Let be the chains/Moore complex functor of the Dold-Kan correspondence.
Let be the standard monoidal category structure given degreewise by the tensor product on Ab and let be the standard monoidal structure on the category of chain complexes.
For two abelian simplicial groups, the Eilenberg-Zilber map or Eilenberg-MacLane map or shuffle map is the natural transformation on chain complexes
defined on two -simplices and by
where the sum is over all -shuffles
and the corresponding degeneracy maps are
The sign is the signature of the corresponding permutation.
This map restricts to the normalized chains complex
The specific maps introduced by Eilenberg-Mac Lane have stronger properties which for simplicial sets make a strong deformation retract of . This is exploited in
- R. Brown, The twisted Eilenberg-Zilber theorem. Simposio di Topologia (Messina, 1964), Edizioni Oderisi, Gubbio (1965), 33–37. pdf
which has been a foundation of the subject of Homological Perturbation Theory. The homotopies play a key role in the formulae and calculations.
See monoidal Dold-Kan correspondence for details.
On normalized chain complexes the EZ map has a left inverse, given by the Alexander-Whitney map :
This is in 29.10 of (May).
For the next statement notice that both and are in fact symmetric monoidal categories.
The EZ map is symmetric in that for all the square
commutes, where denotes the symmetry isomorphism in and .
In the context of filtered spaces and their associated fundamental crossed complex?es there is a natural Eilenberg-Zilber morphism
which is difficult to define directly because of the complications of the tensor product of crossed complexes, but has a direct definition in terms of the associated cubical homotopy –groupoids. This morphism is an isomorphism of free crossed complexes if are the skeletal filtrations of CW-complexes. For more on all this, see the book Nonabelian Algebraic Topology p. 533.
The Eilenberg-Zilber map was introduced in (5.3) of
See also 29.7 of
- Peter May, Simplicial objects in algebraic topology , Chicago Lectures in Mathematics, Chicago, (1967) (djvu)
- A.P. Tonks, On the Eilenberg-Zilber Theorem for crossed complexes. J. Pure Appl. Algebra, 179~(1-2) (2003) 199–220,
and section 11.2 of