂_{0}^{∘}e_{2}^{Q(β′)}^{∘}σ′_{2}(Q(β′)) + γ̂_{0}^{∘}δ′ ∼ e_{2}^{Q(β)}^{∘}σ′_{2}(Q(β))^{∘}γ̂_{0} + γ̂_{0}^{∘}δ′.

Thus δ^{∘}Σγ and γ̂_{0}^{∘}δ′ are homotopic in π_{q′+1}(Q(β′)). Hence the difference of δ_{0}^{∘}Σγ and ΣΩγ̂_{0}^{∘}δ′_{0} is given by a map ϵ_{0} in the homotopy group π_{q′+1}(ΣΩQ(β)) which vanishes in π_{q′+1}(Q(β′)). Thus ϵ_{0} can be lifted to a map ϵ̂_{0} ∈ π_{q′+1}(E^{2}(ΩQ(β))) by the arguments given in Remark 3.6. This implies that ι_{1}^{ΩQ(β)}^{∘}ϵ_{0} is trivial in π_{q′+1}(P^{2}(ΩQ(β))). This yields the following homotopy relative to S^{2}:

P^{2}(Ωγ̂_{0})^{∘}σ_{2}(Q(β′)) ∼ P^{2}(Ωγ̂_{0})^{∘}σ′_{2}(Q(β′)) + P^{2}(Ωγ̂_{0})^{∘}ι_{1}^{ΩQ(β′)}^{∘}δ′_{0}
∼ P^{2}(Ωγ̂′_{0})^{∘}σ′_{2}(Q(β′)) + ι_{1}^{ΩQ(β)}^{∘}δ_{0}^{∘}Σγ + ι_{1}^{ΩQ(β)}^{∘}ϵ_{0}
∼ σ′_{2}(Q(β))^{∘}γ̂_{0} + δ^{∘}Σγ ∼ σ_{2}(Q(β))^{∘}γ̂_{0}.

This completes the proof of Proposition 8.2. QED.

The attaching map ψ(β) of the top cell of E(β) is given by the composition

S^{q}∗S^{1}  C(S^{q})∪S^{q}×S^{2}  Q(β),^{[C(1_{S^{q}}),Σ1_{S^{1}}]^{r}}w  ^{χ̂_{β}}  w

where χ̂_{β} denotes the map defined by μ̂_{β}|_{S^{q}×S^{2}} = μ_{0}^{∘}(β×1_{S^{2}}) and μ̂_{β}|_{C(S^{q})} = χ_{q+1} the characteristic map of the top cell of Q(β). Then a direct calculation shows that the following diagram is strictly commutative:

S^{q′}∗S^{1}  C(S^{q′})∪S^{q′}×S^{2}  Q(β′) S^{q}∗S^{1}  C(S^{q})∪S^{q}×S^{2}  Q(β)_{w}^{[C(1_{S^{q′}}),Σ1_{S^{1}}]^{r}}_{u_{γ∗1_{S^{1}}}}  ^{χ̂_{β′}}  _{w}_{u_{γ̄_{0}}}  _{u_{γ̂_{0}}}_{^{[C(1_{S^{q}}),Σ1_{S^{1}}]^{r}} w}  _{^{χ̂_{β}}  w}  (8.3)

where γ̄_{0} is given by γ̄_{0}|_{C(S^{q′})} = C(γ) and γ̄_{0}|_{S^{q′}×S^{2}} = γ×1_{S^{2}}. Thus we have that γ̂_{0}^{∘}ψ(β^{∘}γ) ∼ ψ(β)^{∘}(γ∗1_{S^{1}}). By Proposition 2.11 (2) and Proposition 8.2, we have the following theorem.

