Involutions on $S[ΩM]$

Abstract

The contribution of this PhD thesis is to construct for each vector bundle $ξ$ an orthogonal ring spectrum $R$, weakly equivalent to $S[ΩM]$, together with an involution on $R$. On the homotopy groups $π_*S[ΩM]$ our involution corresponds to parallel transportation in $ξ$, 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).