A_{n}-SPACES AND PRIMITIVITY

YUTAKA HEMMI Dedicated to Professor Jun-iti Umehara on his 60th birthday

(Received December 6, 1996)

1. Introduction

Curjel conjectured that any finite Hopf spaces have Hopf structures so that the rational cohomologies are primitively generated. The finiteness condition is essential for the conjecture. In fact, Zabrodsky ([3, p. 142]) constructed a mod 3 Hopf space with finitely generated exterior rational cohomology which is never primitively generated for any Hopf structures. His construction can be generalized to all odd primes (Corollary 1.2). Then, we can consider the following problem.

Problem. Is there a mod p A_{n}-space having finitely generated exterior rational cohomology which is never A_{n}-primitively generated for any A_{n}-structures?

In this note we show the following fact.

Theorem 1.1. Let p be a prime. Then for any positive integer n with 2n < p, there exists a mod p A_{2n−1}-space E so that followings hold.

(1) H^{∗}(E; ℚ) is a finitely generated exterior algebra.

(2) The A_{2n−1}-form on E is extendable to an A_{2n}-form.

(3) For any extended A_{2n}-form in (2), H^{∗}(E; ℚ) is never A_{2n}-primitively generated.

The above theorem does not claim that H^{∗}(E; ℚ) is not A_{2n}-primitively generated for any A_{2n}-form on E; it is assuming that an A_{2n−1}-form on E is fixed. In fact, if n ≥ 2, then E has the homotopy type of a loop space. However, if n = 1, then the theorem has no restrictions since an A_{1}-space means just a space. We have the following fact.

Corollary 1.2. For any odd prime p, there exists a mod p Hopf space E so that H^{∗}(E; ℚ) is a finitely generated exterior algebra, but is never primitively generated for any Hopf structures on E.

Partially supported by Grand-in-Aid for Scientific Research C-08640047 from the Ministry of Education

Our construction includes an example of A_{2n}-space so that the A_{2n}-form is extendable to an A_{2n+1}-form, but the rational cohomology is never A_{2n+1}-primitively generated for any extended A_{2n+1}-form. However, the rational cohomology of this space in not an exterior algebra, but a tensor product of a finitely generated exterior algebra and a polynomial algebra with one generator.

To construct the example, we use Harper-Zabrodsky method ([1]). Their method seems very useful for the study of A_{n}-spaces.

The fact proved in this paper is inspired by Jim Lin in the discussion with him about Curjel’s conjecture. The author would like to express his hearty thanks to Jim Lin.

2. Harper-Zabrodsky elements

Let P be the sub Hopf algebra of the mod p Steenrod algebra generated by P^{1}. Then, P is a truncated polynomial algebra generated by P^{1}:

P = ℤ/p[P^{1}]/((P^{1})^{p}).

For any integer k with k ≥ p, we consider the following Hopf algebra over P:

H_{k}^{∗} = Λ(x_{1}, …, x_{p}),

where x_{i} are primitive with deg x_{i} = 2k − 1 + 2(i − 1)(p − 1), and P^{1}x_{i−1} = x_{i}. Let C be the cobar complex of H_{k}^{∗}, that is, C^{i} = (H̃_{k}^{∗})^{⊗i} is the i-fold tensor product of H̃_{k}^{∗}, and d_{i} : C^{i} → C^{i+1} is given by

d_{i} = μ̃ ⊗ 1 ⊗ ⋯ ⊗ 1 − 1 ⊗ μ̃ ⊗ 1 ⊗ ⋯ ⊗ 1 + ⋯ + (−1)^{i−1}1 ⊗ ⋯ ⊗ 1 ⊗ μ̃,

where μ̃ : H̃_{k}^{∗} → H̃_{k}^{∗} ⊗ H̃_{k}^{∗} is the reduced coproduct. Then, Harper-Zabrodsky proved the following fact.

Theorem 2.1 ([1, Prop.6.3]). There exists a class w_{i} ∈ C^{i} for 1 ≤ i ≤ p so that (1) w_{p} = x_{1} ⊗ ⋯ ⊗ x_{1};

(2) d_{i}(w_{i}) = P^{a}w_{i+1} for 1 ≤ i ≤ p − 1, where a = 1 if i is odd, and a = p − 1 if i is even;

(3) for any choice of w_{i} with the above properties, w_{1} equals to x_{1}x_{2} … x_{p} up to unit in ℤ/p.

