Secondary cohomology and the Steenrod square

To Jan–Erik Roos on his sixty–fifth birthday

Hans–Joachim Baues

Abstract

For a field k we choose the Eilenberg–MacLane space Z^{n} = K(k, n) by the realization of the simplicial k–vector space generated by the non–basepoint singular simplices of the n–sphere S^{n} = S^{1} ∧ … ∧ S^{1}. The permutation of the smash product factors S^{1} yields an action of the symmetric group σ_{n} on S^{n} and hence on Z^{n}. Moreover the quotient map S^{n} × S^{m} → S^{n+m} induces a cup product map μ : Z^{n} × Z^{m} → Z^{m+n}; see the Appendix below.

It is well known that the (reduced) cohomology H̃^{n}(X, k) of a pointed space X is the same as the set [X, Z^{n}] of homotopy classes {x} of pointed maps x : X → Z^{n}. Moreover the cup product of the cohomology algebra H^{∗} = H^{∗}(X, k) = H̃^{∗} ⊕ k is induced by the map μ, that is {x} ∪ {y} = {μ(x, y)}. The cohomology algebra is graded commutative in the sense that

{x} ∪ {y} = (−1)^{nm}{y} ∪ {x}

In this paper we replace the homotopy set [X, Z^{n}] by the groupoid [[ X, Z^{n}]] . The objects of this groupoid are the pointed maps x : X → Z^{n} and the morphisms x ⇒ y in [[ X, Z^{n}]] are the homotopy classes of homotopies x ≃ y termed tracks. The set of path components of [[ X, Z^{n}]] is

π_{0} [[ X, Z^{n}]] = [X, Z^{n}] = H̃^{n}

and the group of tracks 0 ⇒ 0 of the trivial map 0 : X → ∗ → Z^{n} in [[ X, Z^{n}]] is

π_{1} [[ X, Z^{n}]] = [X, ΩZ^{n}] = H̃^{n−1}

We associate with [[ X, Z^{n}]] the exact sequence H^{n}(X):

0 → H̃^{n−1} → H^{n}(X)_{1} ⟶^{∂} H^{n}(X)_{0} → H̃^{n} → 0

Here H^{n}(X)_{0} is the set of all pointed maps X → Z^{n} and H^{n}(X)_{1} is the set of pairs (x, H) with H : x ⇒ 0 and ∂(x, H) = x.

The Eilenberg–MacLane spaces Z^{n} have the following basic properties:

(a) Z^{n} is a k–vector space object in the category Top^{∗} of pointed spaces.

(b) The symmetric group σ_{n} acts on Z^{n} via linear automorphisms inducing the sign of a permutation on H_{n}(Z^{n}).

(c) The cup product map μ : Z^{n} × Z^{m} → Z^{n+m} is k–bilinear and equivariant with respect to the inclusion σ_{n} ×σ_{m} ⊂ σ_{n+m}. Moreover μ is associative in the obvious sense and the following diagram commutes.

\xymatrix{Z^{n} × Z^{m} \ar[r]^{μ} \ar[d]_{T} & Z^{m+n} \ar[d]^{τ_{n,m}} \\ Z^{m} × Z^{n} \ar[r]^{μ} & Z^{m+n}}

The map T is the interchange map T (x, y) = (y, x) and τ_{n,m} ∈ σ_{n+m} is the element interchanging the first n–block with the second m–block.

Properties (b) and (c) imply that {Z^{n}, n ≥ 0} is a “symmetric spectrum” in the sense of Hovey–Shipley–Smith [HSS] 1.2.5. We use the poperties (a), (b) and (c) of Z^{n} to show that the graded object

H^{∗}(X) = {H^{n}(X), n ≥ 1}

has the structure of a “secondary algebra” which we call the secondary cohomology of the space X. Using secondary algebras we introduce the third symmetric cohomology SH^{3} of a graded commutative algebra and we show that the secondary cohomology H^{∗}(X) represents an element

<H^{∗}(X)> ∈ SH^{3}(H^{∗}, H̃^{∗}[1])

which is an invariant of the homotopy type of X. There is a natural transformation from the symmetric cohomology SH^{3} to the Hochschild cohomology HH^{3} which carries the class <H^{∗}(X)> to the class

<C^{∗}(X)> ∈ HH^{∗}(H^{∗}, H̃^{∗}[1])

defined by the algebra of cochains C^{∗}(X) of the space X. It is known that the class <C^{∗}(X)> determines all triple Massey products in the cohomology H^{∗}(X, k), see for example Berrick Davydov [BD] or Baues–Minian [BM]. The new class < H^{∗}(X) > in addition determines for k = 𝔽_{2} the Steenrod operations

Sq^{n−1} : H^{n} → H^{2n−1}, n ≥ 1.

The Hochschild cohomology HH^{∗} is defined for algebras and graded algebras in general while the symmetric cohomology SH^{3} is only defined for commutative graded algebras.

1 Secondary modules

Motivated by properties of Eilenberg–MacLane spaces in topology we introduce the algebraic concept of a secondary module. Later we will consider functors from the category of spaces to the categories of secondary modules and secondary algebras respectively.

Let k be a field an let R be a k–algebra with unit i and augmentation ε

k ⟶^{i} R ⟶^{ε} k.  (1.1)

Here i and ε are algebra maps with εi = 1. For example let G be a group together with a homomorphism ε : G → k^{∗} where k^{∗} is the group of units in the field k. Then ε induces an augmentation

ε : k[G] → k  (1)

where k[G] is the group algebra of G. Here k[G] is a vector space with basis G and ε carries the basis element g ∈ G to ε(g). In particular we have for the symmetric group σ_{n} (which is the group of bijections of the set {1, …, n}) the sign–homomorphism

sign : σ_{n} → {1, −1} → k^{∗}  (2)

which induces the sign–augmentation

ε = ε_{sign} : k[σ_{n}] → k  (3)

These examples play a special role in applications of topology below.

For k–vector spaces A, B we use the tensor product

A ⊗ B = A ⊗_{k} B  (1.2)

A homomorphism f : A → B is termed a k–linear map. If A and B are R–modules then the map f is R–linear if in addition f(r · x) = r · f(x) for r ∈ R, x ∈ A. If R and K are k–algebras then also R ⊗ K is a k–algebra with augmentation

ε : R ⊗ K ⟶^{ε⊗ε} k ⊗ k = k

The multiplication in R ⊗ K is defined as usual by (α ⊗ β) · (α′ ⊗ β′) = (αα′) ⊗ (ββ′). Moreover if X is an R–module and Y is a K–module then X ⊗ Y is an R ⊗ K–module by (α ⊗ β) · (x ⊗ y) = (αx) ⊗ (βy). (1.3) Definition. Let R be a k–algebra. A secondary module X = X_{R} over R consists of a diagram

R ⊗ X_{0} ⟶^{Γ} X_{1} ⟶^{∂} X_{0}

where X_{0} and X_{1} are R–modules and ∂ is R–linear and Γ is k–linear such that for r, r′ ∈ R, a ∈ X_{1}, x ∈ X_{0} the following equations hold.

∂Γ(r ⊗ x) = (r − ε(r))x  (1)
Γ(r ⊗ ∂a) = (r − ε(r))a  (2)
Γ((r · r′) ⊗ x) = rΓ(r′ ⊗ x) + ε(r′)Γ(r ⊗ x)  (3)
Γ((r · r′) ⊗ x) = Γ(r ⊗ r′x) + ε(r)Γ(r′ ⊗ x)  (4)

Now let X_{R} and Y_{K} be secondary modules over R and over K respectively. A map between secondary modules

f = f_{h} : X_{R} → Y_{K}  (5)

consists of an augmented algebra map h : R → K and a commutative diagram

\xymatrix{R ⊗ X_{0} \ar[r]^{Γ} \ar[d]_{h⊗f_{0}} & X_{1} \ar[r]^{∂} \ar[d]^{f_{1}} & X_{0} \ar[d]^{f_{0}} \\ k ⊗ Y_{0} \ar[r]^{Γ} & Y_{1} \ar[r]^{∂} & Y_{0}}  (6)

where f_{1} and f_{0} are k–linear and h–equivariant, i. e. f_{i}(r · b) = h(r) · f_{i}(b) for r ∈ R, b ∈ X_{i} and i = 0, 1. Let secmod be the category of secondary modules and let secmod(R) be the subcategory of secondary modules over R and R–equivariant maps f = f_{h} for which h is the identity of R.

One readily checks that secmod(R) is an additive category (in fact an abelian category) with the direct sum X_{R} ⊕ Y_{R} given by

\xymatrix{R ⊗ (X_{0} ⊕ Y_{0}) \ar[r]^{Γ⊕Γ} & X_{1} ⊕ Y_{1} \ar[r]^{∂⊕∂} & X_{0} ⊕ Y_{0}}  (1.4)

Moreover for a map f : X_{R} → Y_{R} in secmod(R) the secondary modules kernel(f) and cokernel(f) are defined in secmod(R) by using kernel(f_{i}) and cokernel(f_{i}) for i = 0, 1 in the obvious way. (1.5) Remark. Let R = k[G] be a group algebra augmented by ε : G → k^{∗} as in (1.1). Then a secondary module X_{R} over R can be identified with a diagram

G × X_{0} ⟶^{Γ} X_{1} ⟶^{∂} X_{0}

where X_{1}, X_{0} are k–vector spaces with an action of G via k–linear automorphisms and where ∂ is k–linear and G–equivariant. Moreover G × X_{0} is the product set and Γ is a function between sets which is k–linear in X_{0} (i. e. for g ∈ G the function X_{0} → X_{1}, x ↦ Γ(g, x) is k–linear). Moreover for g, g′ ∈ G the following equations hold.

∂Γ(g, x) = (g − ε(g))x  (1)
Γ(g, ∂a) = (g − ε(g))a  (2)
Γ(gg′, x) = gΓ(g′, x) + ε(g′)Γ(g, x)  (3)
Γ(gg′, x) = Γ(g, g′x) + ε(g)Γ(g′, x)  (4)

Let I(R) = kernel(ε : R → k) be the augmentation ideal considered as an R–bimodule. For an R–module M the tensor product I(R)⊗_{R} M over R is defined and this tensor product is an R–module by r · (\overline{r} ⊗ m) = (r\overline{r}) ⊗ m for r ∈ R, \overline{r} ∈ I(R) and m ∈ M. We have the equation (\overline{r} · r) ⊗ m = \overline{r} ⊗ (r · m) in I(R) ⊗_{R} M. (1.6) Lemma. A secondary R–module X can be equivalently described by a commutative diagram of R–linear maps:

\xymatrix{I(R) ⊗_{R} X_{1} \ar[r]^{1⊗∂} \ar[d]_{μ} & I(R) ⊗_{R} X_{0} \ar[dl]_{Γ̃} \ar[d]^{μ} \\ X_{1} \ar[r]^{∂} & X_{0}}

Here μ is given by μ(\overline{r} ⊗ m) = \overline{r} · m for m ∈ X_{1} or m ∈ X_{0}.

This characterization of a secondary R–module is somewhat more direct than definition (1.3) which is motivated by topological examples below. Proof of (1.6). Given (1.3) we observe that Γ(1 ⊗ x) = 0 for x ∈ X_{0} by (3). Hence Γ in (1.3) is determined by the restriction

Γ′ : I(R) ⊗ X_{0} ⊂ R ⊗ X_{0} ⟶^{Γ} X_{1}

Now (4) shows that Γ′ induces a map

Γ̃ : I(R) ⊗_{R} X_{0} → X_{1}

which is R–linear by (3). By (1) and (2) we see that the diagram in (1.6) commutes. Conversely given such a diagram we define Γ in (1.3) by

Γ(r ⊗ x) = Γ̃((r − εr) ⊗ x)

Now it is easy to show that equations (1), …,(4) are satisfied. q.e.d.

