Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Can someone point me to a good (hopefully simple and brief) place to read about the basics of homotopy orbits for pointed spaces?

More detail:
As I understand it, in the unpointed case, we use the projection $\mathrm{pr}_X: EG \times X \to X$ to replace the $G$-shaped diagram $X$ with the cofibrant $G$-shaped diagram $EG\times X$; and then the homotopy orbit space is the homotopy colimit of the diagram $X$, which is the categorical colimit of $EG\times X$.

I'd like a description of the pointed case in which it is shown that $EG_+ \wedge X\to X$ is a cofibrant replacement in the appropriate category of diagrams.

Even More (Inspired by T. Goodwillie's comment): This is what I think, too, but I feel like I'm missing something, because in the pointed case, why can't I just use the cone on $G$ instead of $EG$? It is contractible and $G$ acts freely on the nonbasepoint cells, right? So I could just as easily use $CG_+ \wedge X$. but this is clearly wrong!

EDIT: Cone idea is wrong because of the cone point, so I feel pretty good about my understanding of the theory. I'd still like a good reference.

share|improve this question
    
Hi Jeff, could you provide a definition to stimulate our memory juices? –  Mark Grant Jan 8 '11 at 12:00
2  
Of course the details depend on which model structure you use for $G$-spaces (or more generally diagrams of a given shape). If you use the one where equivalences and fibrations are defined objectwise, then any cell complex with $G$ acting freely on the set of (nonbasepoint) cells is cofibrant, and smashing a cell complex with $EG_+$ achieves this. –  Tom Goodwillie Jan 8 '11 at 16:37
2  
$G$ does not act freely on the non-basepoint cells of $CG_+$. You've added another point. –  S. Carnahan Jan 9 '11 at 6:31
    
@Scott: Thanks -- I wasn't thinking clearly, clearly. –  Jeff Strom Jan 9 '11 at 13:41

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.