Involutions on $S[ΩM]$

Details Involutions on $S[ΩM]$ Involutions on $S[ΩM]$

2005-10-11

arxiv.org/abs/math/0510221v1

Mathematics

Algebraic Topology

248 pages, Dissertation for the Degree of Dr.Scient. 2005 at the University of Oslo

Scientific paper

The contribution of this PhD thesis is to construct for each vector bundle $\xi$ an orthogonal ring spectrum $R$, weakly equivalent to $S[\Omega M]$, together with an involution on $R$. On the homotopy groups $\pi_*S[\Omega M]$ our involution corresponds to parallel transportation in $\xi$, and reversing loops in $M$. The main result is theorem 4.3.26. The orthogonal ring spectrum $R$ with involution is intended as input for $LA$, $K$, $TC$ and $THH$, the ultimate goal is to compute homotopy groups of automorphism groups. We take a first step in this direction by considering the definition and a few basic properties of $TC(L)$ and $THH(L)$ for arbitrary orthogonal ring spectra $L$ (with involution).

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