Splitting the Kunneth formula

TL;DR

This paper provides explicit formulas for splitting the Kunneth formula's torsion product, revealing how choices affect naturality and connecting to Massey triple products, with implications for algebraic topology computations.

Contribution

It introduces a method to explicitly split the Kunneth formula using generators and relations, linking the splitting to Massey triple products and analyzing naturality issues.

Findings

Explicit splitting formulas for the Kunneth formula.

Connection between the splitting and Massey triple products.

Analysis of naturality failure in the splitting process.

Abstract

There is a description of the torsion product of two modules in terms of generators and relations given by Eilenberg and Mac Lane. With some additional data on the chain complexes there is a splitting of the map in the Kunneth formula in terms of these generators. Different choices of this additional data determine a natural coset reminiscent of the indeterminacy in a Massey triple product. In one class of examples the coset actually is a Massey triple product. The explicit formulas for a splitting enable proofs of results on the behavior of the interchange map and the long exact sequence boundary map on all the terms in the Kunneth formula. Information on the failure of naturality of the splitting is also obtained.