There is a natural algebraic structure on homology that is dual to the algebra structure on cohomology. It is gotten the same way you get the cup product: use the Kunneth theorem and the diagonal to give you a coalgebra. $H_(X) H_\ast(X) \to H_(X H_\ast(X \times X) \to H_(X) H_\ast(X) \otimes H_(X)$, H_\ast(X)$, the downside is that coalgebras seem strange. 1 There is a natural algebraic structure on homology that is dual to the algebra structure on cohomology. It is gotten the same way you get the cup product: use the Kunneth theorem and the diagonal to give you a coalgebra.$H_(X) \to H_(X \times X) \to H_(X) \otimes H_(X)\$, the downside is that coalgebras seem strange.