Diagonal approximation and the cohomology ring of torus fiber bundles

Abstract

For a torus bundle (S1 × S1) → M → S1, we construct a finite free resolution of ℤ over ℤ[π1(M)] and compute the cohomology groups H*(π1(M); ℤ) and H*(π1(M);ℤp) for a prime p. We also construct a partial diagonal approximation for the resolution, which allows us to compute the cup products in H*(π1(M); ℤ) and H*(π1(M); ℤp).