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

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dc.contributor.advisorDugger, Daniel
dc.contributor.authorMcGinnis, Stewart
dc.date.accessioned2024-12-19T20:06:52Z
dc.date.available2024-12-19T20:06:52Z
dc.date.issued2024-12-19
dc.description.abstractThe 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 $G$.en_US
dc.identifier.urihttps://hdl.handle.net/1794/30293
dc.language.isoen_US
dc.publisherUniversity of Oregon
dc.rightsAll Rights Reserved.
dc.subjectcobordismen_US
dc.subjectequivarianten_US
dc.subjectframeen_US
dc.subjecthomotopyen_US
dc.subjectrepresentationen_US
dc.titleRO(C₂)-graded Stable Stems and Equivariant Framed Bordism
dc.typeElectronic Thesis or Dissertation
thesis.degree.disciplineDepartment of Mathematics
thesis.degree.grantorUniversity of Oregon
thesis.degree.leveldoctoral
thesis.degree.namePh.D.

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