3
$\begingroup$

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?

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.