GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 1, 1996, 27-48

AN ALGEBRAIC MODEL OF FIBRATION WITH THE FIBER K(π, n)-SPACE

N. BERIKASHVILI

Abstract. For a fibration with the fiber K(π, n)-space, the algebraic model as a twisted tensor product of chains of the base with standard chains of K(π, n)-complex is given which preserves multiplicative structure as well. In terms of this model the action of the n-cohomology of the base with coefficients in π on the homology of fibration is described.

1. Introduction

For a fibration F ⟶ E ⟶ B there is the Brown model [1], [2] as a twisted tensor product C_{∗}(B) ⊗_{φ} C_{∗}(F). However, this model gives us no information about the multiplicative structure. The aim of this paper is to construct a model for the particular case of the fiber F = K(π, n) which would inherit the multiplicative structure of C^{∗}(E) as well. The model is given as a twisted tensor product C_{∗}(B) ⊗_{z^{n+1}} C_{∗}^{□}(L(π, n)) of the singular chain complex of the base with the chain complex of the cubical version of the complex K(π, n) [3], [4] and it describes the cubical singular chain complex of the total space (Theorems 5.1 and 6.1 below). It turns out that this model is actually the chain complex of a cubical complex and hence, in addition, it carries a Serre cup product structure as well. In Section 7 the action of the group H^{n}(B, π) on the model is discussed.

This paper is a reply to the needs of obstruction theory and is applied in [5]. The main result was announced in [6].

2. Preliminaries

Let X be a CW -complex and F^{p}X = X^{p} be its filtration by skeletons. If H is the singular homology theory with coefficient group G, then for the

27 first term of the related spectral sequence we have E_{p,q}^{1} = 0, q > 0, and (E_{p,0}^{1}, d^{1}) is the chain complex

1991 Mathematics Subject Classification. 55N10, 55N45, 55R20, 55U99.

Key words and phrases. Simplicial set, cubical set, fibration, K(π, n)-space, cross section, twisted tensor product, perturbed differential, algebraic model, spectral sequence.

1072-947X/96/0100-0027$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

(H_{p}(X^{p}, X^{p−1}, G), d^{1})

with the homology isomorphic to the singular homology of the space X. Of course H_{p}(X^{p}, X^{p−1}, G) is isomorphic to ΣG_{σ}, where σ are p-cells of X and G_{σ} = G. The above chain complex will be referred to as the cell chain complex of the CW -complex.

In some cases the chain map of the cell-chain complex of the CW -complex to the singular or the cubical singular chain complex is defined which induces the sisomorphism of homology stated above.

Recall that the cubical set is a sequence of sets Q = {Q_{0}, Q_{1}, Q_{2} ⋯ } together with the boundary and degeneracy operators

d_{i}^{ϵ} : Q_{n} → Q_{n−1}, 1 ≤ i ≤ n, ϵ = 0, 1,
s_{i} : Q_{n} → Q_{n+1}, 1 ≤ i ≤ n + 1,

subject to the standard equalities (see, e.g., [4], [7]).

Q_{n} are n-cubes of Q; σ^{n} ∈ Q_{n} is said to be degenerate if

σ^{n} = s_{i}τ^{n−1}.

The abbreviation for d_{i}^{ϵ}σ^{n} is σ_{i}^{n,ϵ}, ϵ = 0, 1.

The main example of a cubical set is the cubical singular complex Q(X) of a space X.

Milnor’s notion of the realization of a simplicial set [8] works for a cubical set as well and runs as follows.

Let I^{n} be the standard n-cube and let

e_{i}^{ϵ} : I^{n−1} ⟶ I^{n}, 1 ≤ i ≤ n, ϵ = 0, 1,
p_{i} : I^{n+1} → I^{n}, 1 ≤ i ≤ (n + 1),

be i-face imbeddings and i-projections.

Let Q be a cubical set. Then the realization |Q| is defined as a factor set of the space

⋃_{n}(Q_{n} × I^{n})

by the identifications

(σ^{n}, e_{i}^{ϵ}x) = (d_{i}^{ϵ}(σ^{n}), x), σ ∈ Q_{n}, x ∈ I^{n−1},
(s_{i}(σ^{n}), x) = (σ^{n}, p_{i}x), σ ∈ Q_{n}, x ∈ I^{n+1}, 1 ≤ i ≤ (n + 1).

|Q| is a CW-complex with n-cells in 1–1 correspondence with the nondegenerate n-cubes of Q.

There is a standard continuous map

r : |Q(X)| ⟶ X.

We define the Giever-Hu realization [9], [10] of the cubical set Q and denote it by ||Q||, omitting in the above definition the equality with degeneracy operators. Then ||Q|| has cells in 1–1 correspondence with all cubes of Q. We have the canonical continuous map

||Q|| ⟶ |Q|.

We recall the notion of homology for the cubical set Q. C̃_{∗}^{□}(Q) is the chain complex spanned in each dimension n by all n-cubes of Q, and the boundary operator d is defined by

dσ^{p} = Σ(−1)^{i}d_{i}^{0}σ^{p} − Σ(−1)^{i}d_{i}^{1}σ^{p}.

\overline{C} _{∗}^{□}(Q) is a chain subcomplex of C̃_{∗}^{□}(Q) spanned by the degenerate cubes. The quotient complex

C_{∗}^{□}(Q) = C̃_{∗}^{□}(Q)/\overline{C} _{∗}^{□}(Q)

is said to be the chain complex of Q and its homology is called the homology of Q.

If G is a group of coefficients, then by

C_{□}^{n}(Q, G)

we understand the normalized cochains of Q, i.e., those which are zero on the degenerate cubes.

For the singular cubical set Q(|Q|) of the space |Q| consider the standard imbedding

Q ⊂ Q(|Q|).

It induces the following isomorphisms of homology and cohomology:

H_{n}^{□}(Q, G) = H_{n}^{□}(|Q|, G), H_{□}^{n}(Q, G) = H_{□}^{n}(|Q|, G).

Hence we find that the map r : |Q(X)| → X induces the isomorphisms

H_{n}^{□}(|Q(X)|, G) = H_{n}^{□}(X, G)), H_{□}^{n}(|Q(X)|, G) = H_{□}^{n}(X, G)).

The interaction of simplicial and cubical sets is as follows. Let Δ_{n} be the standard simplex and consider the map

ψ_{n} : I^{n} ⟶ Δ_{n}  (2.1)

from [11] defined by

y_{0} = 1 − x_{1}, ; y_{1} = x_{1}(1 − x_{2}), ; y_{2} = x_{1}x_{2}(1 − x_{3}), ; ⋮ ; y_{n−1} = x_{1}x_{2}x_{3} ⋯ x_{n−1}(1 − x_{n}), ; y_{n} = x_{1}x_{2}x_{3} ⋯ x_{n}.  (2.2)

Clearly, ψ_{n} is the map of the pairs ψ_{n} : (I^{n}, ∂I^{n}) ⟶ (Δ_{n}, ∂Δ_{n}), inducing the homeomorphism I^{n}/∂I^{n} ⟶ Δ_{n}/∂Δ_{n}.

In [11], for a singular simplex σ^{n} : Δ_{n} ⟶ X J. P. Serre considers the composition I^{n} ⟶^{ψ_{n}} Δ_{n} ⟶^{σ^{n}} X as the singular cube tσ^{n} of X, defining the chain map C_{∗}(S(X)) ⟶ C_{∗}^{□}(Q(X)) from the simplicial singular chain complex to the singular normalized cubical chain complex (see the identities below). Hence for an abelian group G we have the cochain map

C_{□}^{∗}(Q(X), G) ⟶ C^{∗}(X, G)  (2.3)

from the normalized singular cubical cochain complex to an ordinary singular cochain complex.