Theorem 8.4 For a map β : S^{q} → S^{3} and a suspension map γ : S^{q′} → S^{q} with 3 ≤ q ≤ q′, we have that E^{3}(Ωγ̂_{0})_{∗}H_{2}^{SS}(ψ(β^{∘}γ)) = (γ∗1_{S^{1}})^{∗}H_{2}^{SS}(ψ(β)) = ±(Σ^{2}γ)^{∗}H_{2}^{SS}(ψ(β))

Corollary 8.4.1 (1) If β^{∘}γ = 0, in π_{q′+1}(S^{3}), then (Σ^{2}γ)^{∗}H_{2}^{SS}(ψ(β)) = {0}.

(2) If β : S^{q} → S^{3} is of finite order ℓ with q ≥ 3, then each element of H_{2}^{SS}(ψ(β)) is also of finite order which divides ℓ.

Fact 7.2, Theorems 7.3, 7.5 and 7.8 imply the following lemma.

Lemma 8.5 Let p be an odd prime, β be the co-H-map α_{1}(3) : S^{2p} → S^{3} and γ be the suspension map α_{2}(2p) = Σ^{2p−3}α_{2}(3) : S^{6p−5} → S^{2p} for the prime p. Then Σ_{∗}H_{2}^{S}(ψ(α_{1}(3)^{∘}α_{2}(2p))) contains ±α_{1}(6)^{∘}α_{2}(2p + 3).

Remark 8.6 The composition α_{1}(5)^{∘}α_{2}(2p + 2) is trivial for all odd primes p except the prime 3. At the prime 3, α_{1}(5)^{∘}α_{2}(8) = −3β_{1}(5) ≠ 0 and α_{1}(7)^{∘}α_{2}(10) = 0 by Lemma 13.8 and Theorem 13.9 in [25].

Proof. Firstly we summarise here some well-known results on odd primary components of stable and unstable homotopy groups of spheres.

By Theorem 6.2 of Oka [17], we know the following fact:

Fact 8.7 Let t = (p − 1) where p is an odd prime. For k < 2(p + 3)t − 4, the p-component of stable homotopy group of sphere spectrum _{p}π_{k}^{S} is trivial unless k = 2rt − 1 (1 ≤ r ≤ p + 2), 2pt − 2 and 2(p + 1)t − 3. In addition, all the non-trivial groups are given as follows:

pπ_{2rt−1}^{S} ≅ ℤ/pℤ generated by α_{r}, r ≠ p,
pπ_{2pt−1}^{S} ≅ ℤ/p^{2}ℤ generated by α′_{p},
pπ_{2pt−2}^{S} ≅ ℤ/pℤ generated by β_{1},
pπ_{2(p+1)t−3}^{S} ≅ ℤ/pℤ generated by α_{1}^{∘}β_{1},

where α_{r} is defined inductively using Toda brackets: α_{r} = ⟨α_{r−1}, pι, α_{1}⟩. Note that α_{p} = pα′_{p}. At p = 3, we remark that β_{1} is given by a Toda bracket ⟨α_{1}, α_{1}, α_{1}⟩.

On the other hand by (13.5) in [25], we know the following fact:

Fact 8.8 π_{2m−1+k}(S^{2m−1})_{(p)} ≅ _{p}π_{k}^{S} if \frac{k+3}{2t} ≤ m.

Since \frac{2pt−2+3}{2t} < p+1 and 2(p+1)−1 = 2t+3, we have π_{2(p+1)t+1}(S^{2t+3})_{(p)} ≅ _{p}π_{2pt−2}^{S} ≅ ℤ/pℤ ∋ β_{1}. Hence there is a generator β_{1}(2t + 3) of π_{2(p+1)t+1}(S^{2t+3})_{(p)} ≅ ℤ/pℤ corresponding to β_{1}.

