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

Abstract

The purpose of this dissertation is to prove fundamental relations in the $RO(C_2)$-graded stable equivariant homotopy groups of spheres $\pi_{*,*}$ 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 $\epsilon \eta = \eta$, $\rho \eta = 1 + \epsilon$, and $24 \nu = 0$ where $\eta$ and $\nu$ are equivariant Hopf maps, $\epsilon$ is a unit in $\pi_{0,0}$, and $\rho$ is the generator of $\pi_{-1,-1}$. We also completely characterize the periodic portion of the topological zero-stem $\pi_{0,*}$ using singular manifold representatives which are the products $C_2 \times D^k$ equipped with various $C_2$-actions. While we focus on $C_2$, most of the theory we develop applies to $RO(G)$-graded homotopy groups for arbitrary finite groups $G$.