We use (1.6) for the following construction of free secondary R–modules. (1.7) Definition. Let d : V → X_{0} be an R–linear map. Then the free secondary R–module X with basis (V, d) is obtained by the following push out in the category of R–modules and R–linear maps:

\xymatrix{I(R) ⊗_{R} V \ar[r]^{1⊗d} \ar[d]_{μ}^{push} & I(R) ⊗_{R} X_{0} \ar[d]^{Γ̃} \ar[ddr]^{μ} \\ V \ar[r]^{i} \ar[drr]_{d} & X_{1} \ar[dr]^{∂} \\  &  & X_{0}}

We also write X_{1} = X_{1}(d) and X = X(d). Since μ(1 ⊗ d) = dμ the R–linear map ∂ is well defined. Moreover we show that X is a well defined secondary R–module: Proof. By (1.6) we have to show that Γ̃(1 ⊗ ∂) = μ on I(R) ⊗_{R} X_{1}. This holds if Γ̃(1 ⊗ ∂)(1 ⊗ i) = μ(1 ⊗ i) on I(R) ⊗_{R} V and Γ̃(1 ⊗ ∂)(1 ⊗ Γ̃) = μ(1 ⊗ Γ̃) on I(R) ⊗_{R} (I(R) ⊗_{R} X_{0}). Now the first equation holds since

Γ̃(1 ⊗ ∂)(1 ⊗ i) = Γ̃(1 ⊗ (∂i)) = Γ̃(1 ⊗ d) = iμ = μ(1 ⊗ i)

Here iμ = μ(1 ⊗ i) holds since i is R–linear. For the second equation we get

Γ̃(1 ⊗ ∂)(1 ⊗ Γ̃) = Γ̃(1 ⊗ (∂Γ̃)) = Γ̃(1 ⊗ μ) = μ(1 ⊗ Γ̃)

Here the last equation holds since for r, \overline{r} ∈ I(R), x ∈ X_{0} we have

Γ̃(1 ⊗ μ)(r ⊗ \overline{r} ⊗ x) = Γ̃(r ⊗ (\overline{r} · x))
= Γ̃((r · \overline{r}) ⊗ x)
= rΓ̃(\overline{r} ⊗ x)
= μ(1 ⊗ Γ̃)(r ⊗ \overline{r} ⊗ x)

Here we use the fact the Γ̃ is R–linear. q.e.d.

One readily checks that the free secondary module X(d) has the following universal property: Let X be an object in secmod(R) and let

\xymatrix{V \ar[rr]^{f} \ar[dr]_{d} &  & X_{1} \ar[dl]^{∂} \\  & X_{0}}  (1.8)

be a commutative diagram of R–linear maps. Then there is a unique map \overline{f} : X(d) → X in secmod(R) of the form

[figure: I(R) ⊗_{R} X_{0}; Γ̃; X_{1}(d); ∂; _{\overline{f}_{1}}; X_{0}; \overline{f}_{0}=identity; I(R) ⊗_{R} X_{0}; Γ̃; X_{1}; ∂; X_{0}]

such that \overline{f}_{1}i = f for i : V → X_{1}(d) defined in (1.7).

2 Examples of secondary modules in topology

We describe examples of secondary modules which arise in topology. Let Top^{∗} be the category of pointed topological spaces with base point. This is a groupoid enriched category in the following sense. For objects X, Y in Top^{∗} the morphism object [[ X; Y ]] is the groupoid given as follows. Objects in [[ X, Y ]] are the pointed maps X → Y and for pointed maps f, g : X → Y the morphisms H : f ⇒ g in [[ X, Y ]] are the tracks from f to g, that is H is a homotopy class of homotopies f ≃ g. The composite of tracks

h ⟸^{G} g ⟸^{H} f

is denoted by Gˆ+H where Gˆ+H is defined by adding homotopies in the usual way. The inverse of the track H is accordingly denoted by ˆ−H : g ⇒ f with (ˆ−H)ˆ+H = ˆ0_{f} and H ˆ+(ˆ−H) = ˆ0_{g} where ˆ0_{f} denotes the identity track of f.

If Y × Z is a product in Top^{∗} then

[[ X, Y × Z]] = [[ X, Y ]] × [[ X, Z]]  (2.1)

is a product of groupoids. This shows that for an algebraic object Y in Top^{∗} the groupoid [[ X, Y ]] is a corresponding algebraic object in the category Grd of (small) groupoids. For example if Y is an abelian group object in Top^{∗} (i. e. an abelian topological group) then [[ X, Y ]] is a abelian group object in the category Grd. A map between abelian group objects which is a homomorphism of the group structure is termed a linear map.

Let C be a category. Then the category of pairs in C denoted by pair(C) is defined. Objects are morphisms f : A → B in C and morphisms (α, β) : f → g in pair(C) are commutative diagrams in C

\xymatrix{A \ar[r]^{α} \ar[d]_{f} & A′ \ar[d]^{g} \\ B \ar[r]^{β} & B′}

Let Ab be the category of abelian groups. (2.2) Proposition. The category of abelian group objects in Grd and linear maps is isomorphic to the category pair(Ab).

Proof. Given an abelian group object G in Grd we obtain the object

∂ : G_{1} → G_{0}

in pair(Ab) as follows. Here G_{0} is the set of objects of G which is an abelian group since G is an abelian group object in Grd. Let 0 ∈ G_{0} be the neutral object in the abelian group G_{0}. Then G_{1} is the set of all morphisms f : a ⇒ 0 in G with a ∈ G_{0} and ∂f = a. The abelian group structure of G_{1} is defined by

(f : a ⇒ 0) + (g : b ⇒ 0) = (f + g, a + b → 0 + 0 = 0)

where the right hand side is defined since G is an abelian group object in Grd.

Conversely given an object ∂ : A_{1} → A_{0} in pair(Ab) we define the abelian group object G(∂) in Grd as follows. The set of objects of G(∂) is the set A_{0}. The set of morphisms of G(∂) is the product set A_{1} × A_{0} where (a_{1}, x) ∈ A_{1} × A_{0} is a morphism (a_{1}, x) : ∂a_{1} + x → x in G(∂). The identity of x is (0, x) : x = ∂0 + x → x. Composition of

x ⟵^{(a_{1},x)} ∂a_{1} + x ⟵^{(b_{1},∂a_{1}+x)} ∂b_{1} + ∂a_{1} + x

is (b_{1} + a_{1}, x) for a_{1}, b_{1} ∈ A_{1} and x ∈ A_{0}. Now it is readily seen that this way one gets an isomorphism of categories. q.e.d.

Now given an abelian group object Y in Top^{∗} the abelian group object G = [[ X, Y ]] in Grd is given via (2.2) by a homomorphism

G_{1} = [[ X, Y ]]_{1} ⟶^{∂} G_{0} = [[ X, Y ]]_{0}  (2.3)

where G_{0} is the set of all pointed maps f : X → Y and where G_{1} is the set of all tracks H : f ⇒ 0 with f ∈ G_{0} and ∂H = f. The group structure of Y induces the group structure on G_{0} and G_{1} in the obvious way. (2.4) Definition. Let R be a k–algebra with augmentation ε : R → k as in (1.1). A track module Y over R is a R–module object Y in Top^{∗} (i. e. a topological R–module) for which each map r : Y → Y given by r ∈ R admits a unique track Γ_{r} : r ⇒ εr where εr : Y → Y is defined by the k–vector space structure of Y. (2.5) Example. Let Z^{n} = K(k, n) be an Eilenberg–MacLane space of the underlying abelian group of the field k with the properties in the introduction, see Appendix A. This shows that for R = k[σ_{n}] the space Z^{n} is a topological R–module, in fact, a track module over R since for r ∈ R there is a unique track r ⇒ εr (with r, εr : Z^{n} → Z^{n}). Here ε is the sign–augmentation as in (1.1)(3). (2.6) Proposition. Let Y be a track module over R. Then for each X in Top^{∗} one obtains canonically a secondary module over R

R ⊗ G_{0} ⟶^{Γ} G_{1} ⟶^{∂} G_{0}

where ∂ is given by the groupoid G = [[ X, Y ]] as in (2.3) and where Γ is defined by the composite

Γ(r ⊗ f) = Γ_{r−εr}f

Here the track Γ_{r−εr} : r − εr ⇒ 0 is given for r − εr ∈ R by (2.4).

Using (2.6) we obtain for each track module Y over R a functor

H(−; Y) : Top^{∗} → secmod(R)  (2.7)

which carries X to the track module H(X; Y) = (G_{1}, G_{0}, ∂, Γ) given by [[ X, Y ]] in (2.6). Of course we have

π_{0}H(X, Y) = cokernel(∂) = [X, Y]  (1)
π_{1}H(X, Y) = kernel(∂) = [X, ΩY]  (2)

Here [X, Y] denotes the set of homotopy classes of pointed maps X → Y and ΩY is the loop space of Y. The elements H ∈ [X, ΩY] are identified with the tracks H : 0 ⇒ 0 where 0 : X → ∗ → Y is the zero map. By (1) and (2) we see that the functor (2.7) carries homotopy equivalences in Top^{∗} to weak equivalences between secondary modules as defined in the next section.

We are mainly interested in the secondary module

H^{n}(X) = H(X, Z^{n})  (2.8)

over R_{n} = k[σ_{n}] given by (2.5) with π_{0}H^{n}(X) = H̃^{n}(X, k) and π_{1}H^{n}(X) = H̃^{n−1}(X, k). This corresponds to the boundary map ∂ in the introduction. Proof of (2.6). By definition of ∂ we have ∂(Γ_{r−εr}f) = (r − εr)f so that (1.3)(1) is satisfied. Now let H : f ⇒ 0 be an element in G_{1} with ∂H = f. Then we get

[figure: •; •; 0; •; 0; •; Γ_{r−εr}; •; r−εr; H; •; f]

so that for Γ = Γ_{r−εr}

Γ ∗ H = 0H ˆ+Γf = Γ0ˆ+(r − εr)H
= ˆ0_{0} ˆ+Γf = ˆ0_{0} ˆ+(r − εr)H
= Γf = (r − εr)H

and this implies (1.3)(2). Next we have by uniqueness of tracks in [[ Y, Y ]] the equations

Γ_{rr′−ε(rr′)} = rΓ_{r−εr′} + ε(r′)Γ_{r−εr}
= Γ_{r−εr}r′ + ε(r)Γ_{r′−εr′}

and these equations imply (1.3)(3),(4). q.e.d. (2.9) Remark. Let Y be given as in (2.6) and let R̂ ⊂ [[ Y, Y ]] be the full subgroupoid with objects given by maps r : Y → Y for r ∈ R. Then we obtain the action

μ_{R} : R̂ × [[ X, Y ]] ⊂ [[ Y, Y ]] × [[ X, Y ]] ⟶^{∘} [[ X, Y ]]

where the second arrow is composition in the groupoid enriched category Top^{∗}. The action μ_{R} determines Γ in the secondary module H(X, Y) given by (2.6) and conversely H(X, Y) determines uniquely the action μ_{R}. In this sense a secondary module is a R̂–module in the category Grd of groupoids. Here R̂ is the groupoid with objects R, path components ε^{−1}(x) with x ∈ k and all automorphism groups in R̂ are trivial. The algebra structure of R yields a corresponding structure of R̂.

3 Weak equivalences

We can consider a secondary module X as a chain complex of k–vector spaces concentrated in degree 0 and 1. The homology of this chain complex is denoted by

π_{0}(X) = cokernel(∂ : X_{1} → X_{0})
π_{1}(X) = kernel(∂ : X_{1} → X_{0}) (3.1)

A map F = F_{h} : X_{R} → Y_{K} between secondary modules is a weak equivalence if h : R → K is an isomorphism and f induces isomorphisms

f_{∗} : π_{0}(X_{R}) ≅ π_{0}(Y_{K})
f_{∗} : π_{1}(X_{R}) ≅ π_{1}(Y_{K})

We point out that π_{0}(X) and π_{1}(X) are also R–modules for which, however, by (1.3)(1),(2) the R–module structure is induced by the augmentation ε, that is r · x = ε(r) · x for r ∈ R, x ∈ π_{0}(X), π_{1}(X). Hence π_{0}(X) and π_{1}(X) are just k–vector spaces with an action of R via ε. Such R–modules are termed ε–modules. If X_{R} is a secondary module with Γ = 0 then X_{0} and X_{1} are also ε–modules. Hence in this case X_{R} is given by a chain complex ∂ : X_{1} → X_{0} of k–vector spaces. We say that X_{R} is of trivial type if Γ = 0 and ∂ = 0 so that in this case X_{0} = π_{0} and X_{1} = π_{1}.

Two secondary modules X_{R}, Y_{R} are weakly equivalent if there exists a chain of R– equivariant weak equivalences

X_{R} ⟵^{∼} X_{1} ⟶^{∼} X_{2} ⟵^{∼} … X_{n} ⟶^{∼} Y_{R}.

(3.2) Proposition. Each secondary module is weakly equivalent to a secondary module of trivial type.

