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\begin{document}

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\section*{Eilenberg-Zilber theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{a_version_for_simplicial_abelian_groups}{A version for simplicial abelian groups:}\dotfill \pageref*{a_version_for_simplicial_abelian_groups} \linebreak
\noindent\hyperlink{cosimplicial_version}{Cosimplicial version}\dotfill \pageref*{cosimplicial_version} \linebreak
\noindent\hyperlink{crossed_complex_version}{Crossed complex version}\dotfill \pageref*{crossed_complex_version} \linebreak
\noindent\hyperlink{the_eilenberg__zilber_theorem_for_simplicial_sets}{The Eilenberg - Zilber theorem for simplicial sets}\dotfill \pageref*{the_eilenberg__zilber_theorem_for_simplicial_sets} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{ProdTopSp}{Homology groups of products of topological spaces}\dotfill \pageref*{ProdTopSp} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The [[Dold-Kan correspondence]] relates [[simplicial groups]] to [[chain complexes]]. The \emph{Eilenberg-Zilber theorem} says how in this context [[double complexes]] and their [[total complexes]] relate to [[bisimplicial sets|bisimplicial groups]] and their [[diagonal of a bisimplicial set|diagonals]]/[[total simplicial sets]].

Analogously there is also a version of the theorem for bi-cosimplicial abelian groups.

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\hypertarget{a_version_for_simplicial_abelian_groups}{}\subsubsection*{{A version for simplicial abelian groups:}}\label{a_version_for_simplicial_abelian_groups}

Let $A : \Delta^{op} \times \Delta^{op} \to Ab$ be a [[bisimplicial object|bisimplicial abelian group]]. Write

\begin{itemize}%
\item $C_\bullet diag A$ for the [[Moore complex]] of its diagonal simplicial group $diag A : \Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{A}{\to} Ab$;


\item $Tot (C A)$ for the [[total complex]] of the [[double complex]] obtained by applying the [[Moore complex]] functor on both arguments of $A$.



\end{itemize}
\begin{utheorem}
\textbf{(Dold-Puppe generalization of Eilenberg-Zilber)}

There is a [[quasi-isomorphism]] (even a chain-[[homotopy equivalence]])

\begin{displaymath}
R : C_\bullet diag (A) \stackrel{\simeq}{\to} Tot C (A)
  \,.
\end{displaymath}
\end{utheorem}
\begin{uremark}
Notice (see the discussion at [[bisimplicial set]]) that the diagonal simplicial set is isomorphic to the [[nerve and realization|realization]] given by the [[coend]]

\begin{displaymath}
diag F_{\bullet,\bullet} 
  \simeq 
   \int^{[n] \in \Delta} \Delta^n \times F_{n,\bullet}
  \,.
\end{displaymath}
\end{uremark}
\hypertarget{cosimplicial_version}{}\subsubsection*{{Cosimplicial version}}\label{cosimplicial_version}

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

\begin{utheorem}
There is a [[natural isomorphism]]

\begin{displaymath}
H^\bullet C^\bullet diag(A)
  \simeq
  H^\bullet Tot C^\bullet(A)
\end{displaymath}
\end{utheorem}
\hypertarget{crossed_complex_version}{}\subsubsection*{{Crossed complex version}}\label{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 $K$ and $L$ be simplicial sets.

\begin{itemize}%
\item There is an Alexander-Whitney diagonal approximation defined as a natural transformation

\end{itemize}
\begin{displaymath}
a_{K,L}: \pi(K\times L)\to \pi K \otimes \pi L.
\end{displaymath}
\begin{itemize}%
\item Using [[shuffles]], one defines an Eilenberg - Zilber map

\end{itemize}
\begin{displaymath}
b_{K,L}:\pi K \otimes \pi L \to\pi(K\times L),
\end{displaymath}
in a somewhat similar way to chain complexes.

\begin{itemize}%
\item The composite

\end{itemize}
\begin{displaymath}
\pi(K\times L)\to \pi K \otimes \pi L\to\pi(K\times L),
\end{displaymath}
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 $\pi(K\times L)$.

\hypertarget{the_eilenberg__zilber_theorem_for_simplicial_sets}{}\subsubsection*{{The Eilenberg - Zilber theorem for simplicial sets}}\label{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 set]]s.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{ProdTopSp}{}\subsubsection*{{Homology groups of products of topological spaces}}\label{ProdTopSp}

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 complex]]es 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]]

\begin{displaymath}
\begin{aligned}
    C_\bullet(X \times Y)
    & \coloneqq
    C_\bullet( \mathbb{Z}[Sing X \times Sing Y] )
    \\
    &=
    C_\bullet( diag \mathbb{Z}[Sing X_\bullet] 
      \otimes \mathbb{Z}[Sing Y_\bullet] )
    \\
    & \underoverset{\simeq}{R}{\longrightarrow}
      Tot C_\bullet(\mathbb{Z}[Sing X]) \otimes C_\bullet(\mathbb{Z}[Sing Y])
    \\
    & = Tot C_\bullet(X) \otimes C_\bullet(Y) 
  \end{aligned}
\end{displaymath}
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.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The original reference is

\begin{itemize}%
\item [[Samuel Eilenberg]], J. Zilber, \emph{On Products of Complexes} , Amer. Jour. Math. 75 (1): 200--204, (1953) .

\end{itemize}
A weak version of the simplicial statement is in theorem 8.1.5 in

\begin{itemize}%
\item [[Charles Weibel]], \emph{An introduction to homological algebra}

\end{itemize}
The stronger version as stated above is in \href{http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi}{chapter 4} of

\begin{itemize}%
\item [[Paul Goerss]], [[Rick Jardine]], \emph{[[Simplicial homotopy theory]]} (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{dvi})

\end{itemize}
The cosimplicial version of the theorem appears as theorem A.3 in

\begin{itemize}%
\item L. Grunenfelder and M. Mastnak, \emph{Cohomology of abelian matched pairs and the Kac sequence} (\href{http://arxiv.org/abs/math/0212124}{arXiv:math/0212124})

\end{itemize}
The crossed complex version is given in

\begin{itemize}%
\item [[Andy Tonks|A.P. Tonks]], \emph{On the Eilenberg-Zilber Theorem for crossed complexes}. J. Pure Appl. Algebra, 179{\tt \symbol{126}}(1-2) (2003) 199--220,

\end{itemize}
(for more detail see Tonks' \href{http://www.maths.bangor.ac.uk/research/tonks/pubs.html}{thesis}),

and on page 360 of [[Nonabelian Algebraic Topology]].



\end{document}