A fundamental problem in combinatorics is to extract the power series coefficients of a generating function, or to estimate them asymptotically. For d-variable generating functions, the Cauchy integral formula gives the coefficient as an integral of a d-form over a small d-torus. Successful estimation of this integral depends crucially on representing this d-cycle in a homology basis forced upon us by Morse theory. The talk concerns progress and problems in effective computation in the relevant homology group.