Hence the only invariants of the weak equivalence class of a secondary module X are π_{0}X and π_{1}X. (3.3) Remark. For the secondary module H^{n}(X) over R_{n} = k[σ_{n}] in (2.7) we know by (3.2) that the weak equivalence type of H^{n}(X) is trivial. This can also be seen by the following topological argument. By Baues [B] there exists a sequence

Z^{n} ⟵^{f} Y_{1} ⟶^{g} Y_{2} ⟵^{h} Y_{3}

of topological R_{n}–modules and R_{n}–linear maps f, g, h with the following properties. The action of R_{n} an Y_{3} satisfies r · y = ε(r) · y for r ∈ R_{n} and y ∈ Y_{3} where ε is the sign augmentation of R_{n}. Moreover f, g and h are homotopy equivalences on Top^{∗}. Hence we obtain weak equivalences of secondary modules over R_{n}

H^{n}(X) = H(X, Z^{n}) ⟵^{∼} H(X; Y_{1}) ⟶^{∼} H(X, Y_{2}) ⟵^{∼} H(X, Y_{3})

where H(X, Y_{3}) is easily seen to be weakly equivalent to a secondary module of trivial type.

For the proof of (3.2) in (3.8) below we need the following pull back construction for secondary modules. Let X_{R} be a secondary module and let Y_{0} be an R–module and let f : Y_{0} → X_{0} be a R–linear map. Then we obtain the following commutative diagram in which the subdiagram ‘pull’ is a pull back in the category of vector spaces.

\xymatrix{R ⊗ Y_{0} \ar[r]_{\overline{Γ}} \ar[dd]^{R⊗f} \ar[rrr]^{γ} & f^{∗}X_{1} \ar[rr]_{\overline{∂}} \ar[dd]^{\overline{f}} &  & Y_{0} \ar[dd]^{f} \\  &  & pull \\ R ⊗ X_{0} \ar[r]^{Γ} & X_{1} \ar[rr]^{∂} &  & X_{0}}

Here \overline{Γ} is defined by \overline{∂} \overline{Γ} = γ with γ(r ⊗ y) = (r − εr) · y and \overline{f} \overline{Γ} = Γ(R ⊗ f). Then f^{∗}X_{1} is an R–module and \overline{∂} is R–linear. Moreover we get the following fact. (3.4) Lemma. The top row Y_{R} = (\overline{∂}, \overline{Γ}) = f^{∗}X_{R} of the diagram is a secondary module over R and (\overline{f}, f) : Y_{R} → X_{R} is a map in secmod(R) which is a weak equivalence if (∂, f) : X_{1} ⊕ Y_{0} → X_{0} is surjective.

The map f^{∗}X_{R} → X_{R} has the following property. Let i : K → R be an augmented map between k–algebras and let g : Z_{K} → X_{R} be an i–equivariant map between secondary modules for which a commutative diagram

\xymatrix{ & Y_{0} \ar[d]^{f} \\ Z_{0} \ar[ur]^{h_{0}} \ar[r]_{g_{0}} & X_{0}}  (1)

is given. Here h_{0} and g_{0} are i–equivariant. Then there exists a unique i–equivariant map h : Z_{K} → f^{∗}X_{R} for which the diagram

\xymatrix{ & f^{∗}X_{R} \ar[d]^{(\overline{f},f)} \\ Z_{K} \ar@{-->}[ur]^{h} \ar[r]_{g} & X_{R}}  (2)

commutes in secmod. Proof of (3.4). The elements of f^{∗}X_{1} are pairs (x_{1}, y) with ∂x_{1} = fy. We define r(x_{1}, y) = (rx_{1}, ry) so that \overline{∂} and \overline{f} are R–linear with \overline{∂}(x_{1}, y) = y and \overline{f}(x_{1}, y) = x_{1}. Moreover

\overline{Γ}(r ⊗ y) = (Γ(r ⊗ fy), (r − εr)y)

Hence (1.3)(1) holds for Y_{R}. Moreover

\overline{Γ}(r ⊗ \overline{∂}(x_{1}, y)) = \overline{Γ}(r ⊗ y)

= (Γ(r ⊗ fy), (r − εr)y) = (Γ(r ⊗ ∂x_{1}), (r − εr)y) = ((r − εr)x_{1}, (r − εr)y) = (r − εr)(x_{1}, y)

This shows (1.3)(2) for Y_{R}. Next we consider (1.3)(3) for Y_{R} and we get

\overline{Γ}(r · r′ ⊗ y) = (Γ(r · r′ ⊗ fy), (r · r′ − ε(rr′))y)
r\overline{Γ}(r′ ⊗ y) + ε(r′)\overline{Γ}(r ⊗ y) = r(Γ(r′ ⊗ fy), (r′ − εr′)y) + ε(r′)(Γ(r ⊗ fy), (r − εr)y)
= (Γ(rr′ ⊗ fy), r(r′ − εr′)y + ε(r′)(r − εr)y)
= (Γ(rr′ ⊗ fg), (rr′ − ε(rr′))y).

Similarly one checks (1.3)(4) for Y_{R}. Hence (f, \overline{f}) : Y_{R} → X_{R} is a well defined map between secondary modules. If (∂, f) is surjective then the pull back is also a push out and therefore (\overline{f}, f) is a weak equivalence. q.e.d.

Now given a secondary module X_{R} we can choose a k–linear section s : π_{0} → X_{0} of the quotient map q : X_{0} → X_{0}/ im(∂) = π_{0}. Hence the R–linear map

f : R ⊗ π_{0} → X_{0}  (3.5)

with f(r ⊗ x) = r · sx is defined with qf(r ⊗ x) = ε(r) · x. Here R ⊗ π_{0} is a free R–module with the action of R given by r · (r′ ⊗ x) = (r · r′) ⊗ x and qf coincides with the R–linear map

qf = ε ⊗ 1 : R ⊗ π_{0} → k ⊗ π_{0} = π_{0}  (1)

For I(R) = ker(ε : R → k) we have ker(ε ⊗ 1) = I(R) ⊗ π_{0}. Using f in (3.5) we get as in (3.4) an R–linear map between secondary modules

(\overline{f}, f) : Y_{R} = f^{∗}X_{R} → X_{R}  (2)

which is a weak equivalence since (∂, f) : X_{1} ⊕ Y_{0} = X_{1} ⊕ R ⊗ π_{0} → X_{0} is surjective. Here Y_{R} is a secondary module which is special in the following sense. We say that a secondary module X_{R} is special if for π_{0} = π_{0}X one has

\begin{cases}X_{0} = R ⊗ π_{0} and \\ im(∂ : X_{1} → X_{0}) = I(R) ⊗ π_{0}. \end{cases}

(3.6) Proposition. A special secondary R–module X_{R} admits an R–linear section t : I(R)⊗ π_{0} → X_{1} of ∂ : X_{1} → I(R) ⊗ π_{0}. Proof. We define t by the map

Γ : R ⊗ X_{0} = R ⊗ R ⊗ π_{0} → X_{1},

namely for r′ ∈ I(R) and x ∈ π_{0} let

t(r′ ⊗ x) = Γ(r′ ⊗ 1 ⊗ x).

Then we have ∂t(r′ ⊗x) = ∂Γ(r′ ⊗1⊗x) = (r′ −εr′)(1⊗x) = r′ ⊗x since ε(r′) = 0. Moreover t is R–linear since for r ∈ R

t(r · r′ ⊗ x) = Γ(r · r′ ⊗ 1 ⊗ x)
= rΓ(r′ ⊗ 1 ⊗ x) + ε(r′)Γ(r ⊗ 1 ⊗ x)
= rΓ(r′ ⊗ 1 ⊗ x)
= r · t(r′ ⊗ x).

q.e.d.

(3.7) Remark. A converse of (3.6) is also true. Let π_{0} and π_{1} be k–vector spaces and let

∂ : X_{1} → I(R) ⊗ π_{0}

be a surjective R–linear map for which π_{1} = ker(∂) is an ε–module and let t be an R–linear section of ∂. Then a special secondary R–module X_{R} is defined in terms of ∂ and t as follows. Let X_{0} = R ⊗ π_{0} and let

Γ : R ⊗ X_{0} = R ⊗ R ⊗ π_{0} → X_{1}

be given by (r, r′ ∈ R, x ∈ π_{0})

Γ(r ⊗ r′ ⊗ x) = (r − εr)t((r′ − εr′) ⊗ x) + ε(r′)t((r − εr) ⊗ x)

Then one can check that Γ satisfies all the axioms in (1.3) so that X_{R} is a well defined special secondary module. Hence by (3.6) special secondary modules are up to isomorphism determined by π_{0} and π_{1} with X_{0} = R ⊗ π_{0} and X_{1} = π_{1} ⊕ I(R) ⊗ π_{0}. (3.8) Proof of (3.2). Let X_{R} be a secondary module. Then we obtain by (3.5) the special secondary module Y_{R} = f^{∗}X_{R} and the weak equivalence Y_{R} ⟶^{∼} X_{R}. Moreover by (3.6) and (3.7) we have Y_{0} = R ⊗ π_{0} and Y_{1} = π_{1} ⊕ I(R) ⊗ π_{0} and

∂ : Y_{1} = π_{1} ⊕ I(R) ⊗ π_{0} → I(R) ⊗ π_{0} ⊂ R ⊗ π_{0} = Y_{0}

is given by the projection and the inclusion. Now we obtain a weak equivalence g with

\xymatrix{Y_{1} \ar[r]^{∂} \ar[d]_{g_{1}} & Y_{0} \ar[d]^{g_{0}} \\ π_{1} \ar[r]^{0} & π_{0}}

where g_{0} = ε ⊗ 1 and g_{1} is the projection. By the definition of Γ in (3.7) in terms of the section t : I(R) ⊗ π_{0} ⊂ Y_{1} we see that g_{1}Γ = 0 so that g is a well defined map between secondary modules where 0 : π_{1} → π_{0} is the secondary module of trivial type given by π_{0} and π_{1}. q.e.d.

4 Tensor products of secondary modules

Here we introduce the tensor product of secondary modules which is needed for the definition of secondary algebras in the next section. For secondary modules X_{R} and Y_{K} the tensor product of the underlying chain complexes is given by the chain complex of k–vector spaces

X_{1} ⊗ Y_{1} ⟶^{d_{2}} X_{1} ⊗ Y_{0} ⊕ X_{0} ⊗ Y_{1} ⟶^{d_{1}} X_{0} ⊗ Y_{0}

\begin{aligned}d_{2}(a ⊗ b) & = (∂a) ⊗ b − a ⊗ (∂b) \\ d_{1}(a ⊗ y) & = (∂a) ⊗ y \\ d_{1}(x ⊗ b) & = x ⊗ (∂b) \end{aligned}

with x ∈ X_{0}, y ∈ Y_{0}, a ∈ X_{1}, b ∈ Y_{1}. Hence d_{1} induces the boundary map

∂_{⊗} : (X_{1} ⊗ Y_{0} ⊕ X_{0} ⊗ Y_{1})/ im(d_{2}) → X_{0} ⊗ Y_{0}  (4.1)

Since k is a field we get by the Künneth formula

π_{0}∂_{⊗} = cok(∂_{⊗}) = π_{0}(X) ⊗ π_{0}(Y)
π_{1}∂_{⊗} = ker(∂_{⊗}) = π_{1}(X) ⊗ π_{0}(Y) ⊕ π_{0}(X) ⊗ π_{1}(Y)

One readily checks that ∂_{⊗} is an R ⊗ K–equivariant k–linear map. (4.2) Definition. We define the tensor product X_{R} ⊗ Y_{K} = (X ⊗ Y)_{R⊗K} of secondary modules X_{R} and Y_{K} by the diagram

R ⊗ K ⊗ X_{0} ⊗ Y_{0} ⟶^{Γ_{⊗}} (X_{1} ⊗ Y_{0} ⊕ X_{0} ⊗ Y_{1})/ im d_{2} ⟶^{∂_{⊗}} X_{0} ⊗ Y_{0}

Here ∂_{⊗} is defined as in (4.1) and Γ_{⊗} is defined by the following formula

Γ_{⊗}(α ⊗ β ⊗ x ⊗ y) = Γ(α ⊗ x) ⊗ (βy) + (ε(α)x) ⊗ Γ(β ⊗ y)
= (αX) ⊗ Γ(β ⊗ y) + Γ(α ⊗ x) ⊗ (ε(β)y)

Here the second equation is a consequence of the first equation since (∂a) ⊗ b = a ⊗ (∂b) by (3.1). (4.3) Lemma. The tensor product X_{R} ⊗ Y_{K} of secondary modules X_{R} and Y_{K} is a well defined secondary module over R ⊗ K. Proof. The map ∂_{⊗} is R ⊗ K–linear and Γ_{⊗} is a well defined k–linear map. Hence we have to check the equations (1) … (4) in (1.3): We first check (1).

