PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 103(117) (2018), 191–198 DOI: https://doi.org/10.2298/PIM1817191P

RELATIONS BETWEEN KERNELS AND IMAGES OF REDUCED POWERS FOR SOME RIGHT A_{p}-MODULES

Theodore Popelensky

Abstract. We investigate the right action of the mod p Steenrod algebra A_{p} on the homology H_{∗}(L^{∧s} , ℤ_{p}) where L = Bℤ_{p} is the lens space. Following ideas of Ault and Singer we investigate the relation between intersection of kernels of the reduced powers P^{p^{i}} and Bockstein element β and the intersection of images of P^{p^{i+1} −1} and of β. Namely one can check that ⋂ _{i=0}^{k} im P^{p^{i+1} −1} ⊂ ⋂ _{i=0}^{k} ker P^{p^{i}} and ⋂ _{i=0}^{k} im P^{p^{i+1} −1} ∩ im β ⊂ ⋂ _{i=0}^{k} ker P^{p^{i}} ∩ ker β. We generalize Ault’s homotopy systems to p > 2 and examine when the reverse inclusions are true.

Introduction

Two natural questions arise when the action of the Steenrod algebra A_{p} on the cohomology H^{∗}(X, ℤ_{p}) of a space X is considered. One is to describe the annihilating ideal I_{X} ⊂ A_{p} which is defined as I_{X} = {ϕ ∈ A_{p} | ϕ(H^{∗}(X, ℤ_{p})) = 0}. The problem appeared to be difficult even for p = 2 and X = ℝP^{∞} , see [1].

The second question is to describe a minimal generating set of the A_{p}-module H^{∗}(X, ℤ_{p}) and the set of elements of the form ϕx, where ϕ ∈ A_{p}, degϕ > 0, x ∈ H^{∗}(X, ℤ_{p}) (or the space spanned by such elements, this is the so called hit problem).

This question is far from being solved even for X = ∏ _{i=1}^{k} ℝP^{∞} , though cases k ⩽ 4 are completely described in the papers [4, 5, 6, 7]

In this note we address the problem which is dual to the second question. Consider the conjugate action of the Steenrod algebra A_{p} on the ℤ_{p}-homology of a space X. We say that x ∈ H_{∗}(X, ℤ_{p}) is P -annihilated iff xP^{m} = 0 for all m > 0 and βP -annihilated iff x is P -annihilated and xβ = 0. The problem is to describe the subspace of all P -anihilated or βP -annihilated elements. Following ideas of Ault and Singer [2, 3], we study subspaces of partially annihilated elements (Δ_{M} (k) and Δ_{M}^{β} (k), see below). It appears that there is a structure (homotopy system, see Definitions 2.1 and 2.2) which relates these spaces with the spaces of so called spike images (I_{M} (k) and I_{M}^{β} (k), see below). The main results are Theorem 2.1 and Corollary 2.1. Also we show that such homotopy systems exist for Λ̃ (see Section 3 and Theorem 4.1). Here Λ̃ is a bigraded space which is given by (1.2), see below.

2010 Mathematics Subject Classification: 55S10; 55R40; 57T25.

Key words and phrases: Steenrod algebra, reduced powers.

This work was funded by Russian Science Foundation 16-11-10069.

Throughout the paper all the (co)homologies are considered with ℤ_{p} coefficients.

The author is grateful to the referees for careful reading of the manuscript and for finding inconsistencies in the preliminary version of it.

1. Lens space and the A_{p}-module Λ̃.

Denote by L the classifying space Bℤ_{p}. The cohomology ring structure is described by the isomorphism H^{∗}(L) ≡ ℤ_{p}[x] ⊗ Λ_{ℤ_{p}}(y), where degx = 2 and degy = 1. The action of the Steenrod algebra A_{p} on H^{∗}(L) is given by the following relations:

P^{k}x^{m} = ( {m \atop k}) x^{k(p−1)+m} ,  βx^{m} = 0,
P^{k}x^{m}y = ( {m \atop k}) x^{k(p−1)+m} y, βx^{m}y = x^{m+1} .