In π_{2(p+2)t+3}(S^{6}), we know the following fact:

Fact 8.9 α_{2}(6)^{∘}β_{1}(4t + 5) = 0 in π_{2(p+2)t+3}(S^{6})_{(p)}, which is obtained by a similar argument given in Page 184 of [25] using (13.8) and Propositions 1.4 and 1.3 in [25]:

α_{2}(6)^{∘}β_{1}(4t + 5) ∈ 2{α_{1}(6), α_{1}(2t + 5), pι_{4t+4}}_{1}^{∘}β_{1}(4t + 5)
= 2α_{1}(6)^{∘}Σ{α_{1}(2t + 4), pι_{4t+3}, β_{1}(4t + 3)}
⊂ 2α_{1}(6)^{∘}{α_{1}(2t + 5), pι_{4t+4}, β_{1}(4t + 4)}_{1},

where {α_{1}(2t + 5), pι_{4t+4}, β_{1}(4t + 4)}_{1} is a subset of π_{2(p+2)t+3}(S^{2t+5})_{(p)} ≅ _{p}π_{2(p+1)t−2}^{S} = 0, since \frac{2(p+1)t−2+3}{2t} < p + 2 and 2(p + 2) − 1 = 2t + 5.

In π_{6t+1}(S^{3}), we also know the following fact:

Fact 8.10 α_{1}(3)^{∘}α_{2}(2t + 2) = 2α_{2}(3)^{∘}α_{1}(4t + 2), which is obtained by a similar argument given in Page 184 of [25] using (3.9) (i) and Propositions 1.4 and 13.6 in [25]:

α_{2}(3)^{∘}α_{1}(4t + 2) ∈ {α_{1}(3), Σ(pι_{2t+1}), Σα_{1}(2t + 1)}_{1}^{∘}Σ^{2}α_{1}(4t)
= α_{1}(3)^{∘}Σ{pι_{2t+1}, α_{1}(2t + 1), α_{1}(4t)},

where the unstable Toda bracket {pι_{2t+1}, α_{1}(2t+1), α_{1}(4t)} ⊂ π_{6t}(S^{2t+1}; p) ≅ _{p}π_{4t−1}^{S} ≅ ℤ/pℤ{α_{2}} corresponds to the stable Toda bracket ⟨pι, α_{1}, α_{1}⟩ = \frac{1}{2}α_{2}, since \frac{4t−1+3}{2t} < 3 ≤ t+1 and 2(t+1)− 1 = 2t+1. Thus \{pι_{2t+1}, α_{1}(2t+1), α_{1}(4t)\} determines \frac{1}{2}α_{2}(2t+1), and hence α_{1}(3)^{∘}α_{2}(2t+2) = 2α_{2}(3)^{∘}α_{1}(4t + 2) in π_{6t+1}(S^{3}; p).

Next we apply these facts on higher Hopf invariants.

By Fact 8.9 we have α_{2}(2t + 2)^{∘}β_{1}(6t + 1) = 0, since t ≥ 2. Then by Theorem 8.4, it implies

\begin{matrix}(Σ^{2}β_{1}(6t + 1))^{∗}H_{2}^{SS}(ψ(α_{1}(3)^{∘}α_{2}(2t + 2))) \\ = ±E^{3}(Ω{ \atop β_{1}(6t + 1)}_{0})_{∗}H_{2}^{SS}(ψ(α_{1}(3)^{∘}α_{2}(2t + 2)^{∘}β_{1}(6t + 1))) = \{0\}. \end{matrix}  (8.4)

By Fact 8.10, we have

E^{3}(Ω{ \atop α_{1}(4t + 2)}_{0})_{∗}H_{2}^{SS}(ψ(α_{1}(3)^{∘}α_{2}(2t + 1)))

= E^{3}(Ω{ \atop α_{1}(4t + 2)}_{0})_{∗}H_{2}^{SS}(ψ(2α_{2}(3)^{∘}α_{1}(4t + 2)))

