Diagonal approximation and the cohomology ring of torus fiber bundles

TL;DR

This paper develops a finite free resolution for the fundamental group of a torus bundle and uses it to compute the cohomology groups and cup product structure, enhancing understanding of the cohomology ring of such bundles.

Contribution

It introduces a finite free resolution for the fundamental group of a torus bundle and computes its cohomology and cup product structure.

Findings

Computed cohomology groups H^*(G,Z) and H^*(G,Z_p) for the torus bundle

Constructed a partial diagonal approximation for the resolution

Determined the cup product structure in the cohomology ring

Abstract

For a torus bundle (S^1 x S^1) -> E -> S^1$, we construct a finite free resolution Z over ZG, where G is the fundamental group of the total space E, and then we compute the cohomology groups H^*(G,Z) and H^*(G,Z_p) for a prime p. We also construct a partial diagonal approximation for the resolution, which allows us to compute the cup product in H^*(G,Z) and H^*(G,Z_p).