There is the conjugate right action of the algebra A_{p} on the reduced homology of L. Denote by [a], a ⩾ 1, the generator of H̃_{2a} (L) and by [â], a ⩾ 0, the generator of H̃_{2a+1} (L). The action of A_{p} on these elements is described by the formulas:

(1.1)  \begin{aligned}P^{k} & = ( {a − (p − 1)k \atop k} ) [a − (p − 1)k], & [a]β & = [{ \atop a − 1}], \\ [â]P^{k} & = ( {a − (p − 1)k \atop k} ) [{ \atop a − (p − 1)k} ], & [â]β & = 0. \end{aligned}

Here we write a homomorphism to the right from its argument.

Consider the bigraded algebra Λ̃ = {Λ̃_{∗,∗} } which is defined by

(1.2)  Λ̃_{n,k} = \{ \begin{matrix}ℤ_{p}, & if n = k = 0; \\ H̃_{k} (L^{∧n} , ℤ_{p}), & if n ⩾ 1; \\ 0, & othervise. \end{matrix}

This algebra is a very natural module on which the right action of A_{p} should be studied.

2. Homotopy system

For a right A_{p}-module M define the following subspaces:

Δ_{M} (k) = ⋂_{i=0}^{k} kerP^{p^{i}} ,  I_{M} (k) = ⋂_{i=0}^{k} imP^{p^{i+1} −1} ,
Δ_{M}^{β} (k) = Δ_{M} (k) ∩ kerβ, I_{M}^{β} (k) = I_{M} (k) ∩ imβ.

The subspaces Δ_{M} (k) and Δ_{M}^{β} (k) are subspaces of partially annihilated elements. The subspaces I_{M} (k) and I_{M}^{β} (k) are subspaces of simultaneous spike images. For M = Λ̃, denote these modules by Δ(k), Δ^{β}(k), I(k), I^{β}(k).

Lemma 2.1. For an A_{p}-module M and k ⩾ 0 one has I_{M} (k) ⊂ Δ_{M} (k) and I_{M}^{β} (k) ⊂ Δ_{M}^{β} (k).

Remark 2.1. Generally I_{M} (k) ⊄ Δ_{M}^{β} (k).

Proof. The statements easily follow from the relations P^{p^{m+1} −1} P^{p^{m}} = 0 and ββ = 0. The last one is a particular case of the general relation P^{pm−1} P^{m} = 0. To check it, note that the Adem relation applies to P^{pm−1} P^{m}:

P^{pm−1} P^{m} = ∑_{j=0}^{m−1} (−1)^{pm−1+j} ( {(p − 1)(m − j) − 1 \atop pm − 1 − pj} ) P^{(p+1)m−1−j} P^{j} .

The inequality (p − 1)(m − j) − 1 < pm − 1 − pj follows easily from the restriction j ⩽ m − 1. Hence all the binomial coefficients in the last sum vanish. □

In some cases one can prove the reverse inclusions Δ_{M} (k) ⊂ I_{M} (k) and Δ_{M}^{β} (k) ⊂ I_{M}^{β} (k). Now we describe an algebraic structure which is responsible for these inclusions.

Definition 2.1. For a right A_{p}-module M define a k-th order homotopy system to be a null space N ⊂ M and a collection of ℤ_{p}-homomorphisms Q_{m} : M → M for 0 ⩽ m ⩽ k such that

(i) Q_{m}(N) ⊂ N for 0 ⩽ m ⩽ k; (ii) for 1 ⩽ m ⩽ k and l < p^{m} the following diagram comutes

\xymatrix{N \ar[r]^{P^{l}} \ar[d]_{Q_{m}} & M \ar[d]^{Q_{m}} \\ N \ar[r]^{P^{l}} & M}

(iii) (x)(P^{p^{m}} Q_{m} − Q_{m}P^{p^{m}} ) = x for any x ∈ N.

