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Effective chain complexes for twisted products

In the paper weak sufficient conditions for the reduction of the chain complex of a twisted product to a free finitely generated chain complex are found.


Introduction
When making algorithmic calculations with simplicial sets in algebraic topology on the level of chain complexes, it is often useful to replace the chain complex C * (X) associated to a simplicial set X with another chain complex EC * where all the groups EC n are finitely generated free abelian. Such chain complexes are called effective. This replacement is usually obtained using a reduction or a strong equivallence.
It is therefore natural to ask if standard topological constructions with simplicial sets are reflected by our replacements. For example by the theorem of Eilenberg and Zilber we know that given simplicial sets X, Y and their effective chain complexes EC * (X), EC * (Y ), the simplicial set X × Y has an effective chain complex EC * (X) ⊗ EC * (Y ).
Let F → E → B be a Kan fibration of simplicial sets. By [6] we may think of the total space E as E = F × τ B, i.e. a twisted cartesian product. We want to find an effective chain complex of the total space E from the knowledge of effective chain complexes of F and B and the twisting operator τ .
In [9] (Theorem 132) the solution of this problem was given in the case when the space B is 1-reduced, which means that the 1-skeleton of B is a point. However, this condition seems to be too restrictive and not necessary. For example if we aim to generalize the results in the paper [1] and construct an equivariant version of the Postnikov tower one cannot assume the base spaces are even 0-reduced (see [2]). In Theorem 10 and Corollary 12 we give weaker conditions under which an effective chain complex for the twisted cartesian product can be found. Our approach is based on the results by Shih as presented in [10] and on the approach from the paper [5].

Basic notions
Let (C * , ∂), (D * , ∂) be chain complexes. The triple of maps ρ = (f, g, h) where f : C * → D * , g : D * → C * are chain homomorphisms and h : C * → C * +1 is a chain homotopy such that gf − id C * = ∂h + h∂, hh = 0, is called a reduction. The chain complex D * is said to be a reduct of C * . We will denote this by C * ⇒ D * . This definition of reduction coincides with the one given in [9], Definition 42 or [5], 2.1. It is easy to observe that a composition of reductions is a reduction. We say, there is a strong equivalence between chain complexes C * and C ′ * if there exists a chain complex D * together with two reductions ρ 1 = (f 1 , g 1 , h 1 ) : D * ⇒ C * and ρ 2 = (f 2 , g 2 , h 2 ) : D * ⇒ C ′ * . We denote this by C * ⇐ D * ⇒ C ′ * or C * ⇔ C ′ * . The following lemma shows that strong equivalences are in some sense composable.
We omit the proof, it can be found in [9]. We will make use of the following "tensor product" of reductions.
Further we will deal only with chain complexes which are formed by free abelian groups. For any simplicial set X there is a canonically associated chain complex C * (X) where the group C n (X) is freely generated by nondegenerate n-simplices of X and the boundary homomorphism ∂ n is induced by face maps in X n as follows Let (C * , ∂) be a chain complex. A collection of maps δ n : C n → C n−1 is called a perturbation if (∂ n + δ n ) 2 = 0 for all n ∈ N. We will now introduce the Basic Perturbation Lemma. It is a powerful tool that enables us to construct new reductions.
Proof. The maps involved in the reduction ρ ′ are given explicitely as follows: The proof can be found in [9], Theorem 50.
On the other hand, if we add a perturbation to the differential of the other chain complex, we easily get the following result: Lemma 4 (Easy Perturbation Lemma). Let ρ = (f, g, h) : (C * , ∂) ⇒ (D * , ∂ ′ ) be a reduction and let δ ′ be a perturbation of the differential ∂ ′ . Then there is a reducion ρ = (f, g, h) : The difficulty with the BPL consists in the fact that it is sometimes difficult to verify the nilpotency assumption. Instead of looking for a description of (hδ) we can find a filtration and check how how the perturbation δ changes the filtration.

