Subject: Half-reply to Goodwillie's question
From: "John Greenlees"
Date: Wed, 28 Feb 2007 17:45:12 -0000
Dear Tom,
Here's some sort of answer to your question,
but only the easy bit. First, the geometric
G-fixed points is
\Phi^G(X)=(X \smash EtildeP)^G
where P is the family of all proper subgroups.
Thus you can obtain a cofibre sequence from the
cofibre sequence
EG_+ ---> S0 ---> EtildeG
by smashing with X and taking (Lewis-May) fixed points. If G is
of prime order this gives your cofibre sequence, and
in general it gives
Homotopy orbits ---> fixed points ---> (X \smash EtildeG)^G
Of course you can then filter EtildeG in some way and hope to
be more explicit on subquotients. The alternative is
to start with
EP_+ ----> S0 ---->EtildeP
and smash with X and take fixed points. If G is of prime order
you get your cofibre sequence again and in general it gives
(EP_+ \smash X)^G---->fixed points --->Geometric fixed points
Again, you can filter EP_+ and hope to be more explicit on subquotients.
For particular groups (cyclic of prime order, cyclic groups
products of these, even nilpotent groups) there are quite nice
filtrations,
but it depends just what you are hoping to get.
John
Subject: question for the list
From: Tom Goodwillie
Date: Wed, 28 Feb 2007 01:02:22 -0500
I have a reference question about stable equivariant homotopy theory.
It's about the relationship between "fixed points" and "geometric fixed
points" for a G-spectrum.
Let G be a finite group and X a G-spectrum.
The thing called "geometric fixed point spectrum" can be defined in a
couple of ways. One of them is "smash X with a certain G-CW complex
\tilde EG and then take fixed points". That based G-space E =\tilde EG
can be characterized by the statement that for every proper subgroup H
the fixed point set E^H is contractible while E^G~S0.
In the case of a group of prime order this leads to the statement that
there is a homotopy (co)fibration sequence of spectra
(homotopy orbits) --> (fixed points) --> (geometric fixed points)
My question is, what is a good generalization of this to general finite
G, and where is a good place to read about it? I can sort of see how one
such answer goes, but I'd rather not be reinventing the wheel.
Tom Goodwillie
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