For map (2.3) at the geometrical level, t : S(X) ⟶ Q(X), we have the identities

\begin{matrix}d_{i}^{1}t(σ^{n}) = t(σ_{i−1}^{n}), 1 ≤ i ≤ n, \\ d_{n}^{0}t(σ^{n}) = t(σ_{n}^{n}), \\ d_{i}^{0}t(σ^{n}) = \underbrace{s_{i}s_{i} ⋯ s_{i}}_{n−i} t(σ_{n,(n−1),··· ,i}^{n}), 1 ≤ i < n. \end{matrix}  (2.4)

We see that (tσ)_{i}^{0} is a degenerate cube if i ≠ n.

The above equalities imply that the following lemma is valid.

Lemma 2.1. Let 0 < k ≤ n and let P be the sum of closed k-cubes of I^{n} which are not degenerated by ψ_{n}. Then P is a deformation retract of I^{n}.

One defines a chain map C_{∗}^{□}(X) ⟶ C_{∗}^{N}(X) of the normalized complexes as follows. Consider the standard triangulation of I^{n} and let s(I^{n}) denote the basic n-dimensional chain of this triangulation as a singular chain of the space I^{n}. Define f by

f(σ^{n}) = σ_{∗}^{n}(s(I^{n})) ∈ C_{n}(X).  (2.5)

If σ^{n} is a degenerate cube, then all n-simplexes in f(s(I^{n})) are degenerate and therefore f defines the map of the normalized complexes.

For cubical cochains J. P. Serre defines the ⌣-product which is certainly valid for general cubical sets too [11]. The definition runs as follows. For the subset

K = (i_{1} < i_{2} < i_{3} ⋯ < i_{k}) ⊂ (1, 2, 3, ⋯, n)

and σ^{n} ∈ Q_{n} we introduce the notation

d_{K}^{ϵ}σ^{n} = d_{i_{1}}^{ϵ}d_{i_{2}}^{ϵ}d_{i_{3}}^{ϵ} ⋯ d_{i_{k}}^{ϵ}(σ^{n}), ϵ = 0, 1.

Let c^{p} ∈ C^{p}(Q, Λ), c^{q} ∈ C^{q}(Q, Λ) be the normalized p- and q-cochains of Q with coefficients in a commutative ring Λ. Define the product c^{p+q} = c^{p} ⌣ c^{q} by

c^{p+q}(τ^{p+q}) = Σ_{(H,K)}(−1)^{a(H,K)}c^{p}(d_{K}^{0}(τ^{p+q}))c^{q}(d_{H}^{1}(τ^{p+q})), (2.6)

where (H, K) is the decomposition of {1, 2, 3, ⋯, (p + q)} into two disjoint subsets.

Note that if τ^{p+q} is of the form t(σ), then on the right side only one summand, precisely that of the decomposition (1, 2, 3, ⋯, p) ⋃(p + 1, p + 2, ⋯, p + q), is not zero, which shows that the cochain map (2.3) preserves multiplicative structure.

For any abelian group π and any positive integer n one introduces the cubical version of the K(π, n)-complex [3], [4]. L(π, n) is a cubical set with p-cubes being n-cocycles, Z^{n}(I^{p}, π), where I^{p} = I×I×⋯×I is the standard cube with standard faces, and the cochain is understood as a cell cochain. The boundary and degeneracy operators are defined similarly to those in the c.s.s. case.

For the complex L(π, n) we have, in C_{∗}^{□}(L(π, n)), the ring structure that converts C_{∗}^{□}(L(π, n)) into a graded differential ring and hence H_{∗}^{□}(L(π, n)) into a graded ring. The multiplication is defined as follows. Given cubes σ^{p} and σ^{q} in L(π, n), i.e., σ^{p} ∈ Z^{n}(I^{p}, π), σ^{q} ∈ Z^{n}(I^{q}, π), we define the (p + q)-cube σ^{p} ∘ σ^{q} = σ^{p+q} ∈ Z^{n}(I^{p+q}, π) as pr_{1}^{∗}σ^{p} + pr_{2}^{∗}σ^{q}, where

pr_{1} : I^{p+q} → I^{p}, pr_{2} : I^{p+q} → I^{q}

are standard projections of I^{p+q} on its first p-cube and its last q-cube. Under this multiplication C̃_{∗}^{□}(L(π, n)) is the differential graded ring and C̄_{∗}^{□}(L(π, n)) is the ideal, and therefore C̃_{∗}^{□}(L(π, n))/\overline{C} _{∗}^{□}(L(π, n)) is the graded differential associative ring. We have C_{0}^{□}(L(π, n))=Z, C_{i}^{□}(L(π, n)) = 0, 0 < i < n, while 1 ∈ Z is the unitary element of the algebra C_{∗}^{□}(L(π, n)).

The Eilenberg–MacLane complex K(π, n) is a simplicial set and C_{∗}(K(π, n)) is the skew-commutative differential ring. The above map

ψ_{n} : I^{n} ⟶ Δ_{n}

enables us to define the chain map

C_{∗}(K(π, n)) ⟶ C_{∗}^{□}(L(π, n))

similarly to that of J. P. Serre. In an obvious manner similar to that used for (2.5) we define

C_{∗}^{□}(L(π, n)) ⟶ C_{∗}^{N}(K(π, n))

which is a homomorphism of differential rings. Hence H_{∗}^{□}(L(π, n)) and H_{∗}(K(π, n)) are isomorphic as rings and thus H_{∗}^{□}(L(π, n)) is a commutative ring.

Recall some notions from the theory of perturbed differentials [12]. Let Y be a filtered left graded differential module over a filtered graded differential algebra A under the pairing

A ⊗ Y ⟶ Y.

Let F^{i}A ⊃ F^{i+1}A and F^{i}Y ⊃ F^{i+1}Y be the above-mentioned filtrations. Fix the integer n. Let

T^{n}(A) = {a ∈ F^{n}A, |a| = +1, d_{A}a = aa};

a is known as a twisting element of the differential algebra.

For a ∈ T^{n}(A) the perturbed differential in Y is defined as

d_{a}x = d_{Y} x + ax, x ∈ Y.

The filtered complex with this perturbed differential will be denoted by Y_{a}. It is a 1-graded and filtered differential chain complex. Consider the set G^{n}(A) of all elements in A of the form

g = 1 + p, p ∈ F^{1}A, d_{A}p ∈ F^{n}A.

It is obvious that this set is the group under the product operation in the algebra A. The group G^{n}(A) acts from the left on the set T^{n}(A) by

ā = (1 + p) ∘ a = (1 + p)^{−1}a(1 + p) + (1 + p)^{−1}d_{A}(1 + p) (2.7)

which is equivalent to the equality

d_{A}(1 + p) = (1 + p)ā − a(1 + p).  (2.8)

The set of orbits under action (2.7) will be denoted by D^{n}(A).

For \overline{a} = (1 + p) ∘ a consider the map φ_{p} : Yā → Y_{a} given by

φ_{p}(x) = (1 + p)x = x + px.  (2.9)

This is a chain map and, obviously, 1–1.

Proposition 2.1. If f : A_{1} ⟶ A is a morphism of filtered differential algebras inducing an isomorphism of the nth terms of spectral sequences, then the induced map D^{n}f : D^{n}(A_{1}) ⟶ D^{n}(A) is 1–1 [12].

3. Auxiliary Complexes

Let B be a topological space and π an abelian group. Let L(π, n) be the K(π, n)-complex of the preceding section and C_{∗}^{□}(L(π, n)) its normalized chain complex. Let G be an abelian group. Consider the chain and cochain complexes C_{∗}^{□}(L(π, n), G) and C_{□}^{∗}(L(π, n), G).

Consider the complexes

\begin{aligned}Y_{∗∗}(B, π, n) & = C_{∗}(B, C_{∗}^{□}(L(π, n))), \\ Y_{∗∗}(B, π, n, G) & = C_{∗}(B, C_{∗}^{□}(L(π, n), G)), \end{aligned}  (3.1)

i.e., the group of singular chains of B with coefficients in the normalized integral chains of the complex L(π, n) and the group of singular chains of B with coefficients in the normalized chains of the complex L(π, n) with coefficients in G. These complexes are bigraded differential modules, the first differential being that of B and the second one that of L(π, n). Both complexes are covariant functors on the category of topological spaces. It is clear that

Y_{∗∗}(B, π, n) = C_{∗}(B) ⊗ C_{∗}^{□}(L(π, n)),
Y_{∗∗}(B, π, n, G) = C_{∗}(B) ⊗ C_{∗}^{□}(L(π, n), G),
Y_{∗∗}(B, π, n, G) = C_{∗}(B, C_{∗}^{□}(L(π, n))) ⊗ G.