∂_{⊗}Γ_{⊗}(α ⊗ β ⊗ x ⊗ y) = ∂Γ(α ⊗ x) ⊗ βy + ε(α)x ⊗ ∂Γ(β ⊗ y)
= (α − ε(α))x ⊗ βy + ε(α)x ⊗ (β − ε(β))y
= αx ⊗ βy − ε(α)ε(β)x ⊗ y
= (α ⊗ β − ε(α ⊗ β))(x ⊗ y)

Next we check (2) for Γ_{⊗}.

Γ_{⊗}(α ⊗ β ⊗ ∂_{⊗}(a ⊗ y)) = Γ_{⊗}(α ⊗ β ⊗ ∂a ⊗ y)
= Γ(α ⊗ ∂a) ⊗ βy + ε(α)∂a ⊗ Γ(β ⊗ y)
= (α − ε(α))a ⊗ βy + ε(α)a ⊗ ∂Γ(β ⊗ y) , see (4.1),
= (α − ε(α))a ⊗ βy + ε(α)a ⊗ (β − ε(β))y
= αa ⊗ βy − ε(α)ε(β)a ⊗ y
= (α ⊗ β − ε(α ⊗ β))(a ⊗ y)

Γ_{⊗}(α ⊗ β ⊗ ∂_{⊗}(x ⊗ b)) = Γ_{⊗}(α ⊗ β ⊗ x ⊗ ∂b)
= Γ(α ⊗ x) ⊗ β∂b + ε(α)x ⊗ Γ(β ⊗ ∂b)
= Γ(α ⊗ x) ⊗ ∂(βb) + ε(α)x ⊗ (β − ε(β))b
= ∂Γ(α ⊗ x) ⊗ βb + ε(α)x ⊗ (β − ε(β))b , see (4.1),
= (α − ε(α))x ⊗ βb + ε(α)x ⊗ (β − ε(β))b
= αx ⊗ βb − ε(α)ε(β)x ⊗ b
= (α ⊗ β − ε(α ⊗ β))(x ⊗ b)

Now we check (3) for Γ_{⊗}.

Γ_{⊗}((α ⊗ β)(α′ ⊗ β′) ⊗ x ⊗ y) = Γ_{⊗}(αα′ ⊗ ββ′ ⊗ x ⊗ y)
= Γ(αα′ ⊗ x) ⊗ ββ′y + ε(αα′)x ⊗ Γ(ββ′ ⊗ y) = (i)

Now we get by (3) that (i)=(ii) coincides with

(ii) = (αΓ(α′ ⊗ x) + ε(α′)Γ(α ⊗ x)) ⊗ ββ′y + ε(αα′)x ⊗ (βΓ(β′ ⊗ y) + ε(β′)Γ(β ⊗ y)) On the other hand we have

(iii) = (α ⊗ β)Γ_{⊗}(α′ ⊗ β′ ⊗ x ⊗ y) + ε(α′ ⊗ β′)Γ_{⊗}(α ⊗ β ⊗ x ⊗ y)
= αΓ(α′ ⊗ x) ⊗ ββ′y + ε(α′)αx ⊗ βΓ(β′ ⊗ y)
+ ε(α′ ⊗ β′)(Γ(α ⊗ x) ⊗ βy + ε(α)x ⊗ Γ(β ⊗ y)

We have to check (ii)=(iii). But this is equivalent to ε(α′)Γ(α ⊗ x) ⊗ ββ′y + ε(αα′)x ⊗ βΓ(β′ ⊗ y) = ε(α′)αx ⊗ βΓ(β′ ⊗ y) + ε(α′ ⊗ β′)Γ(α ⊗ x) ⊗ βy This equation is equivalent to

Γ(α ⊗ x) ⊗ β(β′ − εβ′)y  (α − εα′)x ⊗ βΓ(β′ ⊗ y) Γ(α ⊗ x) ⊗ β∂Γ(β′ ⊗ y)  ∂Γ(α ⊗ x) ⊗ βΓ(β′ ⊗ y)

By (4.1) we know that

Γ(α ⊗ x) ⊗ β∂Γ(β′ ⊗ y) = Γ(α ⊗ x) ⊗ ∂βΓ(β′ ⊗ y)
= ∂Γ(α ⊗ x) ⊗ βΓ(β′ ⊗ y)

This completes the proof that (ii)=(iii) and hence (i)=(iii) and hence (3) holds for X ⊗ Y. Finally we have to check (4). For this we apply (4) to (i) above and we get (i)=(iv) where

(iv) = (Γ(α ⊗ α′x) + ε(α)Γ(α′ ⊗ x)) ⊗ ββ′y + ε(αα′)x ⊗ (Γ(β ⊗ β′y) + ε(β)Γ(β′ ⊗ y)) On the other hand we have

(v) = Γ_{⊗}(α ⊗ β ⊗ α′x ⊗ β′y) + ε(α ⊗ β)Γ_{⊗}(α′ ⊗ β′ ⊗ x ⊗ y)
= Γ(α ⊗ α′x) ⊗ ββ′y + ε(α)α′)x ⊗ Γ(β ⊗ β′y)
+ ε(α ⊗ β)(Γ(α′ ⊗ x) ⊗ β′y + ε(α′)x ⊗ Γ(β′ ⊗ y))

We have to check (iv)=(v). This is the case if and only if the following equation holds.

ε(α)Γ(α′ ⊗ x) ⊗ ββ′y + ε(αα′)x ⊗ Γ(β ⊗ β′y)

= ε(α)α′x ⊗ Γ(β ⊗ β′y) + ε(α ⊗ β)Γ(α′ ⊗ x) ⊗ β′y

This equation holds if and only if the following equation is true

Γ(α′ ⊗ x) ⊗ (β − εβ)β′y  (α′ − ε(α′))x ⊗ Γ(β ⊗ β′y) Γ(α′ ⊗ x) ⊗ ∂Γ(β ⊗ β′y)  ∂Γ(α′ ⊗ x) ⊗ Γ(β ⊗ β′y)

Now again (4.1) shows that this equation is true. Hence we have shown (i)=(v) and this corresponds to equation (4) for X ⊗ Y. Now the proof of the lemma is complete. q.e.d.

(4.4) Lemma. The tensor product of secondary modules is associative and bilinear, that is:

(X_{R} ⊗ Y_{K}) ⊗ Z_{L} = X_{R} ⊗ (Y_{K} ⊗ Z_{L})
(X_{R} ⊕ Y_{R}) ⊗ Z_{L} = X_{R} ⊗ Z_{L} ⊕ Y_{R} ⊗ Z_{L}
Z_{L} ⊗ (X_{R} ⊕ Y_{R}) = Z_{L} ⊗ X_{R} ⊕ Z_{L} ⊗ Y_{R}

We point out that the chain complex k = (0 → k) is a unit for the tensor product, that is

X_{R} ⊗ k = X_{R} = k ⊗ X_{R}  (4.5)

Here we use the obvious identification V ⊗ k = V = k ⊗ V for a k–vector space V.

We shall use the tensor product of secondary modules mainly for the next result. (4.6) Proposition. The cup product map μ : Z^{n} ×Z^{n} → Z^{n+m} induces an i_{n,m}–equivariant map between secondary modules

μ_{∗} : H^{n}(X) ⊗ H^{m}(X) → H^{n+m}(X)

where i_{n,m} : k[σ_{n}] ⊗ k[σ_{m}] → k[σ_{n+m}] is induced by the inclusion σ_{n} × σ_{m} ⊂ σ_{n+m}. Proof. The map μ_{∗} carries f ⊗ g ∈ H_{0}^{n} ⊗ H_{0}^{m} with f : X → Z^{n}, g : X → Z^{m} to the composite μ(f, g) : X → Z^{n} × Z^{m} → Z^{n+m}. Since μ is k–bilinear and i_{n,m}–equivariant μ_{∗} : H_{0}^{n} ⊗ H_{0}^{m} → H_{0}^{n+m} is well defined. Moreover μ_{∗} is induced on (H^{n} ⊗ H^{m})_{1} by the map

μ̄ : H_{0}^{n} ⊗ H_{1}^{m} ⊕ H_{1}^{n} ⊗ H_{0}^{m} → H_{1}^{n+m}  (1)

which carries f ⊗ G with G : g ⇒ 0 ∈ H_{1}^{m} to μ(f, G) : μ(f, g) ⇒ 0 and carries H ⊗ g with H : f ⇒ 0 ∈ H_{1}^{n} to μ(H, g) : μ(f, g) ⇒ 0. Here we use the fact that the bilinearity of μ implies that μ(f, 0) = 0 = μ(0, g). If H : f ⇒ 0 and G : g ⇒ 0 are given then in fact

μ̄(f ⊗ G) = μ̄(H ⊗ g)  (2)

so that μ_{∗} is well defined and i_{n,m}–equivariant. In fact, we have for the track (H, G) : (f, g) ⇒ (0, 0) with (f, g) : X → Z^{n} × Z^{m} given by the homotopy (H_{t}, G_{t}) the formula

\begin{aligned}μ(H, G) & = μ((0, G)ˆ+(H, g)) \\ & = μ((H, 0)ˆ+(f, G)) \end{aligned}  (3)

where ˆ+ denotes addition of tracks. Hence we get

μ(H, g) = 0ˆ+μ(H, g) = μ((0, G))ˆ+μ(H, g)
= μ((0, G)ˆ+(H, g)) = μ(H, G)
= μ((H, 0)ˆ+(f, G))
= μ(H, 0)ˆ+μ(f, G)
= 0ˆ+μ(f, G) = μ(f, G)

and this proves (2). Finally we have to show that μ_{∗} is compatible with the Γ–operator. For this let r ∈ R_{n} and s ∈ R_{m} and let

\begin{aligned}Γ_{r−εr} & : r − εr ⇒ 0 : Z^{n} → Z^{n} \\ Γ_{s−εs} & : s − εs ⇒ 0 : Z^{m} → Z^{m} \\ Γ_{r⊙s−ε(r)·ε(s)} & : r ⊙ s − ε(r ⊙ s) ⇒ 0 : Z^{n+m} → Z^{n+m} \end{aligned}  (4)

where r ⊙ s = i_{n,m}(r, s) ∈ R_{n+m}. We observe that in R_{n+m} we have the following equations

\begin{aligned}(r − εr) ⊙ s + ε(r)(1_{n} ⊙ (s − εs)) & = \\ (r ⊙ s) & − ε(r)(1_{n} ⊙ s) + ε(r)(1_{n} ⊙ s) − ε(r)ε(s)(1_{n} ⊙ 1_{m}) \\ & = r ⊙ s − ε(r ⊙ s) ∈ R_{n+m} \end{aligned} (5)

Let Z^{n}∧Z^{m} = Z^{n}×Z^{m}/Z^{n}×{0}∪{0}×Z^{m} be the smash product. Then the cup product map μ induces a map μ̃ : Z^{n} ∧ Z^{m} → Z^{n+m} and we get the composites a, b, c : Z^{n} ∧ Z^{m} → Z^{n+m} by

\begin{aligned}a & = μ̃(r − εr) ∧ s, \\ b & = ε(r)μ̃(1_{n} ∧ (s − εs)), \\ c & = (r ⊙ s − ε(r ⊙ s))μ̃. \end{aligned}  (6)

Then (5) shows that a + b = c. Hence

\begin{aligned}A & = μ̃(Γ_{r−εr} ∧ s) + ε(r)μ̃(1_{n} ∧ Γ_{s−εs}) and \\ B & = Γ_{r⊙s−ε(r⊙s)}μ̃ \end{aligned}  (7)

are both tracks c ⇒ 0. Now obstruction theory shows that these tracks A and B coincide since the set of homotopy classes [ΣZ^{n} ∧ Z^{m}, Z^{n+m}] is trivial. The equation A = B implies that μ satisfies

μ_{∗}Γ(r ⊗ s ⊗ f ⊗ g) = Γ(r ⊙ s ⊗ μ(f, g))  (8)

by definition on Γ in (2.6) and by formula (2.4), which exactly corresponds to A = B. q.e.d.