twisting cochains
The twisted cartesian product (TCP) is defined as follows: Definition 6. Let B, F be simplicial sets and G a simplicial group with a right action · : F × G → F . A function τ n : B n → G n−1 , n ≥ 1, is said to be a twisting operator, if it satisfies the following properties: ( where e m is the unit element of G m . The twisted cartesian product with the base B, the fiber F and the group G is a simplicial set denoted E or F × τ B where E n = F n × B n has the following face and degeneracy operators: The face and degeneracy operations on E naturally define a differential ∂ τ on the We now introduce the following notation: If X is a simplicial set and x ∈ X n we putd n−i x = d i+1 · · · d n x andd 0 x = x. Given (x, y) ∈ (X × Y ) n we define the Alexander-Whitney operator: For a non-twisted product F × B, there exists a reduction , known as the Eilenberg-Zilber reduction. For the full description of the reduction see [3].
The only difference between the chain complexes (C * (F × τ B), ∂ τ ) and (C * (F × B), ∂) is in their differentials and it is easy to see that So the differential ∂ τ of C * (E) is just the ∂ with the added perturbation Proposition 7 ([9], Theorem 131). Let F × τ B be a twisted product of simplicial sets. Then the Basic Perturbation Lemma can be applied to the reduction data (AW, EML, SH) : According to [10], the perturbation ∂ F τ − ∂ F ⊗ can be seen as a cap product with so called twisting cochain, which is induced by τ . We will now give definitions of those notions.
Let t : C * (B) → C * −1 (G) be a sequence of abelian group homomorphisms t n : We define a few operators that will be used within the construction: Finally, we define the cap product (t∩ ) : . Observe, that the cap product is a homomorphism of graded abelian groups and not of chain complexes. We say that t is a twisting cochain if (∂ F ⊗ + (t∩)) 2 = ∂ F ⊗ (t∩) + (t∩)∂ F ⊗ + (t∩)(t∩) = 0. We saw that the twisting operator τ induces via the BPL a new differential ∂ F τ on the chain complex C * (F ) ⊗ C * (B). Then the same twisting operator τ (this time seen as a part of the twisted cartesian product G × τ B) also induces a differential ∂ G τ on the chain complex C * (G) ⊗ C * (B).
According to [10], the twisting operator induces a twisting cochain t : C * (B) → C * −1 (G) as follows: where e 0 is the unit element of G 0 , λ 0 is a projection on the summand C n−1 (G)⊗C 0 (B) of the sum and p(x ⊗ b) = (εb)x where the map ε : C 0 (B) → Z is the augmentation.
The following proposition was formulated and proved by Shih in [10] and describes the relation between t and ∂ F τ . Proposition 8 ([10], Theorem 2). Let F × τ B be a TCP and let t be the twisting cochain induced by the differential ∂ G τ of the chain complex C * (G)⊗ C * (B). Then Let E = F × τ B be a twisted product of simplicial sets, t be a twisting cochain induced by the differential ∂ G τ on the chain complex C * (G)⊗C * (B) and b ∈ B n , y ∈ F k . Then using the definition of AW and t∩ together with the fact that t(d n b) = 0 we obtain the following formula: Using this formula we can summarize some properties of t∩.

Corollary 9 ([5], Lemma 3.4)
. Let E = F × τ B be a twisted product of simplicial sets and let t be a twisting cochain induced by the differential ∂ G τ on the chain complex C * (G)⊗ C * (B). Then the following holds: (1) The perturbation (t∩) : C * (F ) ⊗ C * (B) → C * (F ) ⊗ C * (B) lowers the filtration degree by at least one. (2) If for all b ∈ B 1 , t(b) = 0, then the perturbation (t∩) lowers the filtration degree by at least two.
Proof. The first part is clear by the formula (1). If t(d n−1 b) = 0 for all b ∈ B n , then which proves the second part.

Effective chain complex for twisted product
We would like to find an answer to the following problem: Let B and F be simplicial sets, G a simplicial group, E = F × τ B a TCP, and ρ B : C * (B) ⇒ EC * (B), ρ F : C * (F ) ⇒ EC * (F ) be reductions to effective chain complexes. Is there a reduction of the chain complex C * (E) to an effective chain complex which can be obtained from ρ B , ρ F and τ by the application of the Basic Perturbation Lemma?
Our aim is to find an answer using the composition of given reductions. Having reductions ρ B , ρ F we can by the Lemma 2 construct the reduction We know that the chain homotopy h F ⊗B from the reduction ρ F ⊗B raises the filtration degree by at most 1. This follows from the fact that h B raises the filtration degree by at most 1 and the proof of Lemma 2. We can use the BPL to construct a reduction ρ E = (f, g, h) : C * (E) ⇒ C * (F )⊗ C * (B). From Corollary 9, the perturbation operator ∂ F τ − ∂ F ⊗ = t∩ lowers the filtration degree by at least one. If the composition h F ⊗B • (∂ F τ − ∂ F ⊗ ) decreased the filtration, it would be nilpotent and hence we could use the BPL on the reduction data ρ F ⊗B and the perturbation ∂ F τ − ∂ F ⊗ to get a reduction ρ t : to an effective chain complex EC * (F )⊗ EC * (B) which is EC * (F ) ⊗ EC * (B) with a new differential obtained from the BPL. However, in full generality h F ⊗B •(∂ F τ −∂ F ⊗ ) = h F ⊗B • (t∩) preserves the filtration degree.
From (1) we see that in the composition h F ⊗B • (t∩)(y ⊗ b), where b ∈ B n , there is only one element with the filtration degree n, namely Now we can establish conditions for (h F ⊗B • (t∩)) i to decrease the filtration and prove the following theorem.
Theorem 10. Let B and F be simplicial sets, G a simplicial group with an action on F , E = F × τ B a TCP, and ρ B : then there is a reduction from the chain complex C * (E) to an effective chain complex EC * (F )⊗ EC * (B) which can be obtained from ρ B , ρ F and τ by the application of the Basic Perturbation Lemma.
can be reduced to an effective chain complex using the BPL.
Proof. If the reduction ρ B is trivial, then the chain homotopy h B is trivial, so h B = 0 and hence b 1 = h B d 0 = 0. To prove the case when G is 0-reduced we compute t(b) where b ∈ B 1 . According to the definition we get . From the Basic Perturbation Lemma we get As the operator SH = 0 on (F × B) 0 the only nonzero term of (∂ If the group G is 0-reduced, τ (b) = e 0 as e 0 is the only element in G 0 and we have t(b) = 0 for b ∈ B 1 . That is why y 1 = g F f F σ(y ⊗ t(d n−1 b)) = 0 and we can apply the previous theorem. Now we turn to strong equivalences.
Corollary 12. Let B and F be simplicial sets, G a simplicial group, E = F × τ B a TCP, and C * (B) ⇔ EC * (B), C * (F ) ⇔ EC * (F ) strong equivalences with effective chain complexes. If G is 0-reduced or ρ B is trivial (i.e. EC * (B) = C * (B) and all reductions are trivial) then C * (F × τ B) is strongly equivalent to an effective chain complex EC * (F )⊗ EC * (B) which can be obtained from the strong equivalences for C * (B) and C * (F ) representing C * (E) and an effective chain complex using the Basic and Easy Perturbation Lemmas.
Proof. By Proposition 7 we have a reduction C * (F × τ B) ⇒ C * (F )⊗ C * (B). Since strong equivalences are composable, it remains to show that there is a strong equiva- Having strong equivalences C * (B) ⇐ D * (B) ⇒ EC * (B) and C * (F ) ⇐ D * (F ) ⇒ EC * (F ) then by Lemma 2 there is a strong equivalence consisting of two reductions: Given the perturbation (t∩) on the chain complex C * (F ) ⊗ C * (B), we can use the Easy Perturbation Lemma on the reduction ρ 1 = (f 1 , g 1 , h 1 ) : where we introduce a perturbation g 1 (t∩)f 1 to the differential of the chain complex D * (F )⊗D * (B) and the reduction data remains unchanged. If the nilpotency condition of the composition (g 1 (t∩)f 1 ) • h 2 was satisfied, we could apply the Basic Perturbation Lemma on the reduction data ρ 2 = (f 2 , g 2 , h 2 ) : . If G is 0-reduced, then the filtration degree of the perturbation g 1 (t∩)f 1 is −2 by Corollaries 9 and 11 and as the the filtration degree of h 2 is +1, the nilpotency condition is satisfied. For ρ B trivial, h 2 is 0 and the nilpotency condition is trivially satisfied.