The cohomology version is

Y^{∗∗}(B, π, n, G) = C^{∗}(B, C_{□}^{∗}(L(π, n), G)).

We have

Y^{∗∗}(B, π, n, G) = Hom(C_{∗}(B), C_{□}^{∗}(L(π, n), G)),
Y^{∗∗}(B, π, n, G) = Hom(Y_{∗∗}(B, π, n), G)).

Consider the graded differential algebra C_{∗}^{□}(L(π, n)) of the preceding section and the bigraded differential algebra

A^{∗}_{∗}(B, π, n) = C^{∗}(B, C_{∗}^{□}(L(π, n)))  (3.2)

of singular cochains of B with coeficients in the normalized chains of L(π, n), where the multiplication is defined as the ⌣-product of B when the coefficients are multiplied by the product in the algebra C_{∗}^{□}(L(π, n)). We obtain

A^{∗}_{∗}(B, π, n) = Hom(C_{∗}(B), C_{∗}^{□}(L(π, n)))

and the above multiplication is the same as that defined by the composition

C_{∗}(B) ⟶^{Δ} C_{∗}(B)⊗C_{∗}(B) ⟶^{x⊗y} C_{∗}^{□}(L(π, n))⊗C_{∗}^{□}(L(π, n)) ⟶^{μ} C_{∗}^{□}(L(π, n))

for x, y ∈ Hom(C_{∗}(B), C_{∗}^{□}(L(π, n))), where Δ is the coproduct of B and μ is the product in C_{∗}^{□}(L(π, n)). By lifting the second index in (3.2) we obtain the bigraded algebra of the fourth quadrant with both differentials increasing the degree of elements by +1:

A^{∗,−∗}(B, π, n) = C^{∗}(B, C_{□}^{−∗}(L(π, n))).

The total complex is the direct product of bihomogeneous components

A^{m} = Π_{p−q=m}^{p,−q}A^{p,−q}

and therefore every element x^{n} is given uniquely as the sum of its components

x^{n} = x^{n,o} + x^{n+1,−1} + x^{n+2,−2} + x^{n+3,−3} + · · · .

The unitary element of the algebra 1 ∈ A^{0,0} ⊂ A^{0} is the cochain of B, c^{0,0}, equal to 1 of C_{∗}(L(π, n)) for every 0-simplex of B. A^{∗,−∗}(B, π, n) is a contravariant functor both on the category of topological spaces and on the category of simplicial sets. The filtration of bialgebra A with respect to the first degree is the decreasing one and is complete in the Eilenberg-Moore sense. The second term of the spectral sequence is

E_{2}^{p,−q} = H^{p}(B, H_{q}(π, n)),

the spectral sequence converging to

H^{p−q}(A^{∗,−∗}(B, π, n))

in the Eilenberg–Moore sense.

Y_{∗∗}(B, π, n, G) = Y^{−∗,−∗}(B, π, n, G) and Y^{∗∗}(B, π, n, G) are the left modules over the bigraded algebra A^{∗,−∗}(B, π, n) to be understood as follows. The pairing

C_{p}^{□}(L(π, n)) ⊗ C_{q}^{□}(L(π, n)) ⟶ C_{p+q}^{□}(L(π, n))  (3.3)

induces the pairings

C_{p}^{□}(L(π, n)) ⊗ C_{q}^{□}(L(π, n)) ⊗ G ⟶ C_{p+q}^{□}(L(π, n)) ⊗ G,  (3.4)
C_{p}^{□}(L(π, n)) ⊗ C_{□}^{p+q}(L(π, n), G) ⟶ C_{□}^{q}(L(π, n), G).  (3.5)

The bicomplex Y_{∗∗}(B, π, n, G) is the module over the algebra A^{∗}_{∗}(B, π, n) via the ⌢-product of B and the product of coefficients (3.3) and (3.4). The bicomplex Y^{∗∗}(B, π, n, G) is the left module over the algebra A^{∗}_{∗}(B, π, n) via the ⌣-product in B and the product of coefficients (3.5).

The spectral sequence arguments show that if f : B_{1} ⟶ B induces an isomorphism of homology, then the induced homomorphisms of complexes

A^{∗,−∗}(B, π, n) ⟶ A^{∗,−∗}(B_{1}, π, n)

and

Y_{∗∗}(B_{1}, π, n) ⟶ Y_{∗∗}(B, π, n)

induce an isomorphism of homology.

In particular, if K is the ordered simplicial complex then the imbedding K ⊂ S(|K|) induces the isomorphism

H(A^{∗,−∗}(|K|, π, n)) ⟶ H(A^{∗,−∗}(K, π, n)).  (3.6)

Consider the algebra A^{∗,−∗}(B, π, n) as a filtered differential algebra with the filtration defined by the first degree and introduce the notation D(A^{∗,−∗}(B, π, n)) = D^{n+1}(A^{∗,−∗}(B, π, n)). Then

D(A^{∗,−∗}(B, π, n)) = T^{n+1}(A^{∗,−∗}(B, π, n))/G^{n+1}(A^{∗,−∗}(B, π, n)),

where T^{n+1}(A^{∗,−∗}(B, π, n)) is the set of all twisting elements of A^{∗,−∗}(B, π, n) of the form

a = a^{n+1,−n} + a^{n+2,−n−1} + a^{n+3,−n−2} ⋯,

and G^{n+1}(A^{∗,−∗}(B, π, n)) is the set of all elements in A^{∗,−∗}(B, π, n) of the form

1 + p = 1 + p^{n,−n} + p^{n+1,−n−1} + p^{n+2,−n−2} + ⋯

(by virtue of C_{i}^{□}(L(π, n)) = 0, 0 < i < n).

Proposition 3.1. If f : B_{1} ⟶ B induces an isomorphism of homology then D(f) : D(A^{∗,−∗}(B, π, n)) ⟶ D(A^{∗,−∗}(B_{1}, π, n)) is the 1–1 map.

Proof. We have E_{2}^{p,−q} = H^{p}(B, H_{q}(π, n)). By assumption, f induces an isomorphism of the second terms and hence of the (n+1)th terms of spectral sequences. To complete the proof apply Proposition 2.1.

4. Transformation of the Functors β : D(A^{∗,−∗}(B, π, n)) → H^{n+1}(B, π) and γ : H^{n+1}(B, π) → D(A^{∗,−∗}(B, π, n))

Consider the homomorphism α : C_{n}(L(π, n)) ⟶ G defined on the n-cubes of L(π, n), τ^{n} = [g], g ∈ G, by α([g]) = g. It induces the homomorphism of cochain complexes β : C^{∗}(B, C_{n}(L(π, n))) ⟶ C^{∗}(B, π). The latter homomorphism defines the map of sets β : A^{∗,−∗}(B, π, n) ⟶ C^{∗}(B, π) by β(a) = β(a^{n+k,−n}) if a = a^{n+k,−n} + a^{n+k+1,−n−1} + · · · . Let a ∈ T (A(B, π, n)). Then |a| = 1, da = aa, and hence δ_{B}a^{n+1,−n} + d_{L}a^{n+2,−n−1} = 0. We see that β(d_{L}a^{n+2,−n−1}) = 0, so that β(δ_{B}a^{n+1,−n}) = 0 and δ_{B}β(a^{n+1,−n}) = 0. Hence, if a ∈ T (A^{∗,−∗}(B, π, n)), then β(a) ∈ Z^{n+1}(B, π).

