Carlsson's spectrum BG^-V

Question

In the appendix to Carlsson's "Equivariant stable homotopy and Segal's Burnside ring conjecture," he introduces a spectrum BG^-V associated to a G-representation V. It is like a Thom spectrum of the vector bundle over BG associated to V, but the stabilization maps come from the Thom space of (V plus a copies of the regular representation) instead of (V plus a trivial bundle).

The Thom space of (V plus regular representation) isn't of the form Thom(V) smash a sphere, so the details of building the spectrum are complicated.

Two questions:

1. Is BG^-V definitely different from the more naive Thom spectrum you could build out of the vector bundle V, where you stabilize using the V+ trivial bundles?

2. What are some other places to read about this spectrum?

Answer 1

You can define the Thom spectrum $X^V$ for any virtual bundle $V$, i.e. any formal difference $V-W$, where $V$, $W$ are vector bundles (or, in your case, representations of $G$ when $X=BG$. Carlsson gives an ad-hoc definition of $BG^{-V}$ (the Thom space of the negative of a representation), which is maybe not the best way to understand the general idea. I recommend May's Equivariant constructions of nonequivariant spectra (1987) for a discussion of the Thom spectrum construction for virtual bundles, with Carlsson's paper in mind, although there are plenty more modern treatments.