5 Squaring operations

Let K, L, R be k–algebras. Then K ⊗ L and L ⊗ K are k–algebras and we consider homomorphisms of k–algebras

K ⊗ L ⟶^{i} R ⟵^{j} L ⊗ K  (5.1)

compatible with augmentations. For α ∈ K and β ∈ L we write

α ⊙ β = i(α ⊗ β) and β ⊙ α = j(α ⊗ β)  (1)

We also consider elements τ, τ̄ ∈ R with ττ̄ = 1 = τ̄τ and

\begin{aligned}τ(α ⊙ β) & = β ⊙ α, \\ τ̄(β ⊙ α) & = α ⊙ β \end{aligned}  (2)

for α ∈ K, β ∈ L. As a special case let R_{n} = k[σ_{n}] be the group algebra of the symmetric group σ_{n}. Then we have the element τ = τ_{n,m} ∈ σ_{n+m} interchanging the n–block {1, …, n} and the m–block {n + 1, …, n + m} such that the diagram

\xymatrix{σ_{n} × σ_{m} \ar[r]^{i_{n,m}} \ar[d]^{T} & σ_{n+m} \ar[d]^{τ} \\ σ_{m} × σ_{n} \ar[r]^{i_{m,n}} & σ_{n+m}}  (3)

commutes. Here T is the interchange map and i_{n,m} is the inclusion. The sign of τ_{n,m} is (−1)^{nm}. We get algebra homomorphisms

R_{n} × R_{m} ⟶^{i} R_{n+m} ⟵^{j} R_{m} ⊗ R_{n}  (4)

induced by the inclusions i_{n,m} and i_{m,n} above and the element τ = τ_{n,m} ∈ R_{n+m} satisfies (2).

The following definition of τ–pairs and τ–squares is part of the concept of a τ–commutative secondary algebra in the next section. (5.2) Definition. A τ–pair consists of secondary modules Z_{K}, Y_{L} and X_{R} and maps

Z_{K} ⊗ Y_{L} ⟶^{μ_{i}} X_{R} ⟵^{μ_{j}} Y_{L} ⊗ Z_{K}  (1)

where i, j have the properties in (5.1) with respect to an element τ ∈ R. We write for z ∈ Z_{0}, y ∈ Y_{0}, z_{1} ∈ Z_{1}, y_{1} ∈ Y_{1},

\begin{matrix}μ_{i}(z ⊗ y) = z · y, μ_{i}(z_{1} ⊗ y) = z_{1} · y, μ_{i}(z ⊗ y_{1}) = z · y_{1}, \\ μ_{j}(y ⊗ z) = y · z, μ_{j}(y ⊗ z_{1}) = y · z_{1}, μ_{j}(y_{1} ⊗ z) = y_{1} · z. \end{matrix}  (2)

Now the element τ (termed the interchange element) satisfies

\begin{aligned}τ(z · y) & = y · z, \\ τ(z_{1} · y) & = y · z_{1}, \\ τ(z · y_{1}) & = y_{1} · z. \end{aligned}  (3)

For K = L and Z_{K} = Y_{L} and i = j and μ = μ_{i} = μ_{j} we call the τ–pair

Y_{L} ⊗ Y_{L} ⟶^{μ} X_{R}  (4)

a τ–square.

For a τ–pair μ_{i}, μ_{j} the induced k–linear maps π_{0}μ_{i}, π_{0}μ_{j}, π_{1}μ_{i}, π_{1}μ_{j} yield

\begin{aligned}π_{0}Z ⊗ π_{0}Y & → π_{0}X ← π_{0}Y ⊗ π_{0}Z \\ π_{0}Z ⊗ π_{1}Y ⊕ π_{1}Z ⊗ π_{0}Y & → π_{1}X ← π_{0}Y ⊗ π_{1}Z ⊕ π_{1}Y ⊗ π_{0}Z \end{aligned}  (5)

These product maps carry u ⊗ v to the products u · v as in (2). Moreover (3) implies for a ∈ π_{0}Y, b ∈ π_{0}Z, a_{1} ∈ π_{1}Y, b_{1} ∈ π_{1}Z,

\begin{aligned}a · b & = ε(τ)b · a, \\ a · b_{1} & = ε(τ)b_{1} · a, \\ a_{1} · b & = ε(τ)b · a_{1}. \end{aligned}  (6)

A map between τ–pairs is a commutative diagram

\xymatrix{Z_{K} ⊗ Y_{L} \ar[d]_{f⊗g} \ar[r]^{μ_{i}} & X_{R} \ar[d]^{d} & Y_{L} ⊗ Z_{K} \ar[l]_{μ_{j}} \ar[d]^{g⊗f} \\ Z′_{K} ⊗ Y′_{L} \ar[r]^{μ′_{i}} & X′_{R} & Y′_{L} ⊗ Z′_{K} \ar[l]_{μ′_{j}}}  (7)

where f, g resp. d are K, L, resp. R–equivariant. Such a map is a weak equivalence if f, g and d are weak equivalences. Of course the product maps (5) are invariants of the weak equivalence type of a τ–pair. A map between τ–squares is accordingly a map (f, g, d) with f = g. Below we describe a squaring operation Sq which is an invariant of the weak equivalence type of a τ–square. Hence though secondary modules X by (5.2) are up to weak equivalence only determined by π_{0}X and π_{1}X a τ–pair or a τ–square have nontrivial weak equivalence types. (5.3) Lemma. For a τ–pair as in (5.2) we have Γ(τ ⊗ (y · z + ε(τ)z · y)) = 0. Proof. Since ττ = 1 we have

0 = Γ(1, z · y) = Γ(ττ, z · y) = Γ(τ ⊗ τ(z · y)) + ε(τ)Γ(τ ⊗ z · y)
= Γ(τ ⊗ y · z) + ε(τ)Γ(τ ⊗ z · y)
= Γ(τ ⊗ (y · z + ε(τ)z · y)).

q.e.d.

(5.4) Lemma. For a τ–square as in (5.2) we have Γ(τ, ∂y_{1} · ∂y_{1}) = (1 − ε(τ))(∂y_{1}) · y_{1}. Proof. We have ∂_{⊗}(y_{1} ⊗ ∂y_{1}) = ∂y_{1} ⊗ ∂y_{1} in Y_{L} ⊗ Y_{L} so that ∂(y_{1} · ∂y_{1}) = ∂y_{1} · ∂y_{1} in X_{R}. Hence

Γ(τ, ∂y_{1} · ∂y_{1}) = Γ(τ, ∂(y_{1} · ∂y_{1}))
= (τ − ε(τ))y_{1} · (∂y_{1})
= (∂y_{1}) · y_{1} − ε(τ)y_{1} · (∂y_{1})

Using (4.1) we see that y_{1} · (∂y_{1}) = (∂y_{1}) · y_{1}. q.e.d. (5.5) Proposition. Let μ : Y_{L} → X_{R} be a τ–square as in (5.2) with ε(τ) = 1. Then there is a well defined k–linear map

Sq : π_{0}(Y) → π_{1}(X)

which carries the element {y} ∈ π_{0}(Y) represented by y ∈ Y_{0} to the element Sq{y} = Γ(τ, y · y). Moreover we have 2 Sq = 0. Proof. We have in X_{R} the equation

∂Γ(τ, y · y) = (τ − ε(τ))(y · y) = τ(y · y) − y · y = 0

since ε(τ) = 1 and τ(y · y) = y · y by (5.2). Hence Γ(τ, y · y) is an element in π_{1}(X) = kernel(∂ : X_{1} → X_{0}). On the other hand we have π_{0}(Y) = cokernel(∂ : Y_{1} → Y_{0}). If we replace y by y + ∂y_{1} ∈ Y_{0} with y_{1} ∈ Y_{1} we get

Γ(τ, (y + ∂y_{1}) · (y + ∂y_{1})) = Γ(τ, y · y) + Δ

where

Δ = Γ(τ, y · (∂y_{1}) + (∂y_{1}) · y) + Γ(τ, (∂y_{1}) · (∂y_{1})).

Since ε(τ) = 1 we see by (5.3) and (5.4) that Δ = 0. Hence the function Sq with Sq{y} = Γ(τ, y · y) is well defined. We obviously have Γ(τ, (r · y)(r · y)) = Γ(τ, r^{2}y · y) = r^{2}Γ(τ, y · y) and we have Γ(τ, (y + z)(y + z)) = Γ(τ, y · y) + Γ(τ, z · z) + Γ(τ, y · z + z · y). Here the last term vanishes by (5.3). Hence we get the equations Sq(ra) = r^{2} Sq(a) and Sq(a + b) = Sq(a) + Sq(b) with a, b ∈ π_{0}(Y). Hence Sq(0) = 0 and then 0 = Sq a + Sq(−a) where Sq(−a) = (−1)^{2} Sq(a) = Sq(a). Hence 2 Sq(a) = 0. q.e.d.

The main examples of τ–pairs and τ–squares arise by the following result. (5.6) Proposition. For a pointed space X we obtain by τ_{n,m} ∈ σ_{n+m} and (4.6) the τ–pair

H^{n}(X) ⊗ H^{m}(X) ⟶^{μ_{∗}} H^{n+m}(X) ⟵^{μ_{∗}} H^{m}(X) ⊗ H^{n}(X)

which is a τ–square if n = m. Proof. This is an easy application of the commutative diagram in property (c) of the cup product map μ in the introduction. q.e.d.

6 Secondary cohomology

Using the tensor product of secondary modules we introduce the notion of a secondary algebra. We define a functor which associates with each space X a secondary cohomology algebra H^{∗}(X).

We consider a sequence R_{∗} of augmented k–algebras R_{n}, n ≥ 0, together with augmented algebra maps

i_{n,m} = ⊙ : R_{n} ⊗ R_{m} → R_{n+m}  (6.1)

carrying α ⊗ β to α ⊙ β such that for γ ∈ R_{k} we have

(α ⊙ β) ⊙ γ = α ⊙ (β ⊙ γ)

in R_{n+m+k}. Since ⊙ is an algebra map we have

(α · α′) ⊙ (β · β′) = (α ⊙ β) · (α′ ⊙ β′)

where α · α′ denotes the product in R_{n}. Let 1_{n} ∈ R_{n} be the unit element of R_{n} with 1_{n} ⊙ 1_{m} = 1_{n+m}. For n = 0 we have R_{0} = k and 1_{0} ∈ R_{0} satisfies 1_{0} ⊙ α = α ⊙ 1_{0} = α. We call R_{∗} = (R_{∗}, ⊙) a coefficient algebra.

Of course we have the trivial coefficient algebra k with R_{n} = k for n ≥ 0. On the other hand we shall use the symmetric coefficient algebra k[σ_{∗}] given by the augmented group algebras R_{n} = k[σ_{n}] where σ_{n} is the symmetric group and R_{n} has the sign augmentation (1.1)(3). Moreover ⊙ = i_{m,n} is induced by the inclusion of groups σ_{n} × σ_{m} ⊂ σ_{n+m} as in (5.1)(4). (6.2) Definition. An algebra V over a coefficient algebra R_{∗} is a sequence of R_{n}–modules V^{n}, n ≥ 0, together with k–linear maps

V^{n} ⊗ V^{m} → V^{n+m}

carrying x ⊗ y to x · y. For z ∈ V^{k} we have in V^{n+m+k}

(x · y) · z = x · (y · z)

and for α ∈ R_{n}, β ∈ R_{m} we have

(αx) · (βy) = (α ⊙ β)(x · y).

We do not assume that the algebra V has a unit. Let V and W be such algebras over R_{∗}. Then a map f : V → W over R_{∗} is given by an R_{n}–linear map f = f^{n} : V^{n} → W^{n}, n ≥ 0, with f(x · y) = f(x) · f(y). This defines the category of algebras over R_{∗}.

If R_{∗} = k is the trivial coefficient algebras then V in (6.2) is just a graded algebra over k. A graded algebra V is commutative if for x ∈ V^{n}, y ∈ V^{m} we have

y · x = (−1)^{nm}x · y  (6.3)

For example the reduced cohomology H̃^{∗}(X, k) of a pointed space X with coefficients in k is a commutative graded algebra. We generalize this notion of commutative algebras as follows. (6.4) Definition. Let R_{∗} be a coefficient algebra and assume elements

τ_{m,n} ∈ R_{n+m} (n, m ≥ 0)

are given with the following properties (m, n, k ≥ 0).

τ_{m,n}τ_{n,m} = 1_{n+m}
τ_{m,0} = τ_{0,m} = 1_{m}
τ_{n,m}(β ⊙ α) = (α ⊙ β)τ_{n,m} for α ∈ R_{n}, β ∈ R_{m}
τ_{m+n,k} = (τ_{m,k} ⊙ 1_{n})(1_{m} ⊙ τ_{n,k})
ε(τ_{m,n}) = (−1)^{m,n} ∈ k

Then we say that an algebra V over R_{∗} is τ–commutative if for x ∈ V^{n}, y ∈ V^{m}

y · x = τ_{n,m}(x · y)

in V^{n+m}.