Definition 2.2. For a right A_{p}-module M define a k-th order β-homotopy system to be a k-th order homotopy system {Q_{m}} with null space N ⊂ M, which is equipped with a ℤ_{p}-homomorphism α : M → M such that

(iv) α(N) ⊂ N and for any x ∈ N one has xαβ + xβα = x.

For a nonnegative integer s, denote by 0 ⩽ d_{j} (s) < p the coefficients of the expansion s = ∑ d_{j} (s)p^{j} in the base p. Recall Lucas’s theorem which states that for any nonnegative integers x and y one has

( {x \atop y}) ≡ ∏_{j} ( {d_{j} (x) \atop d_{j} (x)}) mod p.

Lemma 2.2. For 1 ⩽ a ⩽ p − 1 one has

(P^{p^{0}} )^{a}(P^{p^{1}} )^{a} …(P^{p^{n−1}} )^{a}(P^{p^{n}} )^{a} = cP^{a(p^{0} +p^{1} +···+p^{n−1} +p^{n} )} ,

for some nonzero c ∈ ℤ_{p}.

Proof. First of all we show that for any a such that 1 ⩽ a ⩽ p − 1 one has (P^{1})^{a} = cP^{a} for some nonzero c ∈ ℤ_{p}. For a = 1 the statement is trivial. Assume the statement is true for a = b, where b < p−1. Then (P^{1})^{b+1} = (P^{1})^{b}P^{1} = cP^{b}P^{1}, where c ≠ 0. Apply the Adem relation to P^{b}P^{1}:

P^{b}P^{1} = (−1)^{b}( {p − 2 \atop b} ) P^{b+1} ,

hence (P^{1})^{b+1} = c(−1)^{b}\binom{p−2}{b}P^{b+1} . One easily checks that the binomial coefficient \binom{p−2}{b} is nonzero in ℤ_{p}.

Now we proceed by induction to prove that for any integers a and b such that 1 ⩽ a ⩽ p − 1 and 0 ⩽ b < a one has

P^{a(p^{0} +p^{1} +···+p^{m} )+bp^{m+1}} P^{p^{m+1}} = dP^{a(p^{0} +p^{1} +···+p^{m} )+(b+1)p^{m+1}}

for some nonzero d ∈ ℤ_{p}.

Denote a(p^{0} + p^{1} + ⋯+ p^{m}) + bp^{m+1} by N. Apply the Adem relation to the product P^{N} P^{p^{m+1}} :

P^{N} P^{p^{m+1}} = ∑_{t=0}^{[N/p]}(−1)^{N+t} ( {(p − 1)(p^{m+1} − t) − 1 \atop N − pt} ) P^{N+p^{m+1} −t} P^{t}.

The coefficient for t = 0, by Lucas’s theorem, is equal to (−1)^{N} \binom{p−2}{b}\binom{p−1}{a}^{m+1} and hence is not zero.

It is left to prove that for t > 0, the binomial coefficient vanishes. To apply Lucas’s theorem, it is enough to find j such that d_{j} ((p − 1)(p^{m+1} − t) − 1) < d_{j} (N − pt). Fix 0 < t < [N/p]. For simplicity let a_{j} = d_{j} (t). Choose the smallest j such that a_{j} ≠ 0.

Case 1. If 1 ⩽ a_{j} ⩽ a then d_{j} (N−pt) = a while d_{j} ((p−1)(p^{m+1} −t)−1) = a_{j} −1. But a_{j} − 1 < a.

Case 2. If a_{j+1} ⩾ a_{j} > a then d_{j+1} (N − pt) = p + a − a_{j}, while d_{j+1} ((p − 1)(p^{m+1} − t) − 1) = a_{j+1} − a_{j}. Note that a_{j+1} − a_{j} < p + a − a_{j}.

