HIGHER HOMOTOPY ASSOCIATIVITY

AND THE MOD p KERVAIRE INVARIANT

Yutaka Hemmi

1. Introduction

It is well known that the existence of an element of Hopf invariant one in π_{n−1}^{S}, the stable homotopy group of the sphere of k-stem, is equivalent to the existence of an H-structure on S^{n−1}. This fact is essentially about the mod 2 structure. In fact, any sphere is an H-space if and only if it is a mod 2 H-space. It is easy to imagine that there exists a relation between the existence of an element of mod p Hopf invariant one in π_{2n(p−1)−1}^{S} and the existence of mod p A_{p}-structures on S^{2n−1}. In this paper, we first give a theorem which states an explicit relation between them, then expand it to the one between the existence of an element of mod p Kervaire invariant one and the existence of some kind of extended A_{p}-forms on spheres. Furthermore, as a related problem, we discuss the realization problem of a certain algebra over the mod p Steenrod algebra. We assume all spaces are localized at a prime p.

2. Mod p Hopf invariant

Stasheff [10] introduced the concept of A_{n}-spaces. In his paper, he showed that all odd spheres admit A_{p−1}-forms. One of the easiest way to show it is to consider the double suspension map E^{2} : S^{2n−1} → Ω^{2}S^{2n+1}. Since the homotopy fiber of E^{2} is 2np − 4 connected, we can construct an A_{p−1}-form {μ_{i}}_{i≤p−1} on S^{2n−1} by using the A_{p−1}-form {η_{i}}_{i≤p−1} on Ω^{2}S^{2n+1} defined from the group multiplication so that the following diagram is homotopy commutative:

S^{2n−1}  ⟶^{E^{2}}  Ω^{2}S^{2n+1}
↑^{μ_{i}}  ↑^{η_{i}}
K_{i} × (S^{2n−1})^{i} ⟶^{id×(E^{2})^{i}} K_{i} × (Ω^{2}S^{2n+1})^{i},

where K_{i} is the complex introduced by Stasheff [10]. From this construction we can also see that S^{2n−1} admits an A_{p−1}-form so that E^{2} is an A_{p−1}-map. While S^{2n−1} admits an A_{p}-form if and only if n divides p − 1 (cf. [11],[12]). However, it is not generally true that S^{2n−1} has an A_{p}-form so that E^{2} is an A_{p−1}-map even if n|p − 1. In fact, we can show that the existence of an element of mod p Hopf invariant one is equivalent to the existence of an A_{p}-form on S^{2n−1} of that kind. We show the following

Typeset by AMS-TEX

Theorem 2.1. There exists an A_{p}-form on S^{2n−1} so that E^{2} : S^{2n−1} → Ω^{2}S^{2n+1} is an A_{p−1}-map if and only if there exists an element of mod p Hopf invariant one in π_{2n(p−1)−1}^{S}.

To prove the above theorem, we use the following fact.

Lemma 2.2. For any A_{p−1}-form on S^{2n−1} so that E^{2} : S^{2n−1} → Ω^{2}S^{2n+1} is an A_{p−1}-map, the projective i-space P_{i}(S^{2n−1}) (i ≤ p − 1) of S^{2n−1} has the homotopy type of the i-th James construction J_{i}(S^{2n}).

Proof. Let E : S^{2n} = ΣS^{2n−1} → ΩS^{2n+1} be the suspension map. Then E^{∗} : H^{2n}(ΩS^{2n+1}; ℤ/p) → H^{2n}(S^{2n}; ℤ/p) is an isomorphism. Since E^{2} is an A_{p−1}-map, we have an extension f : P_{p−1}(S^{2n−1}) → ΩS^{2n+1} of E, which can be chosen to satisfy f(P_{i}(S^{2n−1})) ⊂ J_{i}(S^{2n}) for i ≤ p − 1 for dimensional reason. Now

H^{∗}(P_{i}(S^{2n−1}); ℤ/p) ≅ ℤ/p[x]/(x^{i+1}), deg x = 2n,
H^{∗}(J_{i}(S^{2n}); ℤ/p) ≅ Γ[y]/(γ_{j}(y) | j ≥ i + 1), deg y = 2n,

where Γ[y] is the divided polynomial algebra over y, and γ_{j}(y) is the generator of degree 2nj. Here ℤ/p[x]/(x^{i+1}) ≅ Γ[y]/(γ_{j}(y) | j ≥ i + 1) since i ≤ p − 1. Thus f^{∗} : H^{∗}(J_{i}(S^{2n}); ℤ/p) → H^{∗}(P_{i}(S^{2n−1}); ℤ/p) is an isomorphism, and P_{i}(S^{2n−1}) ≃ J_{i}(S^{2n}). □

Proof of Theorem 2.1. If p = 2, then the statement claims that S^{2n−1} is an H-space if and only if there exists an element of mod 2 Hopf invariant one in π_{4n−1}^{S}. This fact is well known. So we assume p ≥ 3.