= 2e(Σ^{2}α_{1}(4t + 2))^{∗}H_{2}^{SS}(ψ(α_{2}(3))) = 2eα_{1}(4t + 4)^{∗}H_{2}^{SS}(ψ(α_{2}(3))),

where e = ±1. Here we see that E^{3}(ΩQ(α_{2}(3))) ≃ ΩQ(α_{2}(3))∗ΩQ(α_{2}(3))∗ΩQ(α_{2}(3)) has the homotopy type of a wedge sum of spheres up to dimension 4t + 5(> 4t + 2). Since the suspension of Whitehead products are trivial, each element of ΣH_{2}^{SS}(ψ(α_{2}(3))) has the form of aα_{2}(6) + bΣι^{∘}α_{1}(2t + 6) for dimensional considerations by Corollary 8.4.1 (2), where a, b are integers and ι : S^{2t+5} ↪ E^{3}(ΩQ(α_{2}(3))) denotes an appropriate map. Since α_{1}(5)^{∘}α_{1}(2t + 4) = 0, we have α_{1}(2t + 6)^{∘}α_{1}(4t + 5) = 0, and hence each element of 2α_{1}(4t + 5)^{∗}Σ_{∗}H_{2}^{SS}(ψ(α_{2}(3))) has the form of 2aα_{2}(6)^{∘}α_{1}(4t + 5) = aα_{1}(6)^{∘}α_{2}(2t + 5). On the other hand, we know

2h_{2}(η_{2}^{∘}α_{2}(3))^{∘}α_{1}(4t + 2) = 2h_{2}(η_{2}^{∘}α_{2}(3)^{∘}α_{1}(4t + 2))
= h_{2}(η_{2}^{∘}α_{1}(3)^{∘}α_{2}(2t + 2)) = α_{1}(3)^{∘}α_{2}(2t + 2) = 2α_{2}(3)^{∘}α_{1}(4t + 2),

and hence Σ^{3}h_{2}(η_{2}^{∘}α_{2}(3))^{∘}α_{1}(4t + 2) = α_{2}(6)^{∘}α_{1}(4t + 5). By the proof of Theorem 7.5, Σ^{3}h_{2}(η_{2}^{∘}α_{2}(3))^{∘}α_{1}(4t + 2) corresponds to α_{1}(4t + 5)^{∗}Σ_{∗}H_{2}^{SS}(ψ(α_{2}(3))) up to sign. Thus we have a = ±1 and 2α_{1}(4t + 5)^{∗}^{∘}Σ_{∗}H_{2}^{SS}(ψ(α_{2}(3))) = {aα_{1}(6)^{∘}α_{2}(2t + 5)}, and hence we have

\begin{matrix}ΣE^{3}(Ω{ \atop α_{1}(4t + 2)}_{0})_{∗}^{∘}Σ_{∗}H_{2}^{SS}(ψ(α_{1}(3)^{∘}α_{1}(2t + 1))) \\ = 2eα_{1}(4t + 5)^{∗}^{∘}Σ_{∗}H_{2}^{SS}(ψ(α_{2}(3))) = \{e′α_{1}(6)^{∘}α_{2}(2t + 5)\}, e′ = ±1. \end{matrix}  (8.5)

By (13.10) in [25], we know that h_{p}(α_{2}(3)) = xα_{1}(2t + 3) for some x ≠ 0 ∈ ℤ/pℤ and there are no other non-trivial James Hopf invariants h_{j}(α_{2}(3)), 1 < j ≠ p for dimensional reasons.

Thus the attaching map of the 4t + 3 cell in ΣΩQ(α_{2}(3)) − ΣΩ(S^{2}) corresponding to that in Q(α_{2}(3)) = S^{2} ∪_{η_{2}∘α_{2}(3)} e^{4t+3} is given by

Σad(η^{∘}α_{2}(3)) = ΣΩη^{∘}Σad(α_{2}(3)) = ΣΩη^{∘}(Σad(α_{2}(3)) − α_{2}(3) + α_{2}(3))
= ΣΩη^{∘}(Σad(α_{2}(3)) − α_{2}(3)) + ΣΩη^{∘}α_{2}(3) = xΣΩη^{∘}α_{1}(2t + 3) + ΣΩη^{∘}α_{2}(3).