Case 3. Assume for some k one has a_{k+1} > a_{k} ⩾ a and a_{k} ⩽ a_{k−1} ⩽ ⋯⩽ a_{j+2} ⩽ a_{j+1} < a_{j}. Then d_{k+1} (N − pt) = p − 1 + a − a_{k} and d_{k+1} ((p − 1)(p^{m+1} −

t) − 1) = −1 + a_{k+1} − a_{k}. Note that p − 1 + a − a_{k} > −1 + a_{k+1} − a_{k}.

Case 4. Assume for some k one has a_{k+1} < a ⩽ a_{k} ⩽ a_{k−1} ⩽ ⋯⩽ a_{j+2} ⩽ a_{j+1} < a_{j}. Then d_{k+1} (N − pt) = p − 1 + a − a_{k} and d_{k+1} ((p − 1)(p^{m+1} − t) − 1) = p − 1 + a_{k+1} − a_{k}. Note that p − 1 + a − a_{k} > p − 1 + a_{k+1} − a_{k}. □

Theorem 2.1. Suppose that for a right A_{p}-module M there exists a k-th order homotopy system {Q_{m}} with a null space N. Then Δ_{M} (k) ∩ N = I_{M} (k) ∩ N. Moreover, if x ∈ Δ_{M} (k)∩N then y = xQ_{m}^{p−1} Q_{m−1}^{p−1} …Q_{0}^{p−1} satisfies yP^{p^{m+1} −1} = cx for some nonzero c ∈ ℤ_{p}.

Proof. Denote for simplicity P^{p^{m}} by P_{m}. Using standard technique for calculations with commutators involving creation-annihilation operators we prove for j = 0, …, m − 1 that

xQ_{m}^{p−1} Q_{m−1}^{p−1} …Q_{j}^{p−1} P_{j}^{p−1} = −xQ_{m}^{p−1} Q_{m−1}^{p−1} …Q_{j+1}^{p−1} .

The relation

xQ_{m}^{p−1} P_{m}^{p−1} = −x

is proved in the same way. Denote xQ_{m}^{p−1} Q_{m−1}^{p−1} …Q_{j+1}^{p−1} by W. Since all Q_{n}’s left the subspace N invariant and P_{j} commutes with Q_{m}, …, Q_{j+1} on the subspace N one has W P_{j} = 0. Note that

[Q_{j}^{n}, P_{j}^{n}] = ∑_{s+t=n−1} P_{j}^{s}[Q_{j}^{n}, P_{j}]P_{j}^{t}.

Apply this relation, for n = p − 1, to the product W (Q_{j})^{p−1} (P_{j})^{p−1} :

W Q_{j}^{p−1} P_{j}^{p−1} = W P_{j}^{p−1} Q_{j}^{p−1} + ∑_{s+t=p−2} W P_{j}^{s}[Q_{j}^{p−1} , P_{j}]P_{j}^{t}.

On the right-hand side, all the summands vanish except W [Q_{j}^{p−1} , P_{j}]P_{j}^{p−2} . Now use the equality

[Q_{j}^{p−1} , P_{j}] = ∑_{s+t=p−2} Q_{j}^{s}[Q_{j} , P_{j}]Q_{j}^{t}

and rewrite the product W [Q_{j}^{p−1} , P_{j}]P_{j}^{p−2} as

W [Q_{j}^{p−1} , P_{j}]P_{j}^{p−2} = ∑_{s+t=p−2} W Q_{j}^{s}[Q_{j} , P_{j}]Q_{j}^{t}P_{j}^{p−2} .