If ā ∼ a, then β(ā) − β(a) ∈ B^{n+1}(B, π). Indeed, by (2.8) we have d_{A}p = ā − a + pā − ap, and therefore δ_{B}p^{n,−n} + d_{L}p^{n+1,−n−1} = ā^{n+1,n} − a^{n+1,−n}. Hence δ_{B}β(p^{n,−n}) = β(ā^{n+1,−n}) − β(a^{n+1,−n}) which is equivalent to δ_{B}β(p) = β(ā) − β(a). The above shows that the map

β : D(A^{∗,−∗}(B, π, n)) ⟶ H^{n+1}(B, π)

defined by β(a) ∈ β(d), d ∈ D(A^{∗,−∗}(B, π, n)), a ∈ d, is correct.

The second transformation

γ : H^{n+1}(B, π) ⟶ D(A^{∗,−∗}(B, π, n))

is more difficult to define, since it requires the construction of the mappings

Z^{n+1}(B, π) → T^{n+1}(A^{∗,−∗}(B, π, n)),
Z^{n+1}(B, π) × C^{n}(B, π) → G^{n+1}(A^{∗,−∗}(B, π, n)),

with suitable properties.

Lemma 4.1. There is a rule which assigns a twisting cochain a(z^{n+1}) ∈ A^{∗,−∗}(B, π, n) to every cocycle z^{n+1} ∈ Z^{n+1}(B, π). If f : B_{1} → B is a map, then a(f^{∗}(z^{n+1})) = f^{∗}(a(z^{n+1})).

Proof. For z^{n+1} ∈ Z^{n+1}(B, π) we first construct the map

κ_{z^{n+1}} : S(B) ⟶ L(π, n),  (4.1)

κ_{z^{n+1}}(σ^{m+1}) being the m-cube τ^{m} ∈ L(π, n) with the following properties:

(i) κ_{z^{n+1}}(σ^{m+1}) if (m + 1) ≤ n is the unique m-cube of L(π, n) (which is degenerate for m > 0);

(ii) κ_{z^{n+1}}(σ^{n+1}) = [g] ∈ C̃_{n}^{□}(L(π, n)), where g = z^{n+1}(σ^{n+1});

(iii)

\begin{aligned}d_{i}^{1}κ_{z^{n+1}}(σ^{m+1}) & = κ_{z^{n+1}}(σ_{i}^{m+1}), 1 ≤ i ≤ m, \\ d_{1}^{0}κ_{z^{n+1}}(σ^{m+1}) & = κ_{z^{n+1}}(σ_{0}^{m+1}), \\ d_{m}^{0}κ_{z^{n+1}}(σ^{m+1}) & = κ_{z^{n+1}}(σ_{m+1}^{m+1}), \\ d_{i}^{0}κ_{z^{n+1}}(σ^{m+1}) & = κ_{z^{n+1}}(σ_{(1)}^{m+1})κ_{z^{n+1}}(σ_{(2)}^{m+1}), 1 < i < m, \end{aligned} (4.2)

where σ_{(1)}^{m+1} and σ_{(2)}^{m+1} are respectively the first i-dimensional and the last (m + 1 − i)-dimensional face of σ^{m+1}.

With such κ_{z^{n+1}} constructed, the twisting element ã_{z} in the auxiliary algebra Ã^{∗}_{∗}(B, π, n) = C^{∗}(B, C̃_{∗}(L(π, n))) is defined as ã = ã^{2,−1} + ã^{3,−2} + ⋯, where ã^{m+1,−m}(σ^{m+1}) = κ_{z^{m+1}}(σ^{m+1}) ∈ C̃_{m}(L(π, n)).

The above equalities imply that dã = ãã. The image of ã, a(z^{n+1}), with respect to the homomorphism Ã^{∗}_{∗}(B, π, n) ⟶ A^{∗}_{∗}(B, π, n) induced by C̃_{∗}^{□}(L(π, n)) ⟶ C_{∗}^{□}(L(π, n)) is a twisting cochain. By virtue of the fact that all cubes of positive dimension < n are degenerate, we find that

a(z^{n+1}) = a^{n+1,−n} + a^{n+2,−n−1} + ⋯

and

β(a) = β(a^{n+1,−n}) = z^{n+1}.

The rest of the proof consists in constructing κ_{z^{n+1}}.

The geometric background for the (algebraic) definition of κ_{z^{n+1}} is as follows. Consider a cross section s of the fibration over the n-skeleton of |S(B)| such that the obstruction cocycle z^{n+1}(s) is equal to z^{n+1}, z^{n+1} =

z^{n+1}(s). For σ^{m+1} ∈ S(B) consider the map I^{m+1} ⟶^{ψ} Δ_{m+1} ⟶^{σ^{m+1}} |S(B)|.

Consider a subcomplex of n-cubes of the cubical complex I^{m+1} which is sent by ψ in the n-simplex of Δ_{m+1}. By virtue of Lemma 2.1 this subcomplex is the retract of I^{m+1} and therefore the cross section over this subcomplex induced by the cross section s^{n} has an extension over I^{m+1}. Consider its restriction over the n-skeleton of I^{m+1}. On the m-cube (0, x_{2}, x_{3}, ⋯, x_{m+1}) it defines an n-dimensional cell-cochain with coefficients in π_{n}(F) = π. Obviously, this cochain is a cocycle. We can define κ_{z}(σ^{m+1}) to be this cocycle regarded as an m-cube of L(π, n).

The reasoning above suggests the definition of κ_{z^{n+1}} as follows. As above, we consider the map ψ : I^{m+1} ⟶ Δ_{m+1} and also we consider I^{m} as the face (0, x_{2}, x_{3}, ⋯, x_{m+1}). The n-cochain a^{m+1,−m}(σ^{m+1}) = λ^{n} of I^{m} is uniquely defined as follows: it is zero on the n-cubes of I^{m} which have a form differing from

u = (0, ε_{1}, ε_{2}, …, ε_{k}, x_{i_{1}}, 1, 1, 1, …, 1, x_{i_{2}}, 1, 1, 1, …, 1, x_{i_{n}}, ν_{1}, ν_{2}, …, ν_{s}).

For the above n-cube we assume ε_{j} to be the last ε equal to zero in the sequence ε_{1}, ε_{2}, …, ε_{k} and consider the (n + 1)-cube in I^{m+1}

(1, …, 1, x_{i_{0}}, 1, …, 1, x_{i_{1}}, 1, …, 1, x_{i_{2}}, 1, …, 1, x_{i_{n}}, ν_{1}, ν_{2}, …, ν_{s}),

where ε_{j} is replaced by x_{i_{0}}. If all ε’s are equal to 1, we consider the (n + 1)- cube

(x_{i_{0}}, 1, …, 1, x_{i_{1}}, 1, …, 1, x_{i_{2}}, 1, …, 1, x_{i_{n}}, ν_{1}, ν_{2}, …, ν_{s}).

This (n + 1)-cube of I^{m+1} defines via the map ψ^{m+1} the nondegenerate (n+1)-simplex v in the simplex Δ^{m+1} and we set

λ(u) = (σ^{m+1})^{∗}(z^{n+1})(v), σ^{m+1} : Δ^{m+1} → B.

λ^{n} is the cocycle. This is proved directly.

Let κ_{z^{n+1}}(σ^{m+1}) be the cocycle λ^{n} regarded as the m-cube of L(π, n). It remains to show that equalities (4.2) are fulfilled. Each of the four

equalities (4.2) is easy to check by using the algebraic definition of κ_{z^{n+1}}. It is obvious that ã and hence a(z^{n+1}) is functorial.

We next define

Z^{n+1}(B, π) × C^{n}(B, π) → G^{n+1}(A^{∗,−∗}(B, π, n)).

Assume that we are given a cocycle z_{B×I}^{n+1} ∈ Z^{n+1}(B × I, π).

By i_{0} : B ⟶ B × I and i_{1} : B ⟶ B × I we denote the imbeddings of B as the lower and the upper bottom of B × I. We introduce the notation

z_{B}^{n+1} = i_{0}^{∗}z_{B×I}^{n+1} z̄_{B}^{n+1} = i_{1}^{∗}z_{B×I}^{n+1}.