For example we have as in (5.1)(3) the interchange elements τ_{n,m} ∈ k[σ_{n+m}] in the symmetric coefficient algebra for which a τ–commutative algebra is the same as a “commutative twisted algebra” in the sense of Stover [St]. On the other hand we can define the interchange elements τ_{n,m} = (−1)^{nm} ∈ k in the trivial coefficient algebra so that in this case a τ–commutative algebra is the same as a commutative graded algebra in (6.3). We now are ready to introduce the notion of a secondary algebra. (6.5) Definition. Let R_{∗} be a coefficient algebra. A secondary algebra H^{∗} over R_{∗} consists of a sequence of secondary modules H^{n} over R_{n}, n ≥ 0, together with i_{n,m}–equivariant maps

μ = μ_{n,m} : H^{n} ⊗ H^{m} → H^{n+m}  (1)

for n, m ≥ 0 which are associative in the sense that the diagram

\xymatrix{H^{n} ⊗ H^{m} ⊗ H^{r} \ar[r]^{1⊗μ} \ar[d]_{μ⊗1} & H^{n} ⊗ H^{m+r} \ar[d]^{μ} \\ H^{n+m} ⊗ H^{r} \ar[r]^{μ} & H^{n+m+r}}  (2)

commutes. Here we use the tensor product of secondary modules. If elements τ_{n,m} ∈ R_{n+m} are given as in (6.4) we say that the secondary algebra H^{∗} is τ–commutative if

H^{n} ⊗ H^{m} ⟶^{μ} H^{n+m} ⟵^{μ} H^{m} ⊗ H^{n}  (3)

is a τ_{n,m}–pair in the sense of (5.2). We study τ–commutative secondary algebras in more detail in section §9 below.

We say that the secondary algebra H^{∗} is w–closed if one has k–linear isomorphisms

w = w^{n} : π_{1}H^{n} ≅ π_{0}H^{n−1}  (4)

for n ≥ 1 which satisfy

\begin{aligned}w^{n+m}(y_{1} · z) & = w^{n}(y_{1}) · z \\ w^{n+m}(y · z_{1}) & = (−1)^{n}y · w^{m}(z_{1}) \end{aligned}  (5)

for y ∈ π_{0}H^{n}, z ∈ π_{0}H^{m}, y_{1} ∈ π_{1}H^{n}, z_{1} ∈ π_{1}H^{m}. In (5) the multiplication is defined by the maps (1) as in (5.2)(5). A map f : H^{∗} → G^{∗} between secondary algebras is given by a sequence f^{n} : H^{n} → G^{n} of R_{n}–equivariant maps between secondary modules such that f^{n} is compatible with μ in (1) and w in (4). Such a map f is a weak equivalence if f^{n} is a weak equivalence in secmod for n ≥ 0.

Let secalg be the category of secondary algebras over the symmetric coefficient algebra k[σ_{∗}] which are τ–commutative and w–closed. For an object H^{∗} in secalg we obtain a commutative graded algebra H^{∗} by

H^{n} = π_{0}H^{n} for n ≥ 0  (6.6)

with the multiplication H^{n} ⊗ H^{m} → H^{n+m} induced by μ in (6.5)(1). As in (5.2)(6) we see that H^{∗} is commutative since H^{∗} is τ–commutative. Moreover μ : H^{n} ⊗H^{n} → H^{2n} for n ≥ 1 is a τ–square which yields the squaring operation

Sq^{n−1} = w^{2n−1} Sq : H^{n} → H^{2n−1}  (6.7)

with 2 Sq^{n−1} = 0 by (5.5). Here we use the assumption that H^{∗} is w–closed.

The next result describes the secondary cohomology algebra H^{∗}(X) of a pointed space X. (6.8) Theorem. There is a functor

H^{∗} : Top^{∗} → secalg

which carries a pointed space X to a secondary algebra H^{∗}(X) which is τ–commutative and w–closed. Moreover the algebra H̃^{∗} = π_{0}H^{∗}(X) is (6.6) coincides with the reduced cohomology algebra H̃^{∗}(X; k) and for k = 𝔽_{2} the squaring operation Sq^{n−1} in (6.7) coincides with the corresponding Steenrod operation. Proof. This is an easy consequence of (3.7) and property (c) of the cup product maps in the introduction. The result on Steenrod squares is a consequence of a result of Kristensen, lemma 2.5 in [K].

7 Crossed modules and Hochschild cohomology

We introduce the concept of a crossed module in the context of algebras and we show that (in the graded case) a crossed module is the same as a secondary algebra over the trivial coefficient algebra R_{∗} = k. A crossed module and equivalently a secondary algebra over k represent an element in the third Hochschild cohomology. This leads to the notion of a characteristic class of a differential algebra.

We here consider the graded and the non–graded case at the same time. A graded vector space V is assumed to be positively graded, i. e. V^{i} = 0 for i < 0.

We use the following notation. An algebra Ã is given by a (graded) k–vector space Ã and a multiplication map Ã ⊗ Ã → Ã which is associative. On the other hand a k–algebra A is an algebra with unit k → A and augmentation ε : A → k. Hence a k–algebra A is an algebra under and over k. Then the augmentation ideal

Ã = kernel(ε : A → k)

is an algebra which determines the k–algebra A = Ã⊕k completely. Moreover an Ã–module is also an A–module and vice versa.

Let A be a (graded) k–algebra. An A–bimodule V is a (graded) k–vector space which is a left and a right A–module such that for a, b ∈ A, x ∈ V we have (a · x) · b = a · (x · b). For example A can be considered as an A–bimodule via the multiplication in A. (7.1) Definition. A crossed module is a map of A–bimodules

∂ : V → A

satisfying ε∂ = 0 and (∂v) · w = v · (∂w) for v, w ∈ V.

Let π_{0}(∂) = cokernel(∂) and π_{1}(∂) = kernel(∂) in the category of (graded) vector spaces. Then the algebra structure of A induces an algebra structure of π_{0}(∂) and the A–bimodule structure of V induces a π_{0}(∂)–bimodule structure of π_{1}(∂). In fact for {a} ∈ π_{0}(∂) the multiplication {a}·v = a·v with v ∈ π_{1}(∂) is well defined since (a+∂w)·v = a·v +(∂w)·v = a · v + w · ∂v = a · v where ∂v = 0. Hence a crossed module yields the exact sequence

0 ⟶ π_{1}(∂) ⟶^{i} V ⟶^{∂} A ⟶^{q} π_{0}(∂) ⟶ 0

in which all maps are A–bimodule morphisms. Here the A–bimodule structure of π_{0}(∂) and π_{1}(∂) is induced by the algebra map q. (7.2) Lemma. A secondary algebra H^{∗} over the trivial coefficient algebra k as defined in (6.5) is the same as a crossed module ∂. Proof. Given H^{∗} we obtain

∂ : H_{1}^{∗} → H_{0}^{∗}

where H_{0}^{∗} is an algebra by the multiplication μ in (6.5)(1). Moreover H_{1}^{∗} is a H_{0}^{∗}–bimodule by the multiplication (6.5)(1). Using (4.1) we see that ∂ yields the crossed module

∂ = (∂, 0) : H_{1}^{∗} → H_{0}^{∗} ⊕ k

where H_{0}^{∗}⊕k is the k–algebra given by H_{0}^{∗}. Conversely it is easy to see that a crossed module (7.1) defines a secondary algebra over k. q.e.d.

We now use crossed modules or equivalently secondary algebras over k for the definition of Hochschild cohomology. (7.3) Definition. Let H be a (graded) k–algebra and let M be an H–bimodule. A crossed n–fold extension of H by M with n ≥ 2 is an exact sequence in the category of (graded) k–vector spaces

E : 0 ⟶ M ⟶^{γ} V_{n−1} ⟶^{∂_{n−1}} … ⟶ V_{1} ⟶^{∂_{1}} A ⟶^{q} H ⟶ 0

where V_{n−1}, …, V_{2} are H–bimodules and ∂_{1} is a crossed module. Moreover all maps are A–bimodule maps with the A–bimodule structure induced by the algebra map q : A → H. A weak equivalence between two such extensions is a commutative diagram

[figure: M; V_{n−1}; f_{n−1}; …; V_{1}; f_{1}; A; f_{0}; H; M; W_{n−1}; …; W_{1}; B; H]

where f_{0} is an algebra map and f_{1} is a f_{0}–biequivariant homomorphism and f_{i} for n − 1 ≥ i ≥ 2 is an H–bimodule map. For n ≥ 2 let HH^{n+1}(H, M) be the set of weak equivalence classes of crossed n–fold extensions of H by M.

In Baues–Minian [BM] we show: (7.4) Proposition. HH^{n+1}(H, M) as defined above coincides naturally with the Hochschild cohomology of H with coefficients in M, n ≥ 2.

The proposition holds in the graded and in the non–graded case.

We have seen in (7.2) that each secondary algebra H^{∗} over k yields a canonical crossed extension

0 ⟶ π_{1}(H^{∗}) ⟶ H_{1}^{∗} ⟶^{∂} H_{0}^{∗} ⊕ k ⟶ π_{0}(H^{∗}) ⊕ k ⟶ 0

Here H = π_{0}(H^{∗}) ⊕ k is a k–algebra and M = π_{1}(H^{∗}) is an H–bimodule. Hence the crossed extension represents an element

<H^{∗} > ∈ HH^{3}(H, M)  (7.5)

which is termed the characteristic class of the secondary algebra H^{∗}. On the other hand a differential algebra C (like the cochain algebra of a space) as well yields a crossed extension representing a characteristic class <C > as in the following example. (7.6) Example. Let C be a differential graded k–algebra, that is, C = {C^{i}, i ≥ 0} with C^{i}C^{j} ⊆ C^{i+j} and d : C → C of degree +1 satisfying d(xy) = (dx)y + (−1)^{|x|}xd(y) and dd = 0 and εd = 0. Then d induces the map of graded k–vector spaces

V = coker(d̃)[1] ⟶^{∂} ker(d) = A  (1)

Here we define for a graded vector space W the shifted graded vector space W [1] by

W^{n} = (W [1])^{n+1}, w ↦ s(w),  (2)

Hence for the cokernel of the differential coker(d) = C̃/ im(d̃) we obtain the shifted object V = coker(d̃)[1]. Since d is of degree +1 the boundary induces ∂ by ∂s{v} = d(v) for {v} ∈ coker(d), v ∈ C. The algebra C induces an algebra structure of A = ker(d). Moreover it induces the structure of an A–bimodule on V by setting

\begin{aligned}a · (s\{v\}) & = (−1)^{|a|}s\{a · v\} \\ (s\{v\}) · b & = s\{v · b\} \end{aligned}  (3)

One can check that ∂ : V → A is a crossed module in the sense of (7.1), see [BM]. This proves that ∂ : V → A is crossed module and therefore we obtain by (7.2) a secondary algebra ∂ : V → Ã over k which is, in fact, w–closed (see (6.5)) by defining

w : π_{1}(∂) = H̃^{∗}(C)[1] ≅ H̃^{∗}(C) = π_{0}(∂)  (4)

with ws(x) = x for x ∈ H^{∗}(C) = ker(d)/ im(d). The equations in (3) for the A–bimodule structure of V correspond exactly to the equations in (6.5)(5).

According to (3) we define for a graded algebra H^{∗} the H^{∗}–bimodule H̃^{∗}[1] by setting

\begin{aligned}a · (sx) & = (−1)^{|a|}s(a · x) \\ (sx) · b & = s(x · b) \end{aligned}  (5)

Then we obtain by (1) and (4) the crossed 2–extension

0 ⟶ H̃^{∗}[1] ⟶ V ⟶^{∂} A ⟶ H^{∗} ⟶ 0

which by (7.4) represents an element

<C > ∈ HH^{∗}(H^{∗}, H̃^{∗}[1])  (6)

where H^{∗} = H^{∗}C is the cohomology algebra of the differential algebra C. A cocycle θ representing <C > is considered in Berrick–Davydov [BD].

As a special case we obtain for a pointed space X the augmented algebra of cochains on X denoted by C^{∗}X for which

H^{∗}(C^{∗}X) = H^{∗}(X)

is the cohomology algebra of X. Hence we get by (7.6)(6) the class

<C^{∗}(X)> ∈ HH^{∗}(H^{∗}(X), H̃^{∗}(X)[1])  (7.7)

which is an invariant of the homotopy type of X in the sense that a pointed map f : X → Y satisfies

(f^{∗})^{∗} <C^{∗}(X)> = (f^{∗}[1])_{∗} <C^{∗}(Y)>

in HH^{3}(H^{∗}(Y), H̃^{∗}(X)[1]) where f^{∗} : H^{∗}(Y) → H^{∗}(X) yields the structure of an H^{∗}(Y)– bimodule on H̃^{∗}(X)[1].