Since W Q_{j}^{s} ∈ N, one has W Q_{j}^{s}[Q_{j} , P_{j}] = −W Q_{j}^{s}. Finally,

W [Q_{j}^{p−1} , P_{j}]P_{j}^{p−2} = ∑_{s+t=p−2} −W Q_{j}^{s}Q_{j}^{t}P_{j}^{p−2} = −(p − 1)W Q_{j}^{p−2} P_{j}^{p−2} .

Proceeding in the same way, we obtain the equality

W Q_{j}^{p−1} P_{j}^{p−1} = (−1)^{p−1} (p − 1)!W = −W.

Here we use simple part of the Wilson theorem which states that for a prime p one has (p − 1)! + 1 ≡ 0 mod p.

By Lemma 2.2, P^{p^{m+1} −1} = d P_{0}^{p−1} …P_{m}^{p−1} for some nonzero d ∈ ℤ_{p}. Then

yP^{p^{m+1} −1} = d xQ_{m}^{p−1} …Q_{1}^{p−1} Q_{0}^{p−1} P_{0}^{p−1} P_{1}^{p−1} …P_{m}^{p−1}
= −d xQ_{m}^{p−1} …Q_{1}^{p−1} P_{1}^{p−1} …P_{m}^{p−1}
= ⋯= (−1)^{m}d xQ_{m}^{p−1} P_{m}^{p−1} = (−1)^{m+1} d x.  □

Corollary 2.1. Suppose that for a right A_{p}-module M there exists a k-th order β-homotopy system {Q_{m}} with a null space N and a ℤ_{p}-homomorphism α. Then Δ_{M}^{β} (k) ∩ N = I_{M}^{β} (k) ∩ N. Moreover, if x ∈ Δ_{M}^{β} (k) ∩ N, then y = xα belongs to N and satisfies yβ = x.

Proof. From xβα + xαβ = x and x ∈ kerβ it follows that yβ = xαβ = x.

From Definition 2.2 one has y ∈ N. □

3. Shift maps in the homology of the smash-product power of the lens space

In this section we consider examples of homotopy systems for certain A_{p}- modules. The reduced homology of L^{∧s} is a vector space with the basis which consists of the elements [u_{1}, …, u_{s}] of degree ∑ degu_{j}, where every u_{j} is a, a ⩾ 1, or â, a ⩾ 0, and dega = 2a, deg â = 2a + 1. Now we describe homotopy systems which appear naturally for H̃_{∗}(L^{∧s} ).

Define the shift map Ψ_{i}^{l} : Λ̃_{s,∗} → Λ̃_{s,∗+l} by formulas [u_{1}, …, u_{i}, …, u_{s}]Ψ_{i}^{l} =

[u_{1}, …, u_{i}+l, …, u_{s}]. Here u_{j} +l denotes a_{j} +l if u_{j} = a_{j} and { \atop a_{j} + l} if u_{j} = â_{j}. Also

define [u_{1}, …, a_{i}, …, u_{s}]α_{i} = 0 and [u_{1}, …, â_{i}, …, u_{s}]α_{i} = [u_{1}, …, a_{i} + 1, …, u_{s}]. Clearly (Ψ_{i}^{p^{m}} )^{k} = Ψ_{i}^{kp^{m}} .

Hereinafter for s = 1 we denote Ψ^{l} = Ψ_{1}^{l} and α = α_{1}.

Lemma 3.1. We have

(a) [a]Ψ^{kp^{m}} P^{n} = [a]P^{n}Ψ^{kp^{m}} for a ⩾ (p − 1)n, p^{m} > n, 1 ⩽ k ⩽ p − 1;

(b) [â]Ψ^{kp^{m}} P^{n} = [â]P^{n}Ψ^{kp^{m}} for a ⩾ (p − 1)n, p^{m} > n, 1 ⩽ k ⩽ p − 1;

(c) [a]Ψ^{kp^{m}} β = [a]βΨ^{kp^{m}} ≠ 0;

(d) [â]Ψ^{kp^{m}} β = 0 = [â]βΨ^{kp^{m}} .

