Splitting the Kunneth formula

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.