Doctor of Philosophy
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.
Summary for Lay Audience
In algebraic topology we study properties of spaces by algebraic means; the homotopy, homology or cohomology theories are algebraic invariants of the spaces that are preserved under continuous deformations. Another approach to study properties of spaces is throughout their symmetries or group actions, which is known as the theory of transformation groups. The equivariant cohomology is an algebraic invariant of both the topology of the space and its given group action. This algebraic invariant will become a module over a polynomial ring in several interesting situations; for example, when involutions or reflections of the space are considered. In this thesis, we characterize topological and algebraic properties of the equivariant cohomology for these particular actions, we relate this results with the progress done by circle and torus actions, and we also provide applications on symplectic manifolds and manifold with corners, that generalize the notions of polytopes and polyhedra.
Chaves Ramirez, Sergio, "Equivariant cohomology for 2-torus actions and torus actions with compatible involutions" (2020). Electronic Thesis and Dissertation Repository. 7049.