Subject: Fixed points (answer to Tom Goodwillie)
From: Peter May <may@math.uchicago.edu>
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