By virtue of Lemma 4.1

a(z_{B}^{n+1}) = i_{0}^{∗}a(z_{B×I}^{n+1}) a(z̄_{B}^{n+1}) = i_{1}^{∗}a(z_{B×I}^{n+1}).

In the lemma below we shall show that a(z_{B}^{n+1}) and a(z̄_{B}^{n+1}) are the equivalent twisting elements of the algebra A^{∗,−∗}(B, π, n).

Let us consider the standard prism construction

w_{1} : C_{∗}(B) ⟶ C_{∗+1}(B × I)

subject to the condition

dw_{1} − w_{1}d = i_{1}^{∗} − i_{0}^{∗}.  (4.3)

To every singular simplex σ^{m} ∈ S(B) this construction assigns the singular (m+1)-chain w_{1}(σ^{m}) which is the image of the main integral (m+1)-chain of the standard triangulation of Δ_{m} ×I by the map σ^{m} ×id : Δ_{m} ×I ⟶ B ×I.

The map

w_{1}^{∗} : A^{∗,−∗}(B × I, π, n) ⟶ A^{∗,−∗}(B, π, n)  (4.4)

can be defined by the composition

C_{B} ⟶^{w_{1}} C_{B×I} ⟶^{x} C_{∗}^{□}(L(π, n)), x ∈ A^{∗,−∗}(B × I, π, n).

From (4.3) we obtain

w_{1}^{∗}d = dw_{1}^{∗} + i_{1}^{∗} − i_{0}^{∗}.  (4.5)

Let

u(z_{B×I}^{n+1}) = w_{1}^{∗}(a(z_{B×I}^{n+1})).

We see that |u| = 0, u ∈ F^{n}(A^{∗,−∗}(B, π, n)).

Lemma 4.2. In the algebra A^{∗,−∗}(B, π, n) there holds the equality

d_{A(B,π,n)}u(z_{B×I}^{n+1}) = a(z̄_{B}^{n+1})−a(z_{B}^{n+1})+u(z_{B×I}^{n+1})a(z̄_{B}^{n+1})−a(z_{B}^{n+1})u(z_{B×I}^{n+1}).

Proof. By Lemma 4.1 we have d_{A(B×I,π,n)}a_{z_{B×I}^{n+1}} = a_{z_{B×I}^{n+1}} a_{z_{B×I}^{n+1}} .

By virtue of the map w_{1}^{∗} and (4.5) the left side of the equality becomes

d(u(z_{B×I}^{n+1})) − a(z̄_{B}^{n+1}) + a(z_{B}^{n+1}).

The right side becomes

u(z_{B×I}^{n+1})a(z̄_{B}^{n+1}) − a(z_{B}^{n+1})u(z_{B×I}^{n+1})  (4.6)

which can be shown as follows. The standard triangulation of Δ_{m} × I has as vertices of the lower bottom those of Δ_{m}, say, b_{0} < b_{1} < b_{2} < ⋯ b_{m}, while the copies of the vertices of Δ_{m}, say, b′_{0} < b′_{1} < b′_{2} < ⋯ b′_{m}, are the vertices of the upper bottom. It is assumed that b(i) < b′(i). Only

(b_{0} < b_{1} < b_{2} < ⋯ b_{i} < b′_{i} < b′_{i+1} < ⋯ b′_{m}), i = 0, 1, ⋯ m

are (m + 1)-dimensional simplices.

By the above definition w_{1}(σ^{m}) is the image of

∑_{0≤i≤m}(−1)^{i}(b_{0} < b_{1} < b_{2} < ⋯ b_{i} < b′_{i} < b′_{i+1} < ⋯ < b′_{m}).

Thus we see that the value of ãã on w_{1}(σ^{m}) is equal to

∑_{0≤i≤m}(−1)^{i}[∑_{j≤i} a′(b_{0} < b_{1} < ⋯ < b_{j}) ·

·a′(b_{j} < b_{j+1} < ⋯ < b_{i} < b′_{i} < b′_{i+1} < ⋯ < b′_{m}) +

+ ∑_{i≤j≤m} a′(b_{0} < b_{1} < b_{2} < ⋯ < b_{i} < b′_{i} < b′_{i+1} < ⋯ < b′_{j}) ·

·a′(b′_{j} < b′_{j+1} < ⋯ < b′_{m})].

Here a′ = (σ^{m} × id)^{∗}ã and the two summands coincide with the two summands in (4.6).

Definition 4.1. Let z_{B}^{n+1} ∈ Z^{n+1}(B, π) and c_{B}^{n} ∈ C^{n}(B, π). On B × I consider the cocycle z_{B×I}^{n+1} = P r^{∗}z_{B}^{n+1}+δ_{B×I}c_{B×I}^{n}, where c_{B×I}^{n} is the cochain c_{B}^{n} imbedded in B × 0. Denote u(c_{B}^{n}, z_{B}^{n+1}) = u(z_{B×I}^{n+1}).

Lemma 4.3. If z_{B}^{n+1} ∈ Z^{n+1}(B, π) and c_{B}^{n} ∈ C^{n}(B, π), then a(z_{B}^{n+1} + δ_{B}c_{B}^{n}) and a(z_{B}^{n+1}) are the equivalent twisting elements, the equivalence being given by the element 1+u(c_{B}^{n}, z_{B}^{n+1}). Hence the map γ : H^{n+1}(B, π) ⟶ D(A^{∗,−∗}(B, π, n)) defined by a(z^{n+1}) ∈ γ(h) if z^{n+1} ∈ h ∈ H^{n+1}(B, π) is correct.

Proof. By the assumptions of the lemma the equality of Lemma 4.2 is identical to

a(z_{B}^{n+1} + δ_{B}c_{B}^{n}) = (1 + u(c_{B}^{n}, z_{B}^{n+1}))^{−1}a(z_{B}^{n+1})(1 + u(c_{B}^{n}, z_{B}^{n+1})) +
+ (1 + u(c_{B}^{n}, z_{B}^{n+1}))^{−1}d_{A}(1 + u(c_{B}^{n}, z_{B}^{n+1})).

5. Algebraic Model

Recall that the bicomplexes Y_{∗∗}(B, π, n) = C_{∗}(B, C_{∗}^{□}(L(π, n))), Y_{∗∗}(B, π, n, G) = C_{∗}(B, C_{∗}^{□}(L(π, n), G)) and Y^{∗∗}(B, π, n, G) = C^{∗}(B, C_{□}^{∗}(L(π, n), G)) are the modules over the bialgebra A^{∗,−∗}(B, π, n) = C^{∗}(B, C_{∗}^{□}(L(π, n))).

Definition 5.1. Let F ⟶ E ⟶ B be a Serre fibration with the fiber F K(π, n)-space. Then π_{i}(F) = 0, i ≠ n and π_{n}(F) = π. Let h^{n+1} ∈ H^{n+1}(B, π) be the obstruction class of the fibration and let z^{n+1} ∈ h^{n+1}. Consider a(z^{n+1}) ∈ T (A^{∗,−∗}(B, π, n)) and the perturbed differential

\begin{matrix}d_{a(z^{n+1})}(y) = d_{Y} (y) + a(z^{n+1})y, \\ y ∈ Y_{∗∗}(B, π, n), Y_{∗∗}(B, π, n, G), Y^{∗∗}(B, π, n, G). \end{matrix}  (5.1)

The complex Y_{∗∗}(B, π, n) with this perturbed differential Y_{z^{n+1}}(B, π, n) = Y_{a(z^{n+1})}(B, π, n) is the (integral) homology model of the fibration. The complex Y_{∗∗}(B, π, n, G) with this perturbed differential Y_{z^{n+1}}(B, π, n, G) = Y_{a(z^{n+1})}(B, π, n, G) is the homology model with coefficients G. The complex Y^{∗∗}(B, π, n, G) with this perturbed differential Y_{z^{n+1}}^{∗∗}(B, π, n, G) = Y_{a(z^{n+1})}^{∗∗}(B, π, n) is the cohomology model of the fibration.

