Timeline for When is $X_{hG} \to X/G$ a weak equivalence for $X$ a free $G$-space, $G$ compact Lie?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 27, 2017 at 13:50 | vote | accept | Alexander Körschgen | ||
| Apr 27, 2017 at 13:50 | answer | added | Alexander Körschgen | timeline score: 3 | |
| Dec 8, 2016 at 2:41 | comment | added | Alexander Körschgen | On a side note: while the Tychonoff property answers 1. (as described in the OP), it is also sufficient for 2. because it is stable under products and EG is Tychonoff (it is normal and Hausdorff). The distinction between the Kelley product and the classical product does not make a difference for the weak homotopy type of $(F \times EG)/G$. | |
| Dec 8, 2016 at 2:36 | comment | added | Alexander Körschgen | Thank you for your comment. As far as I know, the concept of slices only appears in the context of Tychonoff spaces throughout the literature (Bredon, Palais, tom Dieck …). One of the key steps in Bredon's result is the existence of slices for actions on a Tychonoff space. Manifolds are Tychonoff (no smoothness required), so, unfortunately, your condition does not add anything to my list of known results above. | |
| Dec 4, 2016 at 4:35 | comment | added | Tyler Lawson | Are you familiar with the condition (for (2)) that the group action "has slices" (any point $x$ has a set $U$ containing $x$ such that the map $G \times U \to X$ is a homeomorphism onto a neighborhood)? E.g. this is always true for a smooth, free action of a Lie group on a smooth manifold (the slice theorem in differential geometry). This condition is strong enough to make $X \to X/G$ into a fiber bundle with fiber $G$. | |
| Nov 29, 2016 at 18:44 | history | asked | Alexander Körschgen | CC BY-SA 3.0 |