whose suspensions give the attaching maps of cells in

ΩQ(α_{2}(3))∗ΩQ(α_{2}(3))∗ΩQ(α_{2}(3)) − Ω(S^{2})∗Ω(S^{2})∗Ω(S^{2})

up to dimension 8t + 6(> 6t + 4). Hence by (8.5), there is an integer y such that each element of Σ_{∗}H_{2}^{SS}(ψ(α_{1}(3)^{∘}α_{2}(2t + 2))) has the form

e′α_{1}(6)^{∘}α_{2}(2t + 5) + yΣ^{2t+1}Σad(η^{∘}α_{2}(3))
= e′α_{1}(6)^{∘}α_{2}(2t + 5) + y{xΣ^{2t+2}Ωη^{∘}α_{1}(4t + 5) + Σ^{2t+2}Ωη^{∘}α_{2}(2t + 5)}.

If y is non-zero, then each element of β_{1}(6t + 4)^{∗}Σ_{∗}H_{2}^{SS}(ψ(α_{1}(3)^{∘}α_{2}(2t + 2))) has the form

e′α_{1}(6)^{∘}α_{2}(2t + 5)^{∘}β_{1}(6t + 4) + y{xΣ^{2t+2}Ωη^{∘}α_{1}(4t + 5)^{∘} + Σ^{2t+2}Ωη^{∘}α_{2}(2t + 5)}^{∘}β_{1}(6t + 4) = yxΣ^{2t+2}Ωη^{∘}α_{1}(4t + 5)^{∘}β_{1}(6t + 4) ≠ 0

which contradicts to (8.4). Thus we obtain Σ_{∗}H_{2}^{SS}(ψ(α_{1}(3)^{∘}α_{2}(4t + 2))) = {e′α_{1}(6)^{∘}α_{2}(2t+5)}, e′ = ±1. QED.

Using this, we show the following theorem.

Theorem 8.11 There is a series of simply connected closed C^{∞}-manifolds N_{p} indexed by odd primes p ≥ 5 with cat N_{p} = cat (N_{p} − {P }), where P is a point in N_{p}.

Remark 8.12 The manifold N_{p} does not have the property in Theorem 8.11 if we consider cat_{q} , for any prime q ≠ p, instead of cat or cat_{p} .

Proof of Theorem 8.11. We fix the prime p ≥ 5 and let L_{p} = E(α_{1}(3)^{∘}α_{2}(2p)) for the prime p (see Theorem 13.4 in [25]). Then L_{p} is a C^{∞}-manifold with a CW decomposition S^{2} ∪_{α} e^{6p−4}∪_{ψ(β)}e^{6p−2}, where α = η^{∘}β and β = α_{1}(3)^{∘}α_{2}(2p). As is well-known (see [25] or [11]), α_{1}(3) is a co-H-map and α_{2}(2p) is a suspension map, and hence we have h_{2}(η^{∘}α_{1}(3)^{∘}α_{2}(2p)) = α_{1}(3)^{∘}α_{2}(2p). Also by Proposition 13.6 and (13.7) in [25] and by the fact that S^{3} is an H-space, we know that