Now we consider the above Hopf algebra for the case that k = p. It is clear that H^{∗}(SU; ℤ/p) includes H_{p}^{∗} as a sub Hopf algebra over P. Thus we consider w_{i} as elements in the cobar complex of H^{∗}(SU; ℤ/p). We prove the following

Lemma 2.2. For any i with 1 ≤ i ≤ (p + 1)/2, we have

w_{2i−1} ∉ Im d_{2i−2} + Im P^{1}.

Here since d_{0} = 0, the above for i = 1 means that

w_{1} ∉ Im P^{1}.

Proof. The lemma is clear for i = 1. Thus we assume that i ≥ 2.

Suppose contrarily that

w_{2i−1} = d_{2i−2}u_{0} + P^{1}u_{1}.

Now d_{2i−1}(w_{2i−1}−d_{2i−2}u_{0}) = d_{2i−1}w_{2i−1} = P^{1}w_{2i}. Thus, P^{1}u_{1} = w_{2i−1}−d_{2i−2}u_{0} is another choice for w_{2i−1}. Since P^{p−1}P^{1}u_{1} = 0, we can choose w_{j} for j ≤ 2i−2 to satisfy w_{j} = 0 if we choose P^{1}u_{1} for w_{2i−1}. This is a contradiction to Theorem 2.1 (3).

3. Construction of examples

Let w̄_{i} ∈ H̃(SU; ℤ)^{⊗i} be an integral class so that the mod p reduction of it is w_{i}. Fix a positive integer n with 2n < p, and put m = deg w_{2n} + 2n − 2. Let

X = Ht_{m−1}(SU) × K(ℤ, m),

where Ht_{m−1}(SU) is the m−1 homotopy approximation of SU, that is, π_{i}(Ht_{m−1}(SU)) = 0 for i > m − 1 and there is a map SU → Ht_{m−1}(SU) which induces an isomorphism on the ith homotopy group for i ≤ m − 1. Then it is clear that

H^{∗}(X; ℚ) ≅ H^{∗}(SU((m + 1)/2); ℚ).

Let {μ_{i} : K_{i} × X^{i} → X} be the standard A_{∞}-form on X. Define η : K_{2n} × X^{2n} → X as the composition

K_{2n} × X^{2n} ⟶^{Δ} K_{2n} × X^{2n} × K_{2n} × X^{2n} ⟶^{1×pr} K_{2n} × X^{2n} × Σ^{2n−2}X^{∧2n}

⟶^{μ_{2n}×w} X × K(ℤ, m) ⊂ X × X ⟶^{μ_{2}} X,

where X^{∧2n} is the 2n-fold smash product of X, Δ is the diagonal map, pr : K_{2n} × X^{2n} → (K_{2n}/∂K_{2n}) ∧ X^{∧2n} ≈ Σ^{2n−2}X^{∧2n} is the projection, and w is the adjoint of w̄_{2n}.

Lemma 3.1. {μ_{i}}_{i≤2n−1} ∪ {η} is an A_{2n}-form on X.

Proof. The lemma is clear since η = μ_{2n} on ∂K_{2n} × X^{2n} ∪ K_{2n} × X^{[2n]}, where X^{[2n]} is the fat wedge in X^{2n}.

From now on we consider X as an A_{2n}-space with the A_{2n}-form in Lemma 3.1. Let P_{i}X (i ≤ n) be the projective i-space of X. Since the A_{2n}-form on X is the extension of the original A_{2n−1}-form, we have the inclusion

P_{2n−1}X ⊂ Ht_{m}(BSU) × K(ℤ; m + 1).

Let ι ∈ H̃^{m+1}(Ht_{m}(BSU) × K(ℤ; m + 1); ℤ/p) be the class represented by the composition

Ht_{m}(BSU) × K(ℤ; m + 1) → K(ℤ; m + 1) → K(ℤ/p; m + 1),

where the first map is the projecton, and the second one is the mod p reduction. Let z ∈ H̃^{∗}(P_{2n−1}X; ℤ/p) be the restriction of ι.

