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 ______________________________________________________________