Waldhausen $K$-theory for $G$-spaces

Question

I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it?

Let $G$ be a finite group. We know that there is a functor 'equivariant suspension spectrum' $X\mapsto \Sigma^\infty_G(X_+)$ from $G$-spaces to $G$-spectra. It seems that there should be another such functor $A_G$, 'equivariant Waldhausen $K$-theory spectrum'. The fixed point spectrum ought to split up according to conjugacy classes of subgroups of $G$ just as for the suspension spectrum: $$ A_G(X)^G\sim \Pi_H\ A(X^H_{hW_GH}) $$ ($X^H_{W_GH}$ is homotopy orbits for the action of $W_GH=(N_GH)/H$ on fixed points of $H$.)

On the one hand when $X$ is based and connected then there should be a description of $A_G(X)$ that makes it a special case of a $K$-theory of structured $G$-ring-spectra.

On the other hand there should be a splitting $$ A_G(X)\sim \Sigma^\infty_G(X_+)\times Wh^{diff}_G(X) $$ where the other factor is related to equivariant pseudoisotopies and $h$-cobordisms.

Answer 1

I did a little work on this problem in Bielefeld in 1991, based on a suggestion by Waldhausen.

Let $G$ be a finite group. The algebraic $K$-theory $A^G(X)$ of the category of finite retractive $G$-spaces and $G$-maps over and under $X$, with $G$-homotopy equivalences as the weak equivalences, does indeed have a Segal-tom Dieck style factorization in terms of non-equivariant $A$-theory, indexed over the conjugacy classes of subgroups $H$ of $G$, as you describe. This follows from the additivity theorem. However, I was unable to realize this as the $G$-fixed points of a $G$-spectrum $A_G(X)$.

If you naively define the latter as the algebraic $K$-theory of the $G$-category with the same objects as above, but consider all (not necessarily $G$-equivariant) maps under and over $X$ as morphisms, and take the non-equivariant homotopy equivalences as the weak equivalences, then its $G$-fixed part $A_G(X)^G$ is instead the algebraic $K$-theory of the category of finite retractive $G$-spaces and $G$-maps, but with respect to a coarser notion of weak equivalence, namely a $G$-equivarant map whose underlying map is a homotopy equivalence

There was some interest in trying to prove a version of the Segal conjecture for this theory, i.e., to see if $$ A_G(X)^G \to A_G(X)^{hG} $$ is an equivalence after suitable completion. This could then contain the original Segal conjecture/Carlson's theorem about $Q_G(X_+)^G \to Q_G(X_+)^{hG}$ as a retract. I observed that in general there could not be such an equivalence for a version of $A_G(X)$ satisfying the Segal-tom Dieck splitting $$ A_G(X)^G \simeq \prod_{(H)} A(X^H_{hW_GH}) $$ and the other expected formula $$ A_G(X)^{hG} \simeq F(BG_+, A(X)) $$ (for $X = X^G$, probably). The argument used your (Goodwillie's) computation of the derivative of $A \colon X \mapsto A(X)$ in terms of $\Sigma^\infty(\Lambda X)_+$, to see that the derivatives of the two right hand sides above were not equivalent. Basically free loop spaces do not commute with Borel constructions. So my conclusion was while the Segal conjecture could hold for one space, it would then mostly fail in a neighborhood of that space.

Answer 2

For the sake of people googling this, there's a pretty canonical reference avaliable now: Malkiewich and Merling.

Answer 3

Are you aware of the work of Mona Merling? She's a student of Peter May who will be graduating in a year. She's done lots of work on equivariant algebraic K-theory, and I saw her give a talk about this back in April. Her thesis breaks down into two papers. The first is on the arxiv and covers the space-level considerations. The second is more likely to contain what you're asking about, but it hasn't been put online yet. I don't have my notes from that talk on me, but I seem to recall it was a bit subtle and one shouldn't assume everything works out as in the non equivariant case. My recommendation would be to email her and see if this statement is true.