Timeline for Homology of homotopy fixed point spaces
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 2, 2013 at 17:49 | vote | accept | Craig Westerland | ||
| Mar 16, 2011 at 21:48 | answer | added | Jesper Grodal | timeline score: 4 | |
| Mar 15, 2011 at 16:25 | comment | added | John Klein | Addendum to my penultimate comment: if $X$ is a $G$-finitely dominated spectrum then the norm equivalence is valid for all $G$. $G$ finitely dominated means that $X$ is an equivariant retract up to homotopy of a $G$-finite spectrum $Y$, i.e., $Y$ is built up from the trivial spectrum by attaching a finite number of free cells. | |
| Mar 15, 2011 at 11:35 | comment | added | John Klein | Craig: I doubt it. | |
| Mar 15, 2011 at 4:50 | comment | added | Craig Westerland | John, I'd love a version of the latter statement when $X$ is a space, and not a spectrum. I suppose I can get it from the latter when $X$ is an infinite loop space, but is there any hope of that happening when it's not? | |
| Mar 14, 2011 at 16:05 | comment | added | John Klein | "2." has an analog for spaces, namely, the space of sections of the fibration $EG x_G X \to BG$ has a Federer spectra sequence which converges to the homotopy of the function space. Also if $BG$ is finitely dominated and $X$ is a spectrum then one has the norm equivalence $D_G \wedge_G X \simeq X^{hG}$ which expresses the homotopy of $X^{hG}$ has the homotopy of $X$ with coefficients twisted by the dualizing spectrum $D_G$. This can be computed in some cases... | |
| Mar 14, 2011 at 9:47 | comment | added | Craig Westerland | I guess I'm happy starting with $G$ being finite and of order coprime to $p$, whereas $X$ has, say, finite $\mathbb{F}_p$ homology. But in the end, I's like to have a general picture of all of the tools available. | |
| Mar 14, 2011 at 7:31 | comment | added | Tilman | Can you be a bit more explicit what your $G$ is (discrete, Lie, finite, p, p-prime, etc.) and what your $X$ is (finite CW, p-complete, ...)? There is an unstable homotopy fixed point spectral sequence, a version of the Bousfield spectral sequence of a cosimplicial space, coming from looking at $map(EG,X)$ as a cosimplicial $G$-space by the canonical simplicial structure of $EG$. Whether or not that helps depends on your particular case -- in general, it's hard to describe $E^2$, it'll be a fringed spectral sequence, and convergence will be an issue. | |
| Mar 14, 2011 at 6:57 | history | asked | Craig Westerland | CC BY-SA 2.5 |