α_{1}(3)^{∘}α_{2}(2p) ≠ 0, Σ^{1}(α_{1}(3)^{∘}α_{2}(2p)) ≠ 0 but Σ^{2}(α_{1}(3)^{∘}α_{2}(2p)) ∈ π_{6p−3}(S^{5})_{(p)} = 0, which implies that h_{2}(ηα_{1}(3)^{∘}α_{2}(2p)) ≠ 0 and Σ_{∗}H_{2}^{S}(ψ(α_{1}(3)^{∘}α_{2}(2p))) ∋ 0 by Lemma 8.5. Hence by Fact 7.2 and Theorem 7.5, we have cat (S^{2}∪_{α}e^{6p−4}) = 2, cat (S^{2}∪_{α}e^{6p−4})∪_{ψ(β)}e^{6p−2} = 2, 2 ≤ cat L_{p} ≤ 3. If cat L_{p} = 2, then we put N_{p} = L_{p} which satisfies cat (N_{p} − {P }) = cat (S^{2}∪_{α}e^{6p−4})∪_{ψ(β)}e^{6p−2} = 2 = cat L_{p}. Otherwise we have cat L_{p} = 3 and then by Theorem 7.8, we also have cat L_{p}×S^{n} = cat L_{p} = 3, n ≥ 1, and then we put N_{p} = L_{p}×S^{2} which satisfies cat (N_{p} − {P }) = cat(L_{p}×{∗}∪(S^{2}∪_{α}e^{6p−4})×S^{2}) = 3 = cat N_{p}. Thus, in each case, N_{p} satisfies the required property. QED.

Theorem 8.13 There is a simply connected closed manifold M with cat M = 3, cat M×S^{n} = 3 for any n ≥ 2 while we know only 3 ≤ cat M×S^{1} ≤ 4 for n = 1.

Corollary 8.13.1 There is a connected orientable closed manifold N with cat N = cat N×S^{n} for any n ≥ 1.

Remark 8.14 Ganea’s conjecture for the manifold M is true if we consider cat_{p} , for any prime p ≠ 3, instead of cat or cat_{3} .

Proof of Theorem 8.13. Let M = E(α_{1}(3)^{∘}α_{2}(6)) for the prime p = 3 (see [25]). Then M is a C^{∞}-manifold with a CW decomposition S^{2} ∪_{α} e^{14}∪_{ψ(β)}e^{16}, where α = η^{∘}β and β = α_{1}(3)^{∘}α_{2}(6). Also by Theorem 13.4 in [25] and by the fact that S^{5} is an H-space at p = 3, we know that

α_{1}(3)^{∘}α_{2}(6) ≠ 0, Σ^{3}(α_{1}(3)^{∘}α_{2}(6)) ≠ 0 ∈ π_{16}(S^{6})_{(3)} but Σ^{4}(α_{1}(3)^{∘}α_{2}(6)) ∈ π_{17}(S^{7})_{(3)} = 0,

which implies that Σ_{∗}^{3}h_{2}(ψ(ηα_{1}(3)^{∘}α_{2}(2p))) ≠ 0 and Σ_{∗}^{2}H_{2}^{S}(ψ(α_{1}(3)^{∘}α_{2}(2p))) ∋ 0 by Lemma 8.5. Hence by Theorems 7.5 and 7.8, we have that cat M = 3, 3 ≤ cat M×S^{1} ≤ 4 and cat M×S^{n} = 3 for n ≥ 2. QED.

Proof of Corollary 8.13.1. Let M = E(α_{1}(3)^{∘}α_{2}(6)) for the prime p = 3 as in the proof of Theorem 8.13. Then M is a C^{∞}-manifold with cat M = 3, 3 ≤ cat M×S^{1} ≤ 4 and cat M×S^{n} = 3 for n ≥ 2. If cat M×S^{1} = 3, we put N = M which satisfies the required property. Otherwise, we put N = M×S^{1} with ψ′(α_{1}(3)^{∘}α_{2}(6)) as the attaching map of the top cell, where we know the set Σ_{∗}H_{3}^{S}(ψ′(α_{1}(3)^{∘}α_{2}(6))) include the set Σ_{∗}^{2}H_{2}^{S}(ψ(α_{1}(3)^{∘}α_{2}(6))) which contains Σ^{4}α_{1}(3)^{∘}α_{2}(6) = 0. Then Theorem 5.3 implies that N satisfies the required properties. QED.

References

[1] I. Berstein and E. Dror, On the homotopy type of non-simply connected co-H-space, Ill. Jour. Math. 20 (1976), 528–534.

