Witt Vectors and Equivariant Ring Spectra

TL;DR

This paper explores the relationship between equivariant ring spectra and Witt vectors, constructing a ring homomorphism that connects their algebraic structures and analyzing its properties in specific cases.

Contribution

It introduces a novel connection between equivariant ring spectra and Witt vectors, providing a new perspective and tools for understanding equivariant cobordism.

Findings

Constructed a ring homomorphism from G-typical Witt vectors to fixed point spectra.

Proved injectivity of the homomorphism for the periodic unitary cobordism spectrum.

Interpreted results in the context of equivariant cobordism.

Abstract

This paper establishes a connection between equivariant ring spectra and Witt vectors in the sense of Dress and Siebeneicher. Given a commutative ringspectrum T in the highly structured sense, that is, an E-infinity-ringspectrum, with action of a finite group G we construct a ringhomomorphism from the ring of G-typical Witt vectors of the zeroth homotopy group of T to the zeroth homotopy group of the G-fixed point spectrum of T. In the particular case, where T is the periodic unitary cobordism spectrum introduced by Strickland, we show that this ringhomomorphism is injective, and we interpret this in terms of equivariant cobordism.