Subject: Fixed points (answer to Tom Goodwillie)
From: Peter May
Date: Wed, 28 Feb 2007 21:23:53 -0600
Take the family P of proper subgroups and its universal
space EP, so that EP^H is contractible for H proper and
empty for H=G. We have a cofiber sequence
EP_+ >--> S0 >--> \tildeEP.
The righthand space is what you are calling \tildeEG,
but what is usually called \tildeEG is the cofiber in
EG_+ >--> S0 >--> \tildeEG.
The two agree only in the case you cite of a cyclic
group of prime order. If you smash the first cofiber
sequence with X and take G-fixed points, you get the
general cofiber sequence
Something >--> (fixed points) >--> (geometric fixed points).
The first two cofiber sequences above are special cases of
standard cofiber sequences for families that are studied,
for example, in Section 17 of ``Generalized Tate cohomology'',
by Greenlees and myself. We were thinking of a G-spectrum
X (genuine, not naive, there) as the representing spectrum
of a theory, so we called it k_G, and we gave some information
about the ``Something''. For example, even before passing
to fixed points, we proved (Proposition 17.2) that, for any
family F, such as P,
X smash EF_+ is equivalent to F(EF_+,X) smash EF_+.
Thus the construction you are interested in does not know the
difference between X and the function spectrum F(EP_+,X).
Hope that helps a little. John Greenlees can certainly tell
you more.
Peter