Now we consider the following diagram induced by the filtration ΣX ⊂ P_{1}X ⊂ ⋯ ⊂ P_{2n−1}X ⊂ P_{2n}X:

H̃^{∗}(P_{2n−1}X; ℤ/p)  H̃^{∗}(P_{2n}X; ℤ/p) H̃^{∗}(X; ℤ/p)^{⊗2n−1}  H̃^{∗}(X; ℤ/p)^{⊗2n} u  _{h h h hh j}_{β^{∗}} u  ^{ε^{∗}}_{^{’ ’ ’ ’’)}^{α^{∗}}}  _{’ ’ ’ ’’)}

Then we prove the following fact.

Lemma 3.2. β^{∗}(z) = w_{2n}.

Proof. Let β : (K_{2n+2} × X^{2n}, S) → (P_{2n}X, P_{2n−1}X) and γ : (K_{2n+2} × X^{2n}, S) → (P′_{2n}X, P_{2n−1}X) be the relative homeomorphisms defining projective spaces ([2, p.280]), where P′_{2n}X is the projective 2n-space of X for the original A_{∞}-form, and S = ∂K_{2n+2} × X^{2n} ∪ K_{2n+2} × X^{[2n]}. Then β|S and γ|S coincide except for on T = ∂_{3}(3, 2n)(K_{3} × K_{2n}) × X^{2n}. In fact, for (ρ, σ) ∈ K_{3} × K_{2n} and (x_{1}, …, x_{2n}) ∈ X^{2n} we have

β(∂_{3}(3, 2n)(ρ, σ), x_{1}, …, x_{2n}) = (ρ, η(σ, x_{1}, …, x_{2n}))
γ(∂_{3}(3, 2n)(ρ, σ), x_{1}, …, x_{2n}) = (ρ, μ_{2n}(σ, x_{1}, …, x_{2n})).

Let T_{1} and T_{2} be two copies of T. Define δ : (T_{1} ∪ T_{2}) ∨ S → P_{2n−1}X by

δ|T_{1} = β|T, δ|T_{2} = γ|T and δ|S = γ|S,

where T_{1} and T_{2} are attached on ∂_{3}(3, 2n)(∂(K_{3} × K_{2n})) × X^{2n}. We notice that T_{1} ∪ T_{2} is homeomorphic to S^{2n−1} × X^{2n}. Furthermore, δ|T_{1} ∪ T_{2} is homotopic to the composition

S^{2n−1} × X^{2n} ⟶^{pr} Σ^{2n−1}X^{∧2n} ⟶^{w′} ΣX ⟶^{ε′} P_{2n−1}X,

where w′ is the adjoint to the composition Σ^{2n−2}X^{∧2n} ⟶^{w̄_{2n}} K(ℤ, m) ⊂ X and ε′ is the inclusion.

Now it is clear that there is a natural map ϕ : S → (T_{1} ∪ T_{2}) ∨ S so that δϕ ≃ β|S. This implies that

β^{∗}(z) = γ^{∗}(z) + (ε′w′)^{∗}(z),

where γ^{∗} : H̃^{∗}(P_{2n−1}X; ℤ/p) → H̃^{∗}(X; ℤ/p)^{⊗2n} is the natural homomorphism defined by using the cofibration P_{2n−1}X → P′_{2n}X → Σ^{2n}X^{∧2n}. Here, γ^{∗}(z) = 0 since z is the restriction of ι ∈ H̃^{∗}(Ht_{m}(BSU)×K(ℤ; m+1); ℤ/p). Furthermore, (ε′w′)^{∗}(z) = w_{2n}. Thus, β^{∗}(z) = w_{2n}.

Now β^{∗}α^{∗} = d_{2n−1}, and we have

β^{∗}(P^{1}z − α^{∗}w_{2n−1}) = P^{1}w_{2n} − d_{2n−1}w_{2n−1} = 0

by Theorem 2.1 (2). Thus, there exists a class y ∈ H̃^{∗}(P_{2n}X; ℤ/p) so that ε^{∗}y = P^{1}z − α^{∗}w_{2n−1}. Let f : X → K(ℤ/p, m + 2p − 2) be an A_{2n}-map which is the adjoint of the restriction of y : P_{2n}X → K(ℤ/p, m + 2p − 1) to ΣX. Note that f represents P^{1}ι for n > 1, and P^{1}ι − w_{1} for n = 1. Let j : E → X be the homotopy fiber of f. Then, E is an A_{2n}-space with H^{∗}(E; ℚ) ≅ H^{∗}(SU((m + 1)/2); ℚ). Then we prove the following fact which implies Theorem 1.1.

