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.
A version for simplicial abelian groups:
Let be a bisimplicial abelian group. Write
for the Moore complex of its diagonal simplicial group ;
for the total complex of the double complex obtained by applying the Moore complex functor on both arguments of .
(Dold-Puppe generalization of Eilenberg-Zilber)
There is a quasi-isomorphism (even a chain-homotopy equivalence)
Let be a bi-cosimplicial abelian group. And let the Moore cochain complex functor. Write for the double complex obtained by applying to each of the two cosimplicial directions. Then we have natural isomorphisms in cohomology
There is a natural isomorphism
Crossed complex version
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 and be simplicial sets.
- There is an Alexander-Whitney diagonal approximation defined as a natural transformation
- Using shuffles, one defines an Eilenberg - Zilber map
in a somewhat similar way to chain complexes.
is homotopic to the identity on , whilst the other composite is the identity on , thus this is a strong deformation retract of .
The Eilenberg - Zilber theorem for simplicial sets
Cegarra and Remedios have proved a version of the Eilenberg - Zilber theorem for simplicial sets. This is discussed under the entry on bisimplicial sets.
Homology groups of products of topological spaces
This is the original motivating application.
Let and be two topological spaces. Their chain homology complexes and are the Moore complexes of the simplicial abelian groups and . So from the Dold-Puppe quasi-isomorphism 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
- Samuel Eilenberg, J. Zilber, On Products of Complexes , Amer. Jour. Math. 75 (1): 200–204, (1953) .
A weak version of the simplicial statement is in theorem 8.1.5 in
The stronger version as stated above is in chapter 4 of
The cosimplicial version of the theorem appears as theorem A.3 in
- L. Grunenfelder and M. Mastnak, Cohomology of abelian matched pairs and the Kac sequence (arXiv:math/0212124)
The crossed complex version is given in
- A.P. Tonks, On the Eilenberg-Zilber Theorem for crossed complexes. J. Pure Appl. Algebra, 179~(1-2) (2003) 199–220,
(for more detail see Tonks’ thesis),
and on page 360 of Nonabelian Algebraic Topology.