(also nonabelian homological algebra)
The Dold-Kan correspondence relates simplicial groups to chain complexes. The Eilenberg-Zilber theorem says how in this context double complexes and their total complexes relate to bisimplicial groups and their diagonals/total simplicial sets.
Analogously there is also a version of the theorem for bi-cosimplicial abelian groups.
Let $A : \Delta^{op} \times \Delta^{op} \to Ab$ be a bisimplicial abelian group. Write
$C_\bullet diag A$ for the Moore complex of its diagonal simplicial group $diag A : \Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{A}{\to} Ab$;
$Tot (C A)$ for the total complex of the double complex obtained by applying the Moore complex functor on both arguments of $A$.
(Dold-Puppe generalization of Eilenberg-Zilber)
There is a quasi-isomorphism (even a chain-homotopy equivalence)
Notice (see the discussion at bisimplicial set) that the diagonal simplicial set is isomorphic to the realization given by the coend
Let $A : \Delta \times \Delta \to Ab$ be a bi-cosimplicial abelian group. And let $C : Ab^\Delta \to Ch^\bullet$ the Moore cochain complex functor. Write $C(A)$ for the double complex obtained by applying $C$ to each of the two cosimplicial directions. Then we have natural isomorphisms in cohomology
There is a natural isomorphism
A version for crossed complexes is given by Andy Tonks. We give a summary:
First note that there is a tensor product for crossed complexes developed by Brown and Higgins. Letting $K$ and $L$ be simplicial sets.
in a somewhat similar way to chain complexes.
is homotopic to the identity on $\pi(K\times L)$, whilst the other composite is the identity on $\pi K \otimes \pi L$, thus this is a strong deformation retract of $\pi(K\times L)$.
Cegarra and Remedios have proved a version of the Eilenberg - Zilber theorem for simplicial sets. This is discussed under the entry on bisimplicial sets.
This is the original motivating application.
Let $X$ and $Y$ be two topological spaces. Their chain homology complexes $C_\bullet(X)$ and $C_\bullet(Y)$ are the Moore complexes of the simplicial abelian groups $\mathbb{Z}[Sing X]$ and $\mathbb{Z}[Sing Y]$. So from the Dold-Puppe quasi-isomorphism $R$ from above we have a quasi-isomorphism from the singular cohomology of their product topological space
and hence in particular an isomorphism in cohomology.
By following through these maps one can obtain an explicit description of the quasi isomorphism if needs be.
The original reference is
A weak version of the simplicial statement is in theorem 8.1.5 in
The stronger version as stated above, published by Dold and Puppe, in chapter 2 of
where is is ascribed to Pierre Cartier. This result is discussed in chapter 4 of
The cosimplicial version of the theorem appears as theorem A.3 in
The crossed complex version is given in
(for more detail see Tonks’ thesis),
and on page 360 of Nonabelian Algebraic Topology.
Last revised on April 5, 2021 at 07:52:52. See the history of this page for a list of all contributions to it.