Suppose that S^{2n−1} admits a required A_{p}-form. Then, by Lemma 2.2,

P_{p}(S^{2n−1}) = P_{p−1}(S^{2n−1}) ∪ e^{2np} ≃ J_{p−1}(S^{2n}) ∪ e^{2np},

and P^{n}(x) = x^{p} ≠ 0 for a generator x ∈ H^{2n}(P_{p}(S^{2n−1}); ℤ/p). This implies the existence of an element of mod p Hopf invariant one in π_{2n(p−1)−1}^{S}.

On the other hand, the obstruction of an A_{p−1}-form on S^{2n−1} so that E^{2} is an A_{p−1}-map to have an extended A_{p}-form is given as a map θ : Σ^{p−3}S^{(2n−1)p} = S^{2np−3} → S^{2n−1}. Then we have the following homotopy commutative diagram:

S^{2n−1}  ⟶^{E^{2}}  Ω^{2}S^{2n+1}
θ↑  ↑_{θ′}
Σ^{p−3}S^{(2n−1)p} ⟶ Σ^{p−3}(Ω^{2}S^{2n+1} ∧ ⋯ ∧ Ω^{2}S^{2n+1}).

Here θ′ is the obstruction of the A_{p−1}-form on Ω^{2}S^{2n+1} to have an extended A_{p}- form, which is null homotopic. Thus E^{2} ∘ θ ≃ ∗. If there exists an element of mod p Hopf invariant one in π_{2n(p−1)−1}^{S}, the double suspension π_{2np−3}(S^{2n−1}) → π_{2np−1}(S^{2n+1}) is an isomorphism. Thus θ ≃ ∗, and the A_{p−1}-form can be extended to an A_{p}-form. □

From Lemma 2.2 we can prove the following fact, which is proved by Serre [9] for k = 2.

Theorem 2.3. If 2 ≤ k ≤ p − 1, then

ΩJ_{k−1}(S^{2n}) ≃ S^{2n−1} × ΩS^{2nk−1}.

In particular, for k = 2, we have ΩS^{2n} ≃ S^{2n−1} × ΩS^{4n−1}.

Proof. Give an A_{p−1}-form on S^{2n−1} so that the double suspension map is an A_{p−1}- map. For any k ≤ p − 1, we have a homotopy fibration

S^{2n−1} ⟶^{j} S^{2n−1} ∗ k ∗ S^{2n−1} ⟶ P_{k−1}(S^{2n−1}),

where S^{2n−1} ∗ k ∗ S^{2n−1} is the k-fold join of S^{2n−1}, which is homeomorphic to S^{2nk−1}. Thus, by extending the above fibration to the left, we have a homotopy fibration

ΩS^{2nk−1} ⟶^{i} ΩP_{k−1}(S^{2n−1}) ⟶^{ρ} S^{2n−1}.

Since j ≃ ∗, we have a homotopy section s : S^{2n−1} → ΩP_{k−1}(S^{2n−1}) of the above fibration. Thus, we have a homotopy equivalence

μ ∘ (s × i) : S^{2n−1} × ΩS^{2nk−1} → ΩP_{k−1}(S^{2n−1}),

where μ is the loop multiplication. Since P_{k−1}(S^{2n−1}) ≃ J_{k−1}(S^{2n}) by Lemma 2.2, we have the result. □

We remark that Selick [7], [8] discusses decomposition of J_{k}(S^{2n}) for more general cases.

3. Mod p strong Kervaire invariant

In this section we study an analogy to Theorem 2.1. To state our result we need to generalize the concept of A_{n}-spaces.

Definition 3.1. Let f : X → Y be a map. Then an A_{n}-form on X corelative to f is an A_{n−1}-form on X, {M_{i} : K_{i} × X^{i} → X}_{i≤n−1}, together with a map M_{n} : K_{n} × X^{n} → Y satisfying

M_{n}(∂_{k}(r, s)(ρ, σ), x_{1},…, x_{n}) = fM_{r}(ρ, x_{1},…, M_{s}(σ, x_{k},…, x_{k+s−1}),…, x_{n})

for (ρ, σ) ∈ K_{r} × K_{s} and (x_{1},…, x_{n}) ∈ X^{n}, where ∂_{k}(r, s) : K_{r} × K_{s} → K_{r+s−1} is the face operator.