We have

Y_{∗∗z^{n+1}}(B, π, n, G) = Y_{∗∗z^{n+1}}(B, π, n) ⊗ G,
Y_{z^{n+1}}^{∗∗}(B, π, n, G) = Hom(Y_{∗∗z^{n+1}}(B, π, n), G).

The models of fibration depend on the choice of the cocycle in the obstruction class. However, they are isomorphic complexes: if z^{n+1}, z̄^{n+1} are two obstruction cocycles of the same fibration, then there is c^{n} ∈ C^{n}(B, π) with δc^{n} = z̄^{n+1} − z^{n+1}. By virtue of Lemma 4.3 a(z̄^{n+1}) = (1 + u(c^{n}, z^{n+1})) ∘ a(z^{n+1}). Hence the chain map (2.9)

φ_{u(c^{n},z^{n+1})} : Y_{∗∗z̄^{n+1}}(B, π, n, G) ⟶ Y_{∗∗z^{n+1}}(B, π, n, G),
φ_{u(c^{n},z^{n+1})}(y) = (1 + u(c^{n}, z^{n+1}))(y), y ∈ Y_{∗∗}(B, π, n, G)

is an isomorphism.

The same reasoning holds for the cohomology model.

Theorem 5.1. Let F ⟶ E ⟶ B be a Serre fibration with the fiber F K(π, n)-space and let Y_{∗∗z^{n+1}}(B, π, n, G), Y_{z^{n+1}}^{∗∗}(B, π, n, G), z^{n+1} ∈ Z^{n+1}(B, π_{n}(F)), be its homology (resp., cohomology) model. Then:

(i) There are chain and cochain maps defined uniquely up to chain homotopy

Y_{∗∗z^{n+1}}(B, π, n, G) ⟶ C_{∗}^{□}(E, G),
C_{□}^{∗}(E, G) ⟶ Y_{z^{n+1}}^{∗∗}(B, π, n, G)

inducing an isomophism of homology groups.

(ii) if Y_{∗∗z̄^{n+1}}(B, π, n, G), Y_{z̄^{n+1}}^{∗∗}(B, π, n, G) are other models of the fibration and δc^{n} = z̄^{n+1} − z^{n+1}, c^{n} ∈ C^{n}(B, π_{n}(F)), then the triangles

[figure: C_{∗}^{□}(E); Y_{∗∗z^{n+1}}(B, π, n); Y_{∗∗z̄^{n+1}}(B, π, n),; φ_{u(c^{n},z^{n+1})}; C_{□}^{∗}(E, G); Y_{z^{n+1}}^{∗∗}(B, π, n, G); Y_{z̄^{n+1}}^{∗∗}(B, π, n, G); φ_{u(c^{n},z^{n+1})}]

are commutative up to chain homotopy.

The proof below uses the geometric interpretation of the model.

Let K be a simplicial set, z^{n+1} ∈ Z^{n+1}(K, π), and define the cubical complex K ×_{z^{n+1}} L(π, n) as follows. Consider as (p + q)-dimensional cubes of K ×_{z^{n+1}} L(π, n) the pairs (σ^{p}, τ^{q}), where σ^{p} is a p-dimensional simplex of K and τ^{q} is a q-dimensional cube of the cubical set L(π, n).

The face operators are defined by virtue of (2.4) and (4.2) as follows. Let κ_{z^{n+1}} : K ⟶ L(π, n) be map (4.1).

Define

d_{p+i}^{1}(σ^{p}, τ^{q}) = (σ^{p}, d_{i}^{1}τ^{q}), 1 ≤ i ≤ q,
d_{p+i}^{0}(σ^{p}, τ^{q}) = (σ^{p}, d_{i}^{0}τ^{q}), 1 ≤ i ≤ q,
d_{i}^{1}(σ^{p}, τ^{q}) = (σ_{i−1}^{p}, τ^{q}), 1 ≤ i ≤ p,
d_{p}^{0}(σ^{p}, τ^{q}) = (σ_{p}^{p}, τ^{q}),
d_{i}^{0}(σ^{p}, τ^{q}) = (σ_{1}^{p}, κ_{z^{n+1}}(σ_{2}^{p}) ∘ τ^{q}) 1 ≤ i ≤ (p − 1),

where σ_{1}^{p} is the first (i − 1)-face and σ_{2}^{p} is the last (p − i + 1)-face of σ^{p}, while ∘ is the product in the ring C̃_{∗}^{□}(L(π, n)).

The degeneracy operators are defined only partially:

s^{p+i}(σ^{p}, τ^{q}) = (σ^{p}, s^{i}(τ^{q})), 1 ≤ i ≤ (q + 1).

The chain complexes C̃_{∗}^{□}(K ×_{z^{n+1}} L(π, n)), C̄_{∗}^{□}(K ×_{z^{n+1}} L(π, n)), C_{∗}^{□}(K×_{z^{n+1}}L(π, n)), C_{□}^{∗}(K×_{z^{n+1}}L(π, n), G) of K×_{z^{n+1}}L(π, n)) are defined as for the case of cubical sets.

The obvious fact is

Lemma 5.1. The integral chain complex of the cubical complex S(B) ×_{z^{n+1}} L(π, n) is the complex Y_{∗∗z^{n+1}}(B, π, n).

Let |K ×_{z^{n+1}} L(π, n)| be the Milnor realization of this complex (Section 2). Let ||K|| be the Giever–Hu realization of the simplicial set K (i.e., the degeneracy operators are passive). We have the map

α : |K ×_{z^{n+1}} L(π, n)| ⟶ ||K||  (5.2)

defined for every (σ^{p}, τ^{q}) by I^{p+q} ⟶^{proj.} I^{p} ⟶^{ψ} Δ_{p}. The complex |K ×_{z^{n+1}} L(π, n)| is filtered by subcomplexes F^{r} = ∪_{p≤r}|(σ^{p}, τ^{q})|. We have

Lemma 5.2. H_{r+q}(F^{r}, F^{r−1}) = C_{r}(K, H_{q}^{□}(L(π, n))).

Proof. F^{r}/F^{r−1} is a wedge of CW -complexes, one complex K_{σ^{r}} for every σ^{r} ∈ K. Hence H_{r+q}(F^{r}, F^{r−1}) = ∑_{σ^{r}} H_{r+q}(K_{σ^{r}}). The filtration of K_{σ^{r}} by its skeletons gives H_{r+q}(K_{σ^{r}}) = H_{q}^{□}(L(π, n)).

Consider the standard map ||S(B)|| ⟶ B of the Giever–Hu realization of S(B) and also consider the induced fibration E′ ⟶^{pr} ||S(B)||. In the diagram

E ⟵ E′
pr↓  ↓^{pr}
B ⟵ ||S(B)||

the horizontal maps induce an isomorphism of homology. In the induced fibration consider the filtration given by pr^{−1}(||S(B)||^{r}).

Proposition 5.1. There is a commutative diagram

[figure: E′; |S(B) ×_{z^{n+1}} L(π, n)|; ∥S(B)∥ ,; α; pr]

where the upper map is the map of filtered spaces and induces an isomorphism of the first terms of the related spectral sequences.

Proof. The map |S(B) ×_{z^{n+1}} _{z}L(π, n)| ⟶ E′ is constructed by induction on degree of cells. Let s be a cross section over the n-skeleton of ||S(B)|| whose obstruction cocycle is z^{n+1}. The induction steps are as follows:

(0). (σ^{0}, τ^{0}) is a vertex. Let f|(σ^{0}, τ^{0}) = sα|(σ^{0}, τ^{0}) = s(σ^{0}).

(i), i<n. (σ^{j}, τ^{i−j}) are i-dimensional cells. Let f|(σ^{j}, τ^{i−j})=sα|(σ^{j}, τ^{i−j}). (n). (σ^{j}, τ^{n−j}) are n-cells. Let f|(σ^{j}, τ^{n−j}) = sα|(σ^{j}, τ^{n−j}), j < n, and

let f|(σ^{0}, τ^{n}) be the map of this “n-sphere” in a fiber over σ^{0} as an element of π_{n}(F, s(σ^{0})).