Proof. By formulas (1.1), one has

[a]Ψ^{kp^{m}} P^{n} = ( {a + kp^{m} − (p − 1)n \atop n} ) [a + kp^{m} − (p − 1)n],
[a]P^{n}Ψ^{kp^{m}} = ( {a − (p − 1)n \atop n} ) [a + kp^{m} − (p − 1)n].

Two binomial coefficients coincide under the assumptions of Lemma. Other statements are proved in the same way. □

Lemma 3.2. We have

(a) [a]P^{p^{m}} Ψ^{(p−1)p^{m}} − [a]Ψ^{(p−1)p^{m}} P^{p^{m}} = [a] for m ⩾ 0 and a ⩾ (p − 1)p^{m};

(b) [â]P^{p^{m}} Ψ^{(p−1)p^{m}} − [â]Ψ^{(p−1)p^{m}} P^{p^{m}} = [â] for m ⩾ 0 and a ⩾ (p − 1)p^{m};

(c) [a]αβ + [a]βα = [a] for a ⩾ 1;

(d) [â]αβ + [â]βα = [â] for a ⩾ 0.

Proof. By straightforward computation, one can get

[a]P^{p^{m}} Ψ^{(p−1)p^{m}} = ( {a − (p − 1)p^{m} \atop p^{m}} ) [a], [a]Ψ^{(p−1)p^{m}} P^{p^{m}} = ( {a \atop p^{m}} ) [a].

Assume a = ∑ b_{k}p^{k}, where 0 ⩽ b_{k} < p. If b_{m} = p − 1 then \binom{a}{p^{m}} = p − 1 and \binom{a−(p−1)p^{m}}{p^{m}} = 0. If b_{m} < p − 1 then \binom{a}{p^{m}} = b_{m} and \binom{a−(p−1)p^{m}}{p^{m}} = b_{m} + 1. The proof of (b) is similar, and the statements (b) and (c) are clear. □

Lemma 3.3. Assume x = [u_{1}, …, u_{s}], where each u_{j} is a_{j} or â_{j}. If p^{m} > n, a_{i} ⩾ (p − 1)n, 1 ⩽ k ⩽ p − 1, then one has xΨ_{i}^{kp^{m}} P^{n} = xP^{n}Ψ_{i}^{kp^{m}} .

Proof. Without loss of generality one can assume i = 1. The case s = 1 was considered in Lemma 3.1, so assume s > 1. Let y = [u_{2}, …, u_{s}]. Then x = [u_{1}]×y, and by Lemma 3.1 one has

([u_{1}] × y)P^{n}Ψ_{1}^{kp^{m}} = ∑_{l} ([u_{1}]P^{l} × yP^{n−l} )Ψ_{1}^{kp^{m}}

= ∑_{l} (−1)^{k deg y}(xP^{l}Ψ_{1}^{kp^{m}} × yP^{n−l} ) = ∑_{l} (−1)^{k deg y}([u_{1}]Ψ_{1}^{kp^{m}} P^{l} × yP^{n−l} )

= (−1)^{k deg y}([u_{1}]Ψ_{1}^{kp^{m}} × y)P^{n} = xΨ_{1}^{kp^{m}} P^{n}.

Here we have used the standard sign convention. □

Lemma 3.4. Assume x = [u_{1}, …, u_{s}], where each u_{j} is a_{j} or â_{j}. If a_{i} ⩾ (p − 1)p^{m}, then xP^{p^{m}} Ψ_{i}^{(p−1)p^{m}} − xΨ_{i}^{(p−1)p^{m}} P^{p^{m}} = x.