It is clear that the A_{n}-form on X corelative to id_{X} : X → X is just the A_{n}-form on X. Furthermore, if {M_{i}}_{i≤n} is an A_{n}-form on X, then for any map f : X → Y, {M_{i}}_{i≤n−1} ∪ {fM_{n}} is an A_{n}-form corelative to f.

Definition 3.2. Let f : X → Y be a map. Then for any A_{n}-form on X corelative to f, we define the projective i-spaces P_{i}(X; f) (1 ≤ i ≤ n) so that

P_{1}(X; f) = ΣY, P_{i}(X; f) ≃ P_{i−1}(X; f) ∪ C(X ∗ i ∗ X) (2 ≤ i ≤ n).

In fact, P_{i}(X; f) (2 ≤ i ≤ n) is defined by a relative homeomorphism

(P_{i}(X; f), P_{i−1}(X; f)) ≈ (K_{i+2} × X^{i}, ∂K_{i+2} × X^{i} ∪ K_{i+2} × X^{[i]}),

where X^{[i]} is the fat wedge. The detail is similar to the construction of the projective i-space of an A_{n}-space ([10]),

If X has an A_{n}-form corelative f : X → Y, then there is a natural map P_{i}(f) : P_{i}(X) → P_{i}(X; f) (1 ≤ i ≤ n − 1) so that P_{1}(f) = Σf, and (P_{i}(f), P_{i−1}(f)) is a relative homeomorphism for 2 ≤ i ≤ n − 1.

We note that the concept of the corelative A_{n}-space is one of the special case of the A_{n}-pairing defined in [5]. We can also define relative A_{n}-forms. In particular, if Y is an A_{n}-space with the A_{n}-form {M_{i}}_{i≤n}, then for any map f : X → Y, {M_{i}}_{i≤n−1} ∪ {M_{n}(1 × f × ⋯ × f)} is an A_{n}-form on Y relative f. An interesting case is that X = ΣΩY and f is the evaluation map. In this case, the projective n-space is called the modified projective space, which has an interesting structure ([6]).

Now we show an analogous fact to Theorem 2.1 about the mod p strong Kervaire invariant.

Theorem 3.3. Let m be a positive integer, and P^{k}(m) = S^{k−1} ∪_{m} e^{k} the Moore space with H_{k−1}(P^{k}(m); ℤ) ≅ ℤ/m. Let ι : S^{2n−1} → P^{2n}(p) be the inclusion of the bottom cell. Then, there exists an A_{p}-form on S^{2n−1} corelative to ι so that E^{2} : S^{2n−1} → Ω^{2}S^{2n+1} is an A_{p−1}-map if and only if there exists an element of mod p strong Kervaire invariant one in π_{2n(p−1)−2}^{S}.

Proof. Suppose that there exists an A_{p}-form on S^{2n−1} corelative to ι so that E^{2} : S^{2n−1} → Ω^{2}S^{2n+1} is an A_{p−1}-map. Then for the projective space P_{p−1}(S^{2n−1}; ι) we have

(∗)  ΣP_{p−1}(S^{2n−1}; ι) ≃ (S^{2n+1} ∪_{p} e^{2n+2}) ∨ S^{4n+1} ∨ ⋯ ∨ S^{2n(p−1)+1}.

In fact, (P_{i}(ι), P_{i−1}(ι)) : (P_{i}(S^{2n−1}), P_{i−1}(S^{2n−1})) → (P_{i}(S^{2n−1}; ι), P_{i−1}(S^{2n−1}; ι)) is a relative homeomorphism for 2 ≤ i ≤ p − 1, and

ΣP_{p−1}(S^{2n−1}) ≃ ΣJ_{p−1}(S^{2n}) ≃ S^{2n+1} ∨ S^{4n+1} ∨ ⋯ ∨ S^{2n(p−1)+1},

which implies (∗).

Now let

Q = ΣP_{p}(S^{2n−1}; ι)/S^{4n+1} ∨ ⋯ ∨ S^{2n(p−1)+1} = S^{2n+1} ∪_{p} e^{2n+2} ∪ e^{2np+1}.

