RO(C₂)-Graded Stable Stems and Equivariant Framed Bordism

Abstract

The purpose of this dissertation is to prove fundamental relations in the RO(C₂)- graded stable equivariant homotopy groups of spheres π_∗,∗ using geometric methods. The main tool we use is a singular version of the Pontryagin-Thom isomorphism which holds in the equivariant setting. Our work then consists of writing down explicit bordisms between manifold representatives of homotopy classes. Selected relations include ϵη = η, ρη = 1 + ϵ, and 24ν = 0 where η and ν are equivariant Hopf maps, ϵ is a unit in π_0,0, and ρ is the generator of π_−1,−1. We also completely characterize the periodic portion of the topological zero-stem π_0,∗ using singular manifold representatives which are the products C₂ × D^k equipped with various C₂-actions. While we focus on C₂, most of the theory we develop applies to RO(G)-graded homotopy groups for arbitrary finite groups G.