(n+1). For the cell (σ^{n+1}, τ^{0}) the map is already defined for its boundary and the image lies over σ^{n+1}. This n-sphere is homotopic to 0 over σ^{n+1} by virtue of the fact that z^{n+1}(σ^{n+1}) ∈ π_{n}(F) is the class of the n-sphere defined by s on the boundary of σ^{n+1}. Hence f extends from the boundary onto the whole cell and the image lies over σ^{n+1}.

The map f of the boundary of (σ^{1}, τ^{n}) is homotopic to 0 and hence it extends onto the whole cell.

For the rest of the (n + 1)-cells (σ^{i}, τ^{n+1−i}), 1 < i ≤ n, we assume that f|(σ^{i}, τ^{n+1−i}) = sα|(σ^{i}, τ^{n+1−i}).

(m), m > n + 1. (σ^{i}, τ^{j}), i + j = m, f already defined on its boundary is an (m − 1)-sphere over σ^{i} and by the fact that m − 1 > n it is homotopic to 0. f extends over the whole cell. The map is now constructed. By the inductive construction we see that it preserves filtrations.

The first term of both filtrations in the proposition under consideration is C_{p}(S(B), H_{q}(F)), and by virtue of the above map the homomorphism of the first terms of the spectral sequences is an isomorphism.

Proof of Theorem 5.1. The standard imbedding

S(B) ×_{z^{n+1}} L(π, n) ⟶ Q(|S(B) ×_{z^{n+1}} L(π, n)|)

gives the chain map

C_{∗}^{□}(S(B) ×_{z^{n+1}} L(π, n)) ⟶ C_{∗}^{□}(|S(B) ×_{z^{n+1}} L(π, n)|)

inducing an isomorphism of cubical singular homologies. Then by Proposition 5.1 the composition

Y_{∗∗z^{n+1}}(B, π, n, G) = C_{∗}^{□}(S(B) ×_{z^{n+1}} L(π, n), G) ⟶

⟶ C_{∗}^{□}(|S(B) ×_{z^{n+1}} L(π, n)|, G) ⟶^{f_{z{n+1 \atop ∗}}} C_{∗}^{□}(E, G)

induces an isomorphism of homology. Let this chain map be the map of (i).

Consider the fibration F ⟶ E′′ ⟶ B × I induced by projection pr : B × I ⟶ B and a map as in part (i) of the theorem:

E′′ ⟵^{f_{z{n+1 \atop B×I}}} |S(B × I) ×_{z_{B×I}^{n+1}} L(π, n)|

where z_{B×I}^{n+1} = pr∗z_{B}^{n+1}−δc̄^{n} (c̄^{n} is c^{n} placed on B×I) is such that i_{0}^{∗}z_{B×I}^{n+1} = z_{B}^{n+1}, i_{1}^{∗}z_{B×I}^{n+1} = z̄_{B}^{n+1}. We can choose this map so that on the bottoms of B × I it will coincide with f_{z^{n+1}} and f_{}_{z̄^{n+1}}. Consider the imbeddings

C_{∗}^{□}(|S(B × I) ×_{z_{B×I}^{n+1}} L(π, n)|) ⟵ C_{∗}^{□}(|S(B) ×_{z_{B}^{n+1}} L(π, n)|),
C_{∗}^{□}(|S(B × I) ×_{z_{B×I}^{n+1}} L(π, n)|) ⟵ C_{∗}^{□}(|S(B) ×_{}_{z̄_{B}^{n+1}} L(π, n)|).

In view of the chain map

C_{∗}^{□}(E) ⟵ C_{∗}^{□}(E′′) ⟵ C_{∗}^{□}(|S(B × I) ×_{z_{B×I}^{n+1}} L(π, n)|)

it is enough to show that the triangle

[figure: C_{∗}^{□}(|S(B × I) ×_{z_{B×I}^{n+1}} L(π, n)|); φ_{u(c^{n},z{n+1 \atop B})}; C_{∗}^{□}(|S(B) ×_{z_{B}^{n+1}} L(π, n)|); C_{∗}^{□}(|S(B × I) ×_{z_{B×I}^{n+1}} L(π, n)|)]

is commutative up to chain homotopy. To this end consider

C_{∗}^{□}(|S(B×I)×_{z_{B×I}^{n+1}} L(π, n)|) ⟵^{φ_{u(c̄^{n},pr^{∗}z{n+1 \atop B})}} C_{∗}^{□}(|S(B×I)×_{pr^{∗}z_{B}^{n+1}} L(π, n)|).

The two imbeddings of C_{∗}^{□}(|S(B) ×_{z_{B}^{n+1}} L(π, n)|) on the left side complex are chain homotopic and the composition gives the required homotopy. This proves (ii) for an integral homology. Hence we obtain (ii) for the coefficient group G and cohomology.

In particular, if z̄^{n+1} = z^{n+1} and c^{n} = 0, then we deduce that the map in (i) is defined uniquely up to chain homotopy.

6. Multiplicative Structure in the Algebraic Model

We have seen that the cohomology model Y_{z^{n+1}}^{∗∗}(B, π, n, G) of a fibration F → E → B with F a K(π, n)-space can be identified with the cochain complex of the cubical complex S(B) ×_{z^{n+1}} L(π, n). Each cubical complex is endowed with multiplicative structure via the Serre cup product. Hence Y_{z^{n+1}}^{∗∗}(B, π, n, Λ), where Λ is a commutative ring, has a multiplicative structure. It can evidently be described as follows.

Let x^{s}, y^{t} ∈ Y_{z^{n+1}}^{∗∗}(B, π, n, Λ). Being an element of the module C^{p}(B, C^{q}(L(π, n)Λ)), the (p, q)-component, where p + q = s + t, of the product w^{s+t} = x^{s}y^{t} is a p-dimensional cochain of B with coefficients in C^{q}(L(π, n), Λ) defined in the following manner: for the pair (σ^{p}, τ^{q}) consider all decompositions of 1, 2, ⋯, (p + q) in two disjoint sets (H, K) and let

w^{s+t}[(σ^{p}, τ^{q})] = ∑_{(H,K)}(−1)^{a(H,K)}x(d_{K}^{0}(σ^{p}, τ^{q}))y(d_{H}^{1}(σ^{p}, τ^{q})).

Theorem 6.1. The cochain map of Theorem 5.1

C_{□}^{∗}(E, Λ) ⟶ Y_{z^{n+1}}^{∗∗}(B, π, n, Λ)

is multiplicative.

Proof. The above cochain map is induced (see the proof of Theorem 5.1) by the map of cubical sets S(B) ×_{z^{n+1}} L(π, n) ⟶ Q(E).

7. Action of the Group H^{n}(B, π) on the Homology of the Complexes Y_{∗∗z^{n+1}}(B, π, n, G) and Y_{z^{n+1}}^{∗∗}(B, π, n, G)

Consider the space B × Δ_{2} and assume that we are given a cocycle z_{B×Δ^{2}}^{n+1} ∈ Z^{n+1}(B × Δ_{2}, π).

We have three imbeddings i_{0}, i_{1}, i_{2},: B → B × Δ^{2} for three vertices of Δ_{2} and three imbeddings i_{01}, i_{12}, i_{02},: B × I → B × Δ^{2} for three 1-faces of Δ_{2}. For a given cocycle z_{B×Δ^{2}}^{n+1} ∈ Z^{n+1}(B × Δ^{2}) consider six (n+1)-cocycles (three on B and three on B × I):

z_{B,0}^{n+1} = i_{0}^{∗}(z_{B×Δ^{2}}^{n+1}), z_{B×I,01}^{n+1} = i_{01}^{∗}(z_{B×Δ^{2}}^{n+1}),
z_{B,1}^{n+1} = i_{1}^{∗}(z_{B×Δ^{2}}^{n+1}), z_{B×I,12}^{n+1} = i_{12}^{∗}(z_{B×Δ^{2}}^{n+1}),
z_{B,2}^{n+1} = i_{2}^{∗}(z_{B×Δ^{2}}^{n+1}), z_{B×I,02}^{n+1} = i_{02}^{∗}(z_{B×Δ^{2}}^{n+1}).