Then P^{n} : H^{2n+1}(Q; ℤ/p) → H^{2np+1}(Q; ℤ/p) is an isomorphism. This shows the existence of an element of mod p strong Kervaire invariant one in π_{2n(p−1)−2}^{S}.

Next suppose that there exists an element of mod p strong Kervaire invariant one in π_{2n(p−1)−2}^{S}. If p = 2, then by the definition of the Whitehead product, there exists a map f : S^{4n−3} → S^{2n−1} so that the following diagram is homotopy commutative:

S^{4n−3} ⟶ S^{2n−1} ∨ S^{2n−1} ⟶ S^{2n−1} × S^{2n−1}
f↓  ↓^{∇}
S^{2n−1} ⟶_{×2}  S^{2n−1}  ⟶_{ι}  P^{2n}(2),

where horizontal sequences are cofiber sequences and ∇ is the folding map. Thus we have a map M_{2} : S^{2n−1} × S^{2n−1} → P^{2n}(2) so that ι ∘ ∇ ≃ M_{2}|S^{2n−1} ∨ S^{2n−1}, which is an A_{2}-form corelative ι.

Let p ≥ 3, and let L = ∂K_{p} × (S^{2n−1})^{p} ∪ K_{p} × (S^{2n−1})^{[p]}. Give an A_{p−1}-form on S^{2n−1} so that E^{2} is an A_{p−1}-map. Then the obstruction of the A_{p−1}-form to have an extended A_{p}-form is the composition

S^{2np−3} ⟶^{φ} L ⟶^{ψ} S^{2n−1},

where the mapping come of φ is homotopic to K_{p} × (S^{2n−1})^{p} and ψ is defined naturally by the A_{p−1}-form. Since there exists an element of mod p strong Kervaire invariant one in π_{2n(p−1)−2}^{S}, ψ ∘ φ is divided by p, and we have the following homotopy commutative diagram:

S^{2np−3} ⟶^{φ} L ⟶ K_{p} × (S^{2n−1})^{p}
↓  ↓^{ψ}
S^{2n−1} ⟶^{×p} S^{2n−1} ⟶^{ι}  P^{2n}(p)

Thus we have a map K_{p} × (S^{2n−1})^{p} → P^{2n}(p), which shows the existence of an A_{p}-form corelative to ι so that E^{2} is an A_{p−1}-map. □

4. Realization of algebras

Suppose S^{2n−1} is an A_{k}-space. Put P_{i} = P_{i}(S^{2n−1}). Then we have a sequence of spaces

P_{1} = ΣS^{2n−1} = S^{2n} ⊂ P_{2} ⊂ ⋯ ⊂ P_{k}, (4.1)

H^{∗}(P_{i}; ℤ/p) ≅ ℤ/p[x_{2n}]/(x_{2n}^{i+1}),

so that x_{2n} is compatible with the homomorphisms induced by the inclusions. So, we consider the converse problem, that is, we study conditions so that a sequence of spaces of (4.1) exists. Then, we have already known the following fact from section 2.

Theorem 4.2.

(1) If k ≤ p − 1, then (4.1) exists. In fact, we can put P_{i} = J_{i}(S^{2n}).

(2) (4.1) exists for k = p if and only if (4.1) exists for k = ∞. Furthermore, this is equivalent to the condition that n | p − 1.

(3) (4.1) with P_{p−1} = J_{p−1}(S^{2n}) exists for k = p if and only if there exists an elements of mod p Hopf invariant one in π_{2n(p−1)−1}^{S}.

Now we consider an analogy for Kervaire invariant. To do so, we need to review Gray construction ([4]) which is a generalization of James construction.

Let A be a subspace of X. Then we have a cofiber sequence

A ⊂ X → X ∪ CA → ΣX.

Then Gray construction J(X, A) is defined by

J(X, A) = X ∪ X × A ∪ X × A × A ∪ ⋯ / ∼,

where (a_{1},…, a_{k}) ∼ (a_{1},…, â_{i},…, a_{k}) if a_{i} = ∗ for a_{1} ∈ X and a_{j} ∈ A (2 ≤ j ≤ k). Gray proved that J(X, A) has the homotopy type of the homotopy fiber of the projection X ∪ CA → ΣX. By definition, we have

ΣJ(X, A) ≃ Σ(X ∨ (X ∧ A) ∨ (X ∧ A ∧ A) ∨ ⋯).

We note that if A = X, then J(X, A) = J(X).

