MARTINS, S. T. Diagonal approximations and cohomology rings for the fundamental groups of surfaces, torus bundles and some virtually cyclic groups. 2012. Tese

We construct finite free resolutions of Z over Zπ , where π is the fundamental group of a surface distinct from S 2 and RP 2 , and define diagonal approximations for these resolutions. We then proceed to give some applications using the knowledge of those maps.

Communicated by E. ZelmanovFor a torus bundle (S 1 × S 1 ) → M → S 1 , we construct a finite free resolution of Z over Z[π 1 (M )] and compute the cohomology groups H * (π 1 (M ); Z) and H * (π 1 (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 * (π 1 (M ); Z) and H * (π 1 (M ); Zp).

Let G be the fundamental group of a sapphire that admits the Sol geometry and is not a torus bundle. We determine a finite free resolution of Z over ZG and calculate a partial diagonal approximation for this resolution. We also compute the cohomology rings H * (G; A) for A = Z and A = Z p for an odd prime p, and indicate how to compute the groups H * (G; A) and the multiplicative structure given by the cup product for any system of coefficients A.

No abstract