[2] I. Berstein and P. J. Hilton, Category and generalised Hopf invariants, Illinois. J. Math. 12 (1968), 421–432.

[3] J. M. Boardman and B. Steer, On Hopf invariants, Comment. Math. Helv. 42 (1967), 180–221.

[4] C. R. Curjel, A note on spaces of category ≤ 2, Math. Z. 80 (1963), 293–299.

[5] T. Ganea, Cogroups and suspensions, Invent. Math. 9 (1970), 185–197.

[6] T. Ganea, Some problems on numerical homotopy invariants, Symposium on Algebraic Topology, Lect. Notes in Math. 249, Springer Verlag, Berlin (1971) 13–22.

[7] D. Goncalves, Mod 2 homotopy associative H-spaces, Geometric Applications of Homotopy Theory I, (Proc. Conf., Evanston Ill. 1977), Lect. Notes in Math. 657, Springer Verlag, Berlin (1978) 198–216.

[8] K. P. Hess, A proof of Ganea’s Conjecture for Rational Spaces, Topology 30 (1991), 205–214.

[9] J. R. Hubbuck, Two examples on finite H-spaces, Geometric Applications of Homotopy Theory I, (Proc. Conf., Evanston Ill. 1977), Lect. Notes in Math. 657, Springer Verlag, Berlin (1978) 282–291.

[10] N. Iwase, Projective spaces associated with an A_{∞}-structure and Ganea’s conjecture on LS category, GSM Kyushu U. preprint series 1997-17 (1997).

[11] N. Iwase, Ganea’s conjecture on Lusternik-Schnirelmann category, Bull. Lon. Math. Soc., 30 (1998), 623–634.

[12] N. Iwase, A_{∞}-method in Lusternik-Schnirelmann category, GSM Kyushu U. preprint series 1998-13 (1998).

[13] N. Iwase and M. Mimura, Higher homotopy associativity, Algebraic Topology, (Arcata CA 1986), Lect. Notes in Math. 1370, Springer Verlag, Berlin (1989) 193–220.

[14] I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331-348.

[15] I. M. James, Lusternik-Schnirelmann Category, “Handbook of algebraic topology”, 1293-1310, North Holland, Amsterdam, 1995.

[16] B. Jessup, Rational L-S category and a conjecture of Ganea, Trans. Amer. Math. Soc. 317 (1990), 655-660.

[17] S. Oka, The stable homotopy Groups of Spheres I, Hiroshima Math. J. 1 (1971), 305–337.

[18] Y. B. Rudyak, On the Ganea conjecture for manifolds, Proc. Amer. Math. Soc. 125 (1997), 2511–2512.

[19] Y. B. Rudyak, On category weight and its applications, Topology 38 (1999), 37–55.

[20] W. Singhof, Minimal coverings of manifolds with balls, Manuscripta Math. 29 (1979), 385–415.

[21] D. Stanley, Spaces with Lusternik-Schnirelmann category n and cone length n + 1, preprint.

[22] J. D. Stasheff, Homotopy associativity of H-spaces, I, II, Trans. Amer. Math. Soc. 108 (1963), 275–292, 293–312.

[23] M. Sugawara, On a condition that a space is an H-space, Math. J. Okayama Univ. 5 (1956/57), 109–129.

[24] M. Sugawara, A condition that a space is group-like, Math. J. Okayama Univ. 7 (1957), 123–149.

[25] H. Toda, “Composition methods in Homotopy groups of spheres”, Princeton Univ. Press, Princeton N.Y., Ann. of math. studies 49 (1962).

[26] L. Vandembroucq, Adjunction spaces satisfying the Ganea conjecture, preprint.

[27] G. W. Whitehead, “Elements of Homotopy Theory”, Springer Verlag, Berlin, GTM series 61 (1978).

[28] A. Zabrodsky, “Hopf spaces”, Notas de Matemática (59), North-Holland Publ. Co., Amsterdam, North-Holland Mathematics Studies, 22 (1976).