Equivariant cohomology for 2-torus actions and torus actions with compatible involutions
Abstract
The Borel equivariant cohomology is an algebraic invariant of topological spaces with actions of a compact group which inherits a canonical module structure over the cohomology of the classifying space of the acting group. The study of syzygies in equivariant cohomology characterize in a more general setting the torsion-freeness and freeness of these modules by topological criteria. In this thesis, we study the syzygies for elementary 2-abelian groups (or 2- tori) in equivariant cohomology with coefficients over a field of characteristic two. We approach the equivariant cohomology theory by an equivalent approach using group cohomology, that will allow us to distinguish syzygies via the exactness of the Atiyah-Bredon sequence using "shifted" and "virtual" subgroups that overcome the problem of finiteness of the acting group. We apply this characterization to study the equivariant cohomology for locally standard actions of 2- tori and also for torus actions with compatible involution, that will generalize the equivariant formality for Hamiltonian torus actions on symplectic manifolds.