If and are two chain complexes , then the tensor product is the chain complex with
, where .
With that it is declared on producers and the bill
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.
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 .
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 .
Individual evidence
↑ Robert M. Switzer: Algebraic Topology - Homotopy and Homology , Springer Grundlehren der Mathematischen Wissenschaften, Volume 212 (1975), ISBN 3-540-06758-2 , Theorem 13.30
↑ Edwin H. Spanier: Algebraic Topology , Springer-Verlag (1966), ISBN 0-387-90646-0 , Chapter 5, §3, Theorem 6