Vector fields
We will now deal with the case in which we have more information about the reduction ρ B : C * (B) ⇒ EC * (B). In particular, ρ B is obtained via a discrete vector field. A discrete vector field V on a simplicial set X is a set of ordered pairs (σ, τ ), where σ, τ are nondegenerate simplices of X, σ = d i τ for exactly one index i and for every two distinct pairs (σ, τ ), (σ ′ , τ ′ ) we have σ ′ = σ, τ ′ = τ, σ ′ = τ and τ ′ = σ. By writing V (σ) = τ , we mean (σ, τ ) ∈ V . Given a dicrete vector field V , the nondegenerate simplices of X are divided into three subsets S, T , C as follows: • S is the set of source simplices i.e. the simplices σ such that (σ, τ ) ∈ V , • T is the set of target simplices i.e. the simplices τ such that (σ, τ ) ∈ V , • C is the set of critical simplices i.e the remaining ones, not occuring in any edge of V . A discrete vector field V on a simplicial set X induces a reduction ρ X = (h X , f X , g X ) : C * (X) ⇒ D * (X) (see [7], [4]). We will concentrate on the properties of the induced chain homotopy h X . It turns out that h X (σ) ∈ ZT for any σ and more importantly h X (σ) = 0 whenever σ ∈ C ∪ T .
Definition 13. Let X be a simplicial set. For any nondegenerate simplex σ ∈ X n we will consider the following condition: We say that a discrete vector field V on a simplicial set satisfies ( * ) if all nondegenerate simplices of X satisfy ( * ).
Corollary 14. Let B and F be simplicial sets, G a simplicial group, E = F × τ B a TCP and ρ B : C * (B) ⇒ EC * (B), ρ F : C * (F ) ⇒ EC * (F ) be reductions to effective chain complexes. If the reduction ρ B is induced by a vector field satisfying ( * ), then there exists a reduction from the chain complex C * (E) to an effective chain complex which can be obtained from ρ B , ρ F and τ .
Proof. We show that (h B d 0 ) 2 = 0. Under our conditions for any b ∈ B n we have b 1 = h B (d 0 b) ∈ ZT . As h B satisfies ( * ), we see that d 0 b 1 ∈ Z(C ∪ T ) and consequently, b 2 = h B d 0 b 1 = 0 and we can apply Theorem 10.
Now we can classify the simplices: • σ ∈ S has the form [a 1 | . . . |a n ], where a n = 1 and n > 0.
• σ ∈ C is [] and [1]. It is easy to check that the vector field V EML satisfies ( * ). Note that Corollary 14 implies that for any E = F × τ K(Z, 1) there is a reduction C * (E) ⇒ EC * (E) to an effective chain complex if there is a reduction C * (F ) ⇒ EC * (F ) to an effective chain complex.