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

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.

-
add comment

1 Answer

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.

-
Another good person to ask is Angélica Osorno. Also Anna Marie Bohmann. I'll ask her tomorrow if I remember...Oh- Clark Barwick has also probably thought about this. That exhausts my list of names of related humans that I know of... –  Dylan Wilson May 17 '13 at 5:45
Thanks, Dylan. If none of them can help me, I'll ask you about non-humans. –  Tom Goodwillie May 17 '13 at 10:54
Tom, that's a great question and a fine thesis topic or two. Mona is working on several related things, one of which is to understand equivariant Waldhausen theory. She knows how to construct a genuine G-spectrum from suitable Waldhausen G-categories, but not yet how to show that it is an Omega G-spectrum, if it is. As David writes, her motivation is equivariant algebraic K-theory (and number theory). Equivariant Waldhausen A-theory is as yet unexplored as far as I know, and you are right that it should be interesting. We are also working on structured G-ring spectra, of various types. –  Peter May May 18 '13 at 3:39
add comment