Timeline for (∞, 1)-categorical description of equivariant homotopy theory
Current License: CC BY-SA 2.5
9 events
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| Oct 29, 2009 at 14:35 | history | edited | David Treumann | CC BY-SA 2.5 |
whitehead's theorem
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| Oct 29, 2009 at 2:56 | comment | added | Reid Barton | Yes, in fact every smooth manifold with smooth action of a compact Lie group G admits the structure of a G-CW complex; see Illman, "Equivariant singular homology and cohomology for actions of compact Lie groups" (MR0377858). | |
| Oct 29, 2009 at 1:28 | comment | added | David Treumann | I'm sorry, I had in mind that the equivariant homotopy type of a G-space just was its diagram of fixed-point spaces. This equivariant Whitehead theorem is almost very satisfying, but I feel like there is something hiding in the definition of G-CW complex. Is every G-manifold (with some tameness conditions if you like, like the manifold and group action are real analytic) strongly G-homotopy equivalent to a G-CW complex? That would seal the deal. | |
| Oct 29, 2009 at 1:27 | comment | added | Charles Rezk | To make it more concrete: let C be a small category, Psh(C)= presheaves of spaces on C. Put a model category structure on Psh(C), where the generating cofibrations are built using all quotients of representable functors. Will this be equivalent to some other model category of presheaves? (In your case, Reid, C was the group G, and the quotients of representables are the G/H.) | |
| Oct 29, 2009 at 1:09 | comment | added | Reid Barton | Yes, another way to say it is that the model category--in particular, the class of weak equivalences--is determined by the generating cofibrations and acyclic cofibrations I wrote down; it doesn't seem tautological to me that the result is a diagram category. | |
| Oct 29, 2009 at 0:58 | comment | added | Charles Rezk | I believe I can actually recover X itself from its fixed point diagram. Of course, when I do so, I'm thinking "1-categorically", and not "infty-categorically", so perhaps that's not germane here ... I'm also not sure why it's a tautology. I thought the point was the equivariant Whitehead theorem: a map f:X->Y between G-CW complexes is an equivariant homotopy equivalence (i.e., there is G-map g:Y->X such that gf and fg are each homotopic to identity through G-maps) iff X^H -> Y^H is a weak equiv. of spaces for each subgroup H. | |
| Oct 29, 2009 at 0:41 | comment | added | David Treumann | I think you are recovering the homotopy type of X, not X itself. In fact you can recover, tautologically, the equivariant homotopy type of G->Aut(X) out of the diagram O(G)->Spaces. Is there some other natural way of saying "equivariant homotopy type" so that this tautology becomes an interesting theorem? | |
| Oct 29, 2009 at 0:22 | comment | added | Charles Rezk | I don't know why you would say that this is "all that algebraic topology can see in a G-space". Any G-space X can be reconstructed from its fixed point diagram O(G)->Spaces, so that diagram must keep all the information about X. | |
| Oct 28, 2009 at 22:58 | history | answered | David Treumann | CC BY-SA 2.5 |