Theorem 3.3. For any A_{2n}-form on E so that j is an A_{2n−1}-map, any non decomposable class in H^{m}(E; ℚ) can not be represented by an A_{2n}-primitive class.

Proof. Suppose contrarily that there is an A_{2n}-primitive non decomposable class in H^{m}(E; ℚ). Since H^{∗}(E; ℤ) is torsion free in ∗ < m, there is an A_{2n}-primitive non decomposable class v ∈ H^{m}(E; ℤ/p). Here the m-dimensional indecomposable module of H^{∗}(E; ℤ/p) is one dimensional with a generator represented by j^{∗}ι′, where ι′ ∈ H^{∗}(X; ℤ/p) is defined similarly to ι. Thus, we have that

v = j^{∗}ι′ + d (d : decomposable)

up to unit in ℤ/p. Here j^{∗}ι′ is A_{2n−1}-primitive since it has a lift j^{∗}z ∈ H^{m+1}(P_{2n−1}E; ℤ/p). Thus, if n ≥ 2, then d is primitive, and so we have d = 0 since H^{∗}(E; ℤ/p) has no non-trivial decomposable primitives. Let u ∈ H^{m+1}(P_{2n−1}E; ℤ/p) ∩ Im ε^{∗} be a lift of v. Then, it is clear from the form of H^{∗}(P_{2n−1}E; ℤ/p) that

u = j^{∗}z + d′ + α^{∗}ξ

for some ξ ∈ H̃^{∗}(E; ℤ/p)^{⊗2n−1} and a decomposable class d′. We note that if n = 1, then u = α^{∗}v, z = α^{∗}ι′, d′ = 0 and ξ = d. Since β^{∗}d′ = 0, we have β^{∗}(j^{∗}z + α^{∗}ξ) = β^{∗}u = 0. Thus, β^{∗}(j^{∗}P^{1}z + α^{∗}P^{1}ξ) = 0. On the other hand, by Lemma 3.2 and Theorem 2.1 (2),

β^{∗}j^{∗}P^{1}z = j^{∗}P^{1}w_{2n} = j^{∗}d_{2n−1}w_{2n−1}.

Thus, we have

d_{2n−1}(j^{∗}w_{2n−1} + P^{1}ξ) = 0. (∗)

Here j^{∗} : H^{∗}(X; ℤ/p) → H^{∗}(E; ℤ/p) is an isomorphism in ∗ < m + 2p − 3. Thus, the equation (∗) still holds in H^{∗}(X; ℤ/p). Furthermore, by considering the homomorphism induced by the composition SU → Ht_{m−1}(SU) ⊂ X, we can see that (∗) also holds in H^{∗}(SU; ℤ/p)^{⊗2n}; more precisely, there is a class ξ′ ∈ H̃^{∗}(SU; ℤ/p)^{⊗2n−1} so that

d_{2n−1}(w_{2n−1} + P^{1}ξ′) = 0.

Thus w_{2n−1} + P^{1}ξ′ represents a cocycle in the cobar complex of H^{∗}(SU; ℤ/p). It is easily proved that there is no non-trivial class of the corresponding dimension in Cotor_{H^{∗}(SU;ℤ/p)}^{∗,∗}(ℤ/p, ℤ/p), and so we have

w_{2n−1} + P^{1}ξ′ ∈ Im d_{2n−2},

which is a contradiction to Lemma 2.2.

In the construction, we can use w_{2n+1} instead of w_{2n}. However, in this case m = deg w_{2n+1} + 2n − 1 is even. Thus, the rational cohomology of the resulting space E is not an exterior algebra, but a tensor product of an exterior algebra and a polynomial algebra with one generator.

References

1. J. Harper and A. Zabrodsky, Evaluating a p-th order cohomology operation, Pub. Mat. UAB 32 (1988), 61–78.

2. J. D. Stasheff, Homotopy associativity of H-spaces, I, Trans. Amer. Math. Soc. 108 (1963), 275– 292.

3. A. Zabrodsky, Hopf spaces, Math. Studies 22 (1976), North-Holland.

Kochi University Department of Mathematics Faculty of Science Kochi 780, Japan E-mail: hemmi@math.kochi-u.ac.jp