We consider the case that X = P^{2n+1}(p) and A = S^{2n} to get

S^{2n+1}{p} ≃ J(P^{2n+1}(p), S^{2n}),

where S^{2n+1}{p} is the homotopy fiber of a self map on S^{2n+1} of degree p. Thus, in particular,

ΣS^{2n+1}{p} ≃ ΣJ(P^{2n+1}(p); S^{2n}) ≃ Σ(P^{2n+1}(p) ∨ P^{4n+1}(p) ∨ ⋯).

Now we consider the following sequence:

P_{1} = S^{2n} ⊂ P_{2} ⊂ ⋯ ⊂ P_{k} with (4.3)

H^{∗}(P_{i}; ℤ/p) ≅ ℤ/p[x_{2n}] ⊗ Λ(x_{2n+1})/(x_{2n}^{i+1}, x_{2n}^{i−1}x_{2n+1}), β(x_{2n}) = x_{2n+1},

where β is the Bockstein operation, and x_{2n} and x_{2n+1} are compatible with the homomorphisms induced by the inclusions. Then we prove the following fact.

Theorem 4.4.

(1) If k ≤ p − 1, then (4.3) exists. In fact, we can put P_{i} as the 2ni-skeleton of J_{i}(P^{2n+1}(p); S^{2n}).

(2) (4.3) so that P_{p−1} is the 2n(p − 1)-skeleton of J_{p−1}(P^{2n+1}(p); S^{2n}) exists for k = p if and only if there exists an elements of mod p Kervaire invariant one in π_{2n(p−1)−2}^{S}.

(1) and (2) in the above theorem correspond to (1) and (3) in Theorem 4.2, respectively. For the corresponding fact to (2) in Theorem 4.2, Cooke [3] shows that if n | p(p−1), then P_{∞} exists. He conjectured that the condition n | p(p−1) is indeed necessary for the existence of P_{∞}. Aguadé-Broto-Notbohm [1], [2] studied some related problems.

Proof of Theorem 4.4. (1) This fact is clear since H^{∗}(S^{2n+1}{p}; ℤ/p) ≅ Γ(x_{2n}) ⊗ Λ(x_{2n+1}) with β(x_{2n}) = x_{2n+1}, where dim x_{i} = i.

(2) If (4.3) with the desired property exists for k = p, then

\begin{aligned}ΣP_{p} & ≃ ΣJ_{p−1}(P^{2n+1}(p); S^{2n}) ∪ e^{2np+1} \\ & ≃ (P^{2n+2}(p) ∨ ⋯ ∨ P^{2n(p−2)+2}(p) ∨ S^{2n(p−1)+1}) ∪ e^{2np+1}. \end{aligned}

Thus, by collapsing a suitable subcomplex of ΣP_{p}, we have a space P^{2n+2}(p)∪e^{2np+1} so that P^{n} acts non-trivially on H^{2n+1}(P^{2n+2}(p) ∪ e^{2np+1}; ℤ/p), which shows the existence of an element of mod p strong Kervaire invariant one in π_{2n(p−1)−2}^{S}.

Next suppose there exists an element of mod p strong Kervaire invariant one in π_{2n(p−1)−2}^{S}. Then there exists a map g : S^{2np} → P^{2n+2}(p) so that P^{n} ≠ 0 on H^{2n+1}(P^{2n+2}(p) ∪_{g} e^{2np+1}; ℤ/p). Let f : S^{2np−1} → ΩΣS^{2n+1}{p} be the adjoint of the composition of g and the inclusion P^{2n+2(p)} ⊂ ΣS^{2n+1}{p}. Now S^{2n+1} is an H-space and a map of degree p on S^{2n+1} is an H-map. Thus, S^{2n+1}{p} is an H-space, and we have a map h : ΩΣS^{2n+1}{p} → S^{2n+1}{p} so that h ∘ i ≃ id, where i : S^{2n+1}{p} → ΩΣS^{2n+1}{p} is the adjoint of the identity map on ΣS^{2n+1}{p}. The composition i ∘ f : S^{2np−1} → S^{2n+1}{p} factors thorough J_{p−1}(P^{2n+1}(p); S^{2n}) for dimensional reason. Since Σ(i ∘ f) is homotopic to the composition S^{2np} → P^{2n+2}(p) ⊂ ΣS^{2n+1}{p}, the mapping cone of the map S^{2np−1} → J_{p−1}(P^{2n+1}(p); S^{2n}) is the desired space P_{p}. □

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Department of Mathematics, Faculty of Science, Kochi University, Kochi 780, Japan

  E-mail address: hemmi@math.kochi-u.ac.jp