Proof. Again one can consider only i = 1. For s = 1 the statement coincides with Lemma 3.2, so assume s > 1. Let y = [u_{2}, …, u_{s}]; then x = [u_{1}] × y. By Lemma 3.1, one has

(3.1) ([u_{1}] × y)P^{p^{m}} Ψ_{1}^{(p−1)p^{m}}

= ∑_{l=0}^{p^{m} −1} ([u_{1}]P^{l} × yP^{p^{m} −l} )Ψ_{1}^{(p−1)p^{m}} + ([u_{1}]P^{p^{m}} × y)Ψ_{1}^{(p−1)p^{m}}

= ∑_{l=0}^{p^{m} −1} ([u_{1}]P^{l}Ψ_{1}^{(p−1)p^{m}} × yP^{p^{m} −l} ) + ([u_{1}]P^{p^{m}} Ψ_{1}^{(p−1)p^{m}} × y);

(3.2) ([u_{1}] × y)Ψ_{1}^{(p−1)p^{m}} P^{p^{m}}

= ∑_{l=0}^{p^{m} −1} ([u_{1}]Ψ_{1}^{(p−1)p^{m}} P^{l} × yP^{p^{m} −l} ) + ([u_{1}]Ψ_{1}^{(p−1)p^{m}} P^{p^{m}} × y).

Take the difference of (3.1) and (3.2) and note that the sums on the right-hand sides cancel by Lemma 3.1. The difference of the last summands in (3.1) and (3.2) is equal to x by Lemma 3.2. □

Lemma 3.5. Let x = [u_{1}, …, u_{s}]. Then xβα_{i} + xα_{i}β = x.

Proof. Again one can consider only i = 1. For s = 1 the statement coincides with Lemma 3.2(b,c), so assume s > 1. Let y = [u_{2}, …, u_{s}]. Then x = [u_{1}] × y, and by Lemma 3.1 one has

(3.3)  ([u_{1}] × y)βα_{1} = ((−1)^{deg y}[u_{1}]β × y + [u_{1}] × yβ)α_{1}
= [u_{1}]βα_{1} × y + (−1)^{deg yβ} [u_{1}]α_{1} × yβ.
(3.4)  ([u_{1}] × y)α_{1}β = (−1)^{deg y}([u_{1}]α_{1} × y)β
= [u_{1}]α_{1}β × y + (−1)^{deg y}[u_{1}]α_{1} × yβ.

Summing (3.3) and (3.4), we obtain

(x)βα_{1} + (x)α_{1}β = [u_{1}]βα_{1} × y + [u_{1}]α_{1}β × y = ([u_{1}]βα_{1} + [u_{1}]α_{1}β) × y

which by Lemma 3.2(b,c) is equal to [u_{1}] × y = x. □

4. Examples

For some i = 1, …, s denote by N_{s}(i, k) the subspace in Λ̃_{s,∗} = H̃_{∗}(L^{∧s} ) spanned by [u_{1}, …, u_{s}], where u_{i} = a_{i} or u_{i} = â_{i} and a_{i} ⩾ (p − 1)p^{k}.

Lemma 4.1. For s ⩾ 1 and k ⩾ 0 the collection of maps {Q_{m} = Ψ_{i}^{(p−1)p^{m}} | m ⩽ k} forms a k-th homotopy system for Λ̃_{s,∗} with the null space N_{s}(i, k).

Proof. The subspace N_{s}(i, k) is stable under the action of Q_{m}. The properties (ii) and (iii) from Definition 2.1 are checked in Lemmas 3.3 and 3.3. The property (iv) from Definition 2.2 is checked in Lemma 3.5. □

Theorem 4.1. Fix s ⩾ 1 and k ⩾ 0. Assume x ∈ N_{s}(i, k) for all 1 ⩽ i ⩽ s. Then x ∈ Δ(k) iff x ∈ I(k) and x ∈ Δ^{β}(k) iff x ∈ I^{β}(k).

Proof. The statement follows immediately from Lemma 4.1, Theorem 2.1, and Corollary 2.1. □

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Moscow State Lomonosov University

Department of Mechanics and Mathematics

Moscow, Russia

    popelens@mech.math.msu.su