We now compare the class (7.7) with the secondary cohomology algebra H^{∗}(X) in (6.8). (7.8) Proposition. By forgetting structure we obtain from the secondary cohomology H^{∗}(X) a secondary algebra over k denoted by H^{∗}(X)_{(k)}. Then the classes

<C^{∗}X > = <H^{∗}(X)_{(k)} > ∈ HH^{3}(H^{∗}(X), H̃^{∗}(X)[1])

given by (7.5) and (7.7) coincide. Proof. Using (3.3) and the definition of Y_{3} in Baues [B] we see that <H(X, Y_{3})> = <C^{∗}X > for a simplicial set X. Here we use the universal property of Y_{3} which says that a simplicial map X → Y_{3} can be identified with a cocycle in C^{∗}X. q.e.d.

8 τ–crossed modules for commutative graded algebras

We have seen in §7 that a secondary algebra over the trivial coefficient algebra k is the same as a crossed module. We here show that in a similar way a τ–commutative secondary algebra over R_{∗} is the same as a τ–crossed module. Weak equivalence classes of τ–crossed modules yield an abelian group which we call the third symmetric cohomology.

Let R_{∗} be a coefficient algebra with interchange elements τ_{m,n} ∈ R_{m+n}, for example let R_{∗} = k[σ_{∗}] be the symmetric coefficient algebra. An R_{∗}–module V is a sequence of (left) R_{n}–modules V^{n}, n ≥ 0. A map or an R_{∗}–linear map f : V → W between R_{∗}–modules is given by a sequence of R_{n}–linear maps f^{n} : V^{n} → W^{n} for n ≥ 0. The field k (concentrated in degree 0) is an R_{∗}–module. Moreover using the augmentation ε of R_{n}, n ≥ 0, we see that each graded k–vector space M is an R_{∗}–module which we call an ε–module. For x ∈ M^{m} we write |x| = m where |x| is the degree of x.

Given R_{∗}–modules V_{1}, …, V_{k} we define the R_{∗}–tensor product V_{1}\overline{⊗} … \overline{⊗}V_{k} by

(V_{1}\overline{⊗} … \overline{⊗}V_{k})_{n} = ⨁_{n_{1}+...+n_{k}=n} R_{n} ⊗_{R_{n_{1}}⊗...⊗R_{n_{k}}} V_{1}^{n_{1}} ⊗ … ⊗ V_{k}^{n_{k}}  (8.1)

where we use the algebra map ⊙ : R_{n_{1}} ⊗ … ⊗ R_{n_{k}} → R_{n} given by the structure of the coefficient algebra R_{∗} in (6.1). One readily checks associativity

(V_{1,1}\overline{⊗} … \overline{⊗}V_{1,k_{1}})\overline{⊗} … \overline{⊗}(V_{s,1}\overline{⊗} … \overline{⊗}V_{s,k_{s}}) = V_{1,1}\overline{⊗} … \overline{⊗}V_{1,k_{1}} \overline{⊗} … \overline{⊗}V_{s,1}\overline{⊗} … \overline{⊗}V_{s,k_{s}} (1)

Compare Stover [St] 2.9. Moreover the interchange element τ in R_{∗} yields the isomorphism

T : V \overline{⊗}W ≅ W \overline{⊗}V  (2)

which carries v ⊗ w to τ_{w,v}w ⊗ v where

τ_{w,v} = τ_{m,n} ∈ R_{m+n}

for w ∈ W^{m}, v ∈ V^{n}. Of course we have k\overline{⊗}V = V = V \overline{⊗}k. (8.2) Definition. An algebra A over R_{∗} is given by an R_{∗}–linear map μ : A\overline{⊗}A → A, μ(a⊗ b) = a · b, which is associative in the sense that the diagram

\xymatrix{A\overline{⊗}A\overline{⊗}A \ar[r]^{1\overline{⊗}μ} \ar[d]_{μ\overline{⊗}1} & A\overline{⊗}A \ar[d]^{μ} \\ A\overline{⊗}A \ar[r]^{μ} & A}

commutes. Moreover A is τ–commutative if

[figure: A\overline{⊗}A; μ; T; A; A\overline{⊗}A; μ; A]

commutes. One readily checks that this coincides with the notation in (6.2) and (6.3). We say that A is a k–algebra over R_{∗} if algebra maps k ⟶^{i} A ⟶^{ε} k are given with εi = 1. Such a k–algebra A over R_{∗} is completely determined by the algebra Ã over R_{∗} with Ã = kernel(ε : A → k) and A = k ⊕ Ã.

For an R_{∗}–module V let V_{(k)} be the underlying graded k–vectorspace. If A is a k–algebra over R_{∗} then A_{(k)} is a k–algebra (over k) in the sense of §7 above.

(8.3) Definition. Given an algebra A over R_{∗} we say that an R_{∗}–module V is an A–module if a map m : A\overline{⊗}V → V is given such that

\xymatrix{A\overline{⊗}A\overline{⊗}V \ar[d]_{μ\overline{⊗}1} \ar[r]^{1\overline{⊗}μ} & A\overline{⊗}V \ar[d]^{μ} \\ A\overline{⊗}V \ar[r]^{μ} & V}

commutes. Hence for a · x = μ(a ⊗ x) with a ∈ A, x ∈ V we have (αa) · (βx) = (α ⊙ β)(a · x) and (a · b) · x = a · (b · x). If A is a k–algebra over R_{∗} we also assume that 1 · x = x for 1 ∈ k, x ∈ V. Then the A–module V is an Ã–module and vice versa. In particular the algebra A is also an A–module in the obvious way. (8.4) Lemma. Let A be a τ–commutative k–algebra over R_{∗} and let V be an A–module. Then V_{(k)} is a A_{(k)}–bimodule by defining

a · x · b = a · τ_{b,x}(b · x)

for a, b ∈ A, x ∈ V. Proof. We write 1_{x} = 1_{n} ∈ R_{n} for x ∈ V^{n}. Now we have for a, b ∈ A

(a · x) · b = τ_{b,a·x}b · (a · x) = τ_{b,a·x}(b · a) · x
= τ_{b,a·x}(τ_{a,b}a · b) · x
= τ_{b,a·x}(τ_{a,b} ⊙ 1_{x})(a · b · x)
a · (x · b) = a · τ_{b,x}(b · x) = (1_{a} ⊙ τ_{b,x})(a · b · x)

Here we have τ_{b,a·x}(τ_{a,b} ⊙ 1_{x}) = 1_{a} ⊙ τ_{b,x} by one of the equations in (7.4). q.e.d.

If A and V in (8.4) are ε–modules then (8.4) corresponds to the following special case. (8.5) Lemma. Let H be a commutative graded k–algebra and let M be an H–module. Then M is an H–bimodule by defining

a · x · b = a · (−1)^{|b||x|}b · x

for a, b ∈ H and x ∈ M.

For an R_{∗}–module V we obtain as in (1.6) the R_{∗}–linear map

I(R_{∗}) ⊙_{R_{∗}} V ⟶^{μ} V

Here the left hand side is the R_{∗}–module given in degree n by I(R_{n}) ⊗_{R_{n}} V^{n} ond μ carries a ⊗ x to a · x. (8.6) Definition. Let A be a τ–commutative k–algebra over R_{∗}. A τ–crossed module ∂ is given by a commutative diagram of R_{∗}–linear maps

\xymatrix{I(R_{∗}) ⊙_{R_{∗}} V \ar[d]_{μ} \ar[r]^{1⊙∂} & I(R_{∗}) ⊙_{R_{∗}} A \ar[dl]_{Γ̃} \ar[d]^{μ} \\ V \ar[r]^{∂} & A}

with the following properties (1) and (2). The R_{∗}–module V is an A–module and ∂ is an A–module morphism, that is ∂(a · x) = a · (∂x) for a ∈ A, x ∈ V. Moreover ε∂ = 0 and for x, y ∈ V

(∂x) · y = τ_{∂y,x}(∂y) · x.  (1)

The R_{∗}–linear map Γ̃ satisfies for β ∈ I(R_{∗}) and a, b ∈ A the equation

a · Γ̃(β ⊗ b) = Γ̃(1 ⊙ β ⊗ a · b)  (2)

Equation (2) shows that Γ̃ is a map of left A–modules. Moreover (1) and (2) imply that for α ∈ I(R_{∗}) the following equation holds.

Γ̃(α ⊗ a) · b = Γ̃(α ⊙ 1 ⊗ a · b)  (3)

Here the right hand action of b on x = Γ̃(α ⊗ a) ∈ V is defined as in (8.4) by x · b = τ_{b,x}b · x. Using this notation (1) is equivalent to (∂x) · y = x · (∂y), compare (7.1). Proof of (3). For x ∈ Γ̃(α ⊗ a) we have |x| = |a| and hence τ_{b,x} = τ_{b,a}. Therefore we get

Γ̃(α ⊗ a) · b = τ_{b,x}b · Γ̃(α ⊗ a)
= τ_{b,x}Γ̃(1 ⊙ α ⊗ b · a)
= Γ̃(τ_{b,a}(1 ⊙ α) ⊗ b · a)
= Γ̃((α ⊙ 1)τ_{b,a} ⊗ b · a)
= Γ̃((α ⊙ 1) ⊗ τ_{b,a}b · a)
= Γ̃((α ⊙ 1 ⊗ a · b).

(8.7) Lemma. A τ–crossed module ∂ as in (8.6) yields for the underlying k–vector spaces a crossed module in the sense of (7.1)

∂ : V_{(k)} → A_{(k)}

where we use (8.4). Moreover π_{0}(∂) = H is a commutative graded k–algebra and π_{1}(∂) is an H–module with the H–bimodule structure in (8.5).

The lemma is based on the crucial property of a τ–crossed module, namely that π_{0}(∂) = cokernel(∂) and π_{1}(∂) = kernel(∂) are only ε–modules though V and A are R_{∗}–modules. A τ–crossed module yields the exact sequence

0 ⟶ π_{1}(∂) ⟶^{i} V ⟶^{∂} A ⟶^{q} π_{0}(∂) ⟶ 0  (8.8)

in which all maps are A–module morphisms. Here the A–module structures of π_{0}(∂) and π_{1}(∂) are induced by the algebra map q. Moreover for the underlying k–vector spaces this is by (8.7) a crossed extension as in (7.1), (7.3).

The next result generalizes lemma (7.2) on crossed modules. (8.9) Lemma. A τ–commutative secondary algebra H^{∗} over the coefficient algebra R_{∗} as defined in (6.5) is the same as a τ–crossed module ∂ in (8.6).

Proof. Given H^{∗} we obtain

∂ : H_{1}^{∗} → H_{0}^{∗} ⊕ k = A

where A is a τ–commutative k–algebra by the multiplication μ in (6.5)(1). Moreover H_{1}^{∗} = V is an A–module and one now readily checks by (4.2) and (1.6) that ∂ satisfies the properties of a τ–crossed module. Conversely given (∂, Γ̃) as in (8.6) we obtain the secondary module H^{n} over R_{n} by the commutative diagram (see (1.6))

\xymatrix{I(R_{n}) ⊗_{R_{n}} V^{n} \ar[r]^{1⊗∂} \ar[d]_{μ} & I(R_{n}) ⊗_{R_{n}} Ã^{n} \ar[dl]_{Γ̃} \ar[d]^{μ} \\ V^{n} \ar[r]^{∂} & Ã^{n}}

which we deduce from the diagram in (8.6). Moreover we define

μ : H^{n} ⊗ H^{m} → H^{n+m}

by the multiplication

Ã^{n} ⊗ Ã^{m} = H_{0}^{n} ⊗ H_{0}^{m} ⟶^{μ} Ã^{n+m} = H_{0}^{n+m}

of the algebra Ã over R_{∗} and by the map

[figure: (H^{n} ⊗ H^{m})_{1}; μ; H_{1}^{n+m}; (V^{n} ⊗ Ã^{m} ⊕ Ã^{n} ⊗ V^{m})/ im d_{2}; μ; V^{n+m}]

with μ(x ⊗ a) = x · a and μ(b ⊗ y) = b · y where x · a = τ_{a,x}a · x. By (8.6)(1) this map is trivial on im(d_{2}). Now it is easy to show that the multiplication μ on H^{∗} is associative (compare the proof on (8.4)) and τ–commutative. Finally we have to check that μ on H^{∗} is compatible with the equation in (4.2). This follows from (8.7)(2),(3) since for α′ = α + α_{k} ∈ I(R^{n}) ⊕ k = R^{n} and β′ = β + β_{k} ∈ I(R^{m}) ⊕ k = R^{m} we have

μΓ_{⊗}(α′ ⊗ β′ ⊗ x ⊗ y) = Γ̃(α ⊙ β ⊗ x · y) + β_{k}Γ̃(α ⊙ 1 ⊗ x · y) + α_{k}Γ̃(1 ⊙ β ⊗ x · y)

Here we have α ⊙ β = (α ⊙ 1)(1 ⊙ β) and hence in I(R^{n+m}) ⊗_{R^{n+m}} V^{n+m} we have

α ⊙ β ⊗ x · y = (α ⊙ 1)(1 ⊙ β) ⊗ x · y
= α ⊙ 1 ⊗ (1 ⊙ β)x · y
= α ⊙ 1 ⊗ x · βy

Therefore we get by (8.7)(2),(3)

μΓ_{⊗}(α′ ⊗ β′ ⊗ x ⊗ y) = Γ̃(α ⊗ x) · (βy + β_{k}y) + α_{k}x · Γ̃(β ⊗ y)
= μ(Γ(α′ ⊗ x) ⊗ β′y + (ε(α′)x) ⊗ Γ(β′ ⊗ y))

Hence μ is compatible with the equation in (4.2). q.e.d.