The left cocycles define on B three twisting elements a(z_{B,0}^{n+1}), a(z_{B,1}^{n+1}), a(z_{B,2}^{n+1}), while the right cocycles define on B three 0-elements as in Lemma 4.2:

u_{01}(z_{B×I,01}^{n+1}), u_{12}(z_{B×I,12}^{n+1}), u_{02}(z_{B×I,02}^{n+1}) ∈ A^{∗,−∗}(B, π, n) =
= C^{∗}(B, C_{∗}^{□}(L(π, n))).

Consider the standard map

w_{2} : C_{∗}(B) ⟶ C_{∗+2}(B × Δ_{2})

subject to the condition

d_{B×Δ_{2}}w_{2} − w_{2}d_{B} = (i_{01}^{∗} + i_{12}^{∗} − i_{02}^{∗})w_{1}.  (7.1)

To every singular simplex σ^{m} ∈ S(B) this map assigns the singular (m+2)- chain w_{2}(σ^{m}) which is the image of the main integral (m + 2)-chain of the standard triangulation of Δ_{m}×Δ_{2} by the map σ^{m}×id : Δ_{m}×Δ_{2} ⟶ B×Δ_{2}.

Define the map

w_{2}^{∗} : A^{∗,−∗}(B × Δ_{2}, π, n) ⟶ A^{∗,−∗}(B, π, n)

by the composition

C_{∗}(B) ⟶^{w_{2}} C_{∗}(B × Δ_{2}) ⟶^{x} C_{∗}^{□}(L(π, n)), x ∈ A(B × Δ_{2}, π, n).

Let v(z_{B×Δ_{2}}^{n+1}) = w_{2}^{∗}(a(z_{B×Δ_{2}}^{n+1})). We see that |v| = −1 and v = v^{0,−1} + v^{1,−2} + v^{2,−3} + · · ·.

Lemma 7.1. In the above notation we have the equality

d_{A(B,π,n)}v = u_{01} + u_{12} + u_{01}u_{12} − u_{02} + va(z_{B,0}^{n+1}) + a(z_{B,2}^{n+1})v

in the algebra A^{∗,−∗}(B, π, n).

Proof. The lemma is proved similarly to Lemma 4.2, using equality (7.1).

In what follows we abbreviate Y_{∗∗z^{n+1}}(B, π, n) to Y_{z^{n+1}}.

In the situation we are considering we have the triangle of chain isomorphisms

[figure: Y_{z_{B,0}^{n+1}}; Y_{z_{B,1}^{n+1}}; Y_{z_{B,2}^{n+1}} .; φ_{u_{12}}; φ_{u_{02}}; φ_{u_{01}}]

Corollary 7.1. The above triangle is commutative up to chain homotopy.

Proof. By virtue of Lemma 7.1 the map F : Y_{z_{B,0}^{n+1}} → Y_{z_{B,2}^{n+1}} defined by F (x) = vx yields a chain homotopy between the composition φ_{u_{12}}φ_{u_{01}} and φ_{u_{02}}.

Corollary 7.2. Let z_{B}^{n+1} ∈ Z^{n+1}(B, π) and c^{n}, c_{1}^{n} ∈ C^{n}(B, π). Then the triangle

[figure: Y_{z^{n+1}+δ(c^{n}+c_{1}^{n})}; Y_{z^{n+1}+δc^{n}}; Y_{zn+1}; φ_{u(c^{n},z^{n+1})}; φ_{u(c^{n}_{1}+c^{n},z^{n+1})}; φ_{u(c^{n}_{1},z^{n+1}+δc^{n})}]

is commutative up to chain homotopy.

Proof. Let P r : B × Δ^{2} → B be the projection and consider the cocycle of B × Δ^{2} z_{B×Δ^{2}}^{n+1} = P r^{∗}(z_{B}^{n+1}) + δ_{B×Δ^{2}}(c̄^{n}) + δ_{B×Δ^{2}}(¯c̄^{n} + ¯ c̄_{1}^{n}), c̄^{n} being c^{n} identified as the cochain of the subcomplex B × 1 and zero otherwise, and ¯c̄^{n} + ¯ c̄_{1}^{n} being c^{n} + c_{1}^{n} identified as the cochain of B × 2 and zero otherwise. The rest follows from Corollary 7.1.

Corollary 7.3. Let z_{B}^{n+1} ∈Z^{n+1}(B, π) and 0∈C^{n}(B, π); then φ_{u(0,z^{n+1})} : Y_{z_{B}^{n+1}} → Y_{z_{B}^{n+1}} is homotopic to the identity.

Proof. From 0 = 0 + 0 and Corollary 7.1 we see that φ_{u(0,z^{n+1})}φ_{u(0,z^{n+1})} ∼ φ_{u(0,z^{n+1})}; hence by the fact that φ_{u(0,z^{n+1})} is surjective, φ_{u(0,z^{n+1})} is homotopic to the identity (in fact, φ_{u(0,z^{n+1})} is the identity, but we do not need this).

Corollary 7.4. For z_{B}^{n+1} ∈ Z^{n+1}(B, π) and c^{n} ∈ C^{n}(B, π) the triangle

[figure: Y_{z^{n+1}}; Y_{z^{n+1}−δc^{n}}; Y_{zn+1}; φ_{u(c^{n},z^{n+1})}; id; φ_{u(−c^{n},z^{n+1}−δc^{n})}]

is commutative up to chain homotopy.

Proof. By virtue of Corollary 7.2 the composition in question is homotopic to φ_{u(c^{n}−c^{n},z^{n+1})} = φ_{u(0,z^{n+1})}. Corollary 7.3 accomplishes the proof.

Corollary 7.5. For z_{B}^{n+1} ∈ Z^{n+1}(B, π) and c^{n−1} ∈ C^{n−1}(B, π) the map φ_{u(δc^{n−1},z^{n+1})} : Y_{z_{B}^{n+1}} → Y_{z_{B}^{n+1}} is homotopic to the identity.

Proof. On the complex B×I consider an n-cochain k^{n} = δ_{B×I}c^{n−1}−δ_{B}c^{n−1}; here c^{n−1} and δ_{B}c^{n−1} are identified as cochains in B ×0 and zero otherwise. Then δ_{B×I}k^{n} = δ_{B×I}(δ_{B}c^{n−1}). Identify k^{n} as a cochain of B × Δ^{2} via i_{01} : B × I → B × Δ^{2}. Consider z_{B×Δ^{2}}^{n+1} = P r^{∗}z_{B}^{n+1} − δ_{B×Δ^{2}}k^{n}. The restrictions of this cocycle on (B×I)_{02} and (B×I)_{12} are equal to P r^{∗}(z_{B}^{n+1}) and the restriction on (B ×I)_{01} is P r^{∗}z^{n+1} −δ_{B×I}(δ_{B}c^{n}). Hence Corollaries 7.2 and 7.4 carry the proof to the end.

Theorem 7.1. For every z^{n+1} ∈ Z^{n+1}(B, π) and every abelian group G the cohomology group H^{n}(B, π) acts from the left on the homology and cohomology groups H_{q+1}(Y_{z^{n+1}}, G), H^{q+1}(Y_{z^{n+1}}, G). The action is given by the chain map

φ_{u(c^{n},z^{n+1})} : Y_{z^{n+1}} ⟶ Y_{z^{n+1}}, φ_{u(c^{n},z^{n+1})}(y) = (1 + u(c^{n}, z^{n+1}))(y), c^{n} ∈ Z^{n}(B, π), y ∈ Y_{z^{n+1}}.

Proof. We readily prove the theorem, using the above corollaries.

Acknowledgement

The research described in this publication was made possible in part by Grant No. RVC000 from the International Science Foundation.

For the improvement of the proof of Proposition 5.1 the author is obliged to S. Saneblidze.

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(Received 21.09.1994)

Author’s address:

A. Razmadze Mathematical Institute, Georgian Academy of Sciences 1, Z. Rukhadze St., Tbilisi 380093, Republic of Georgia