Theorem by Eilenberg-Zilber

The set of Eilenberg-Zilber , named after S. Eilenberg and YES Zilber is, a set of the mathematical branch of algebraic topology . It establishes a connection between the singular homology groups of a Cartesian product of two topological spaces and homology groups of the spaces themselves.

Tensor products of chain complexes

If and are two chain complexes , then the tensor product is the chain complex with ${\displaystyle (C',d')}$ ${\displaystyle (C'',d'')}$ ${\displaystyle (C',d')\otimes (C'',d'')}$ ${\displaystyle (C,d)}$

${\displaystyle C_{n}=\bigoplus _{i+j=n}C_{i}'\otimes C_{j}''}$

${\displaystyle d_{n}(c_{i}'\otimes c_{j}''):=d_{i}'c_{i}'\otimes c_{j}''+(-1)^{i}c_{i}'\otimes d_{j}''c_{j}''}$, where . ${\displaystyle c_{i}'\in C_{i}',c_{j}''\in C_{j}'',i+j=n}$

With that it is declared on producers and the bill ${\displaystyle d_{n}}$

${\displaystyle d_{n-1}d_{n}(c_{i}'\otimes c_{j}'')=d_{n-1}(d_{i}'c_{i}'\otimes c_{j}'')+(-1)^{i}d_{n-1}(c_{i}'\otimes d_{j}''c_{j}'')}$

${\displaystyle =\underbrace {d_{i-1}'d_{i}'c_{i}'} _{=0}\otimes c_{j}''+\underbrace {(-1)^{i-1}d_{i}'c_{i}'\otimes d_{j}''c_{j}''+(-1)^{i}d_{i}'c_{i}'\otimes d_{j}''c_{j}''} _{=0}+(-1)^{i}(-1)^{i-1}c_{i}'\otimes \underbrace {d_{j-1}''d_{j}''c_{j}''} _{=0}=0}$

shows that there is indeed a chain complex again.

If the boundary operators or should not be mentioned in particular, one writes simply , this is especially true for singular chain complexes of topological spaces , for which the boundary operators are given. ${\displaystyle d',d''}$ ${\displaystyle d}$ ${\displaystyle C=C'\otimes C''}$ ${\displaystyle S(X)}$ ${\displaystyle X}$

Formulation of the sentence

If and are topological spaces, then the singular chain complex of the product space is chain-homotopy equivalent to the tensor product . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle S(X\times Y)}$ ${\displaystyle S(X)\otimes S(Y)}$

meaning

Because of the homotopy equivalence and have the same homology groups. The calculation of the singular homology groups of a product space is therefore traced back to a problem of homological algebra , namely the calculation of the homology of a tensor product of chain complexes. This algebraic problem is solved by Künneth's theorem . ${\displaystyle S(X\times Y)}$ ${\displaystyle S(X)\otimes S(Y)}$

Individual evidence

1. ↑ Robert M. Switzer: Algebraic Topology - Homotopy and Homology , Springer Grundlehren der Mathematischen Wissenschaften, Volume 212 (1975), ISBN 3-540-06758-2 , Theorem 13.30
2. ↑ Edwin H. Spanier: Algebraic Topology , Springer-Verlag (1966), ISBN 0-387-90646-0 , Chapter 5, §3, Theorem 6