We now use τ-crossed modules (or by (8.9) equivalently τ–commutative secondary algebras) for the following definition of symmetric cohomology which is a symmetric analogue of Hochschild cohomology in (7.3). (8.10) Definition. Let R_{∗} be a coefficient algebra with interchange elements τ for example let R_{∗} = k[σ_{∗}] be the symmetric coefficient algebra. Let H be a commutative graded k– algebra and let M be an H–module. A τ–crossed extension E of H by M is an exact sequence of graded k–vector spaces

E : 0 ⟶ M ⟶ V ⟶^{∂} A ⟶^{q} H ⟶ 0

Here ∂ is a τ–crossed module as in (8.6). Moreover all maps are A–module morphisms with the A–module structure induced by the algebra map q : A → H. A weak equivalence between two such τ–crossed extensions is a commutative diagram

[figure: 0; M; V; f_{1}; A; f_{0}; H; 0; 0; M; W; B; H; 0]

Here f_{0} is a morphism of k–algebras over R_{∗} and f_{1} is a f_{0}–equivariant homomorphism such that (f_{0}, f_{1}) is compatible with Γ̃. Let SH^{3}(H, M) be the set of weak equivalence classes of τ–crossed extensions of H by M. Below we show that SH^{3}(H, M) is a well defined set with the structure of a k–vector space. We call SH^{3}(H, M) the third symmetric cohomology of H with coefficients in M.

At this moment we do not know a “cohomology theory” for commutative graded algebras H which yields the symmetric cohomology SH^{3}(H, M) above.

We have the canonical natural homomorphism

SH^{3}(H, M) → HH^{3}(H, M)  (8.11)

which carries the weak equivalence class of the τ–crossed extension E to the weak equivalence class of the underlying crossed extension E_{(k)} given by (8.7), (8.4) and (8.5). We need the following “free” objects. (8.12) Definition. Let V be a graded k–vector space. Then the free R_{∗}–module R_{∗} ⊙ V generated by V is given by

(R_{∗} ⊙ V)^{n} = R_{n} ⊗ V^{n}.

For an R_{∗}–module W we obtain the tensor algebra over R_{∗} by

\overline{T} (W) = ⨁_{n≥0} W^{\overline{⊗}n}

where W^{\overline{⊗}0} = k and W^{\overline{⊗}n} is the n–fold \overline{⊗}–product W \overline{⊗} … \overline{⊗}W defined in (8.1). For the tensor algebra T (V) over k we get

\overline{T} (R_{∗} ⊙ V) = R_{∗} ⊙ T (V)

so that R_{∗} ⊙ T (V) is the free k–algebra over R_{∗} generated by V with the multiplication

(α ⊗ x) · (β ⊗ y) = α ⊙ β ⊗ x · y

for α, β ∈ R_{∗} and x, y ∈ T (V). Let

K_{τ} ⊂ R_{∗} ⊙ T (V) = A

be the R_{∗}–submodule generated by elements 1 ⊗ y · x − τ_{x,y} ⊗ x · y for x, y ∈ T (V). Then K_{τ} generates the ideal A · K_{τ} · A and the R_{∗}–quotient module

Λ = A/A · K_{τ} · A

is the free τ–commutative k–algebra over R_{∗}.

Given a τ–commutative k–algebra A over R_{∗} and a k–linear map d : V → A with εd = 0 we obtain the following push out diagram in the category of R_{∗}–modules

\xymatrix{I(R_{∗}) ⊙ V \ar[r]^{1⊙d} \ar[d]_{i⊙1}^{push} & I(R_{∗}) ⊙_{R_{∗}} A \ar[d]^{i′} \ar[ddr]^{μ} \\ R_{∗} ⊙ V \ar[r] \ar[drr]_{d′′} & Y \ar[dr]^{d′} \\  &  & A}

Here d′′ is defined by d′′(α ⊗ x) = α · d(x). The pair (μ, d′′) induces the R_{∗}–linear map d′ which thus determines the map of A–modules ∂′ and ∂ in the following commutative diagram

[figure: I(R_{∗}) ⊙_{R_{∗}} A; Γ̃; i′; μ; A; Y; i′′; d′; A; A\overline{⊗}Y; ∂′; A; X_{1}(d); A\overline{⊗}Y/U; ∂; A]

with Γ^{≈} = i′′i′ and Γ̃ = pΓ^{≈} = pi′′i′. Here i′′ is defined by i′′(y) = 1 ⊗ y and ∂′ is defined by ∂′(a ⊗ y) = a · d′(y). Let U be the A–submodule of A\overline{⊗}Y generated by the elements

(∂′x) · y − τ_{∂′y,x}(∂′y) · x,
a · Γ^{≈}(β ⊗ b) − Γ^{≈}(1 ⊙ β ⊗ a · b),

with x, y ∈ A\overline{⊗}Y and a, b ∈ A, β ∈ I(R_{∗}). One readily checks that U is in the kernel of ∂′ so that ∂′ induces the A–module map ∂ on the quotient X_{1}(d) = A\overline{⊗}Y/U. We claim that (∂, Γ̃) is a well define τ–crossed module which is the free τ–crossed module with basis (V, d).

Finally we define for a commutative graded k–algebra the free H–module with basis V by H ⊗ V.

(8.13) Definition. Let H be a commutative graded k–algebra and consider a long exact sequence

R : … ⟶ C_{2} ⟶ C_{1} ⟶^{∂} Λ ⟶^{q} H ⟶^{0}

Here Λ is a free τ–commutative k–algebra over R_{∗} and (∂, Γ̃) is a free τ–crossed module and C_{i}, i ≥ 2, is a free H–module. All maps in the sequence are Λ–module morphisms where C_{i} and H are Λ–modules via the algebra map q. Then we call R a free τ–crossed resolution of H.

It is easy to see that free τ–crossed resolutions exist. Moreover given a τ–crossed extension E as in (8.10) we can choose a commutative diagram (n ≥ 2)

[figure: C_{3}; ∂; C_{2}; f_{2}; C_{1}; f_{1}; Λ; f_{0}; H; 0; E :; 0; M; V; A; H; 0]

so that one gets a weak equivalence

(f_{2})_{∗}R_{2} → E

where R_{2} is the τ–crossed extension

0 → C_{2}/∂C_{3} → C_{1} → Λ → H → 0

given by R. This implies that SH^{3}(H, M) is actually a set. In fact, the function

ψ : Hom_{H}(C_{2}/∂C_{3}, M) → SH^{3}(H, M)  (8.14)

which carries f_{2} : C_{2}/∂C_{3} → M to the weak equivalence class of (f_{2})_{∗}R_{2} is surjective. Using this surjection it is possible to define the k–vector space structure of SH^{3}(H, M).

Appendix: Eilenberg–MacLane spaces

Let k be a commutative ring, for example a field. For the definition of Z^{n} = K(k, n) in (2.5) we shall need the following categories and functors; compare Goerss–Jardine [GJ]. Let Set and Mod be the category of sets and k–modules respectively and let ΔSet and ΔMod be the corresponding categories of simplicial objects in Set and Mod respectively. We have functors

Top^{∗} ⟶^{Sing} (ΔSet)^{∗} ⟶^{| |} Top^{∗}

given by the singular set functor Sing and the realization functor | |. Moreover we have

ΔSet ⟶^{R} ΔMod ⟶^{Φ} (ΔSet)^{∗}

where k carries the simplicial set X to the free k–module generated by X and where Φ is the forgetful functor which carries the simplicial module A to the underlying simplicial set. Moreover we need the Dold–Kan functors

Ch_{+} ⟶^{Γ} ΔMod ⟶^{N} Ch_{+}

where Ch_{+} is the category of chain complexes in Mod concentrated in degree ≥ 0. Here N is the normalization functor which by the Dold–Kan theorem is an equivalence of categories with inverse Γ. For a pointed space V let

K(V) = |ΦS(V)| with S(V) = \frac{k Sing(V)}{k Sing(∗)}

Hence K : Top^{∗} → Top^{∗} carries a pointed space to a topological k–module. We define the binatural map

¯⊗ : K(V) × K(W) → K(V ∧ W)  (*)

as follows. We have Sing(V × W) = Sing(V) × Sing(W)

and this bijection induces a commutative diagram in ΔMod

[figure: R Sing(V) ⊗ R Sing(W); R Sing(V × W); S(V) ⊗ S(W); Λ; S(V ∧ W)]

The vertical arrows are induced by quotient maps. For k–modules A, B let ⊗ : A×B → A⊗_{k} B be the map in Set which carries (a, b) to the tensor product a⊗b. Of course this map ⊗ is bilinear. Moreover for A, B in ΔMod the map ⊗ induces the map ⊗ : Φ(A×B) → Φ(A⊗B) in Set and the realization functor yields

| ⊗ | : |ΦA| × |ΦB| = |Φ(A × B)| → |Φ(A ⊗ B)|

Hence for A = S(V) and B = S(W) we get the composite

|ΦS(V)| × |ΦS(W)| ⟶^{|⊗|} |Φ(S(V) ⊗ S(W))| ⟶^{|ΦΛ|} |ΦS(V ∧ W)|

and this is the map ¯⊗ above. One readily checks that ¯⊗ is bilinear with respect to the topological k–module structure of K(V), K(W) and K(V ∧ W) respectively. Moreover the following diagram commutes

[figure: K(V) × K(W); ¯⊗; K(V ∧ W); |SingV | × |SingW |; h×h; |Sing(V × W)|; |Sing(V ∧ W)|; h]  (**)

Here the Hurewicz map h is the realization of the map in ΔSet

Sing(V) → Φk Sing(V) → ΦS(V)

which carries an element x in Sing(V) to the corresponding generator in k Sing(V).

Let S^{n} = S^{1} ∧ … ∧ S^{1} be the n–fold smash product of the 1–sphere S^{1}. Then the symmetric group σ_{n} acts on S^{n} by permuting the factors S^{1}. It is well known that this action of σ_{n} on S^{n} induces the sign–action of σ_{n} on the homology H_{n}(S^{n}) = k. We define the Eilenberg–MacLane space Z^{n} by

Z^{n} = K(S^{n}) = |Φ\frac{kSing(S^{n})}{kSing(∗)}|

Since K is a functor we see that σ_{n} also acts on K(S^{n}) via k–linear automorphisms. We define the multiplication map μ_{m,n} by

μ : Z^{m} × Z^{n} = K(S^{m}) × K(S^{n}) ⟶^{¯⊗} K(S^{m} ∧ S^{n}) = Z^{m+n}

where S^{m} ∧ S^{n} = S^{m+n} and where we use (*). Diagram (**) implies that μ induces the cup product in cohomology. The Eilenberg–MacLane spaces Z^{n} together with the multiplication map μ satisfy the axioms of Karoubi [Ka].

References

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[BM] Baues, H.–J. and Minian, E. G.: Crossed extensions of algebras and Hochschild cohomology. Preprint (2000)

[GJ] Goerss, P. and Jardine, J. F.: Simplicial homotopy theory. Birkhäuser Verlag, Progress in Math. 174, (1999) 510 pages

[H] Huebschmann, J.: Crossed n–fold extensions of groups and cohomology. Comment. Math. Helvetici 55 (1980) 302–314

[HSS] Hovey, M. and Shipley, B. and Smith, J.: Symmetric spectra. Journal of the AMS 13 (2000) 149–208

[K] Kristensen, L.: On secondary cohomology operations, Math. Scand. 12 (1963) 57–82

[Ka] Karoubi, M.: Formes differentielle non commutatives et operation de Steenrod. Topology 34 (1995) 699–715

[St] Stover, C. R.: The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring. Journal of pure and applied Algebra 86 (1993) 289–326

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Hans–Joachim Baues Max–Planck–Institut für Mathematik Vivatsgasse 7 D–53111 Bonn Germany

E–Mail: baues@mpim-bonn.mpg.de