Timeline for Relation between moduli spaces and classifying spaces
Current License: CC BY-SA 3.0
12 events
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Sep 27, 2015 at 20:05 | vote | accept | user51223 | ||
Jul 30, 2015 at 11:28 | comment | added | Dmitri Pavlov | @user51223: I am not sure what your definition of “moduli space” is in the first place, but Madsen and Weiss's M(F) corresponds to my F(pt) (not F[pt]). | |
Jul 30, 2015 at 8:27 | comment | added | user51223 | Actually, as I understand it, the bordism group that you mentioned is not what we get as a moduli space just from plane definition; compare to the definition on Second page of Madsen and Weiss. In fact, I think, in the particular case of surfaces, the discussions on the first pages of Madsen-Weiss answers my question. But, I am not sure that in general, at least in the case of higher dimensional manifolds, $F[pt]$ becomes the moduli space. | |
Jul 29, 2015 at 17:05 | comment | added | Dmitri Pavlov | @user51223: The general framework in the paper of Galatius and Randal-Williams is the same as in the GMTW paper, which I described above. | |
Jul 29, 2015 at 17:04 | comment | added | Dmitri Pavlov | @user51223: In particular, this gives you the homotopy equivalence (1) that you asked for. As for (2), F[pt] in this case is simply the bordism group of (d−1)-manifolds, which I presume is what you mean by M_C. | |
Jul 29, 2015 at 16:47 | comment | added | Dmitri Pavlov | @user51223: Galatius—Madsen—Tillmann—Weiss apply this construction to several different stacks, see the very first paragraph in §4 of their paper. They show that the concordance classifying space of D_d gives the Madsen—Tillmann spectrum and all three maps induce weak equivalences of concordance classifying spaces, which proves their theorem. | |
Jul 28, 2015 at 16:02 | comment | added | user51223 | +as you may know, a work of Galatius and Randal-Williams has a ``weaker'' analogue of Mumford's conjecture where they replace $\Gamma_\infty^+$ with the diffeomorphism group of $g$-fold connected sum of $D^{n}\times D^n$ if I am not mistaken. Of course, I understand that one difficulty is that we don't know about the classification of higher dimensional manifolds, so talking about the moduli space of these objects maynot give the right feeling/construction, whereas in the case of dimension $2$ one knows that oriented surfaces are classified by genus. Shouldn't they consider the above approach? | |
Jul 28, 2015 at 15:56 | comment | added | user51223 | So, for $X=pt$, $F[X]$ has to be the moduli space that one has to look for. Working backwards, given a category $\mathbf{C}$ related to manifolds at least, a possible strategy to get a positive answer to the above question is that one may look for a sheaf or stack $F$ so that 1) there is a homotopy equivalence $CF(pt)\to B\mathbf{C}$; 2) there is a homotopy equivalence $F[pt]\to \mathcal{M}_\mathbf{C}$. Of course, I am still thinking about the work of Randal-Williams and Ebert who showed that higher dimensional analogue of Mumford's conjecture does not hold. How would you comment on that? | |
Jul 28, 2015 at 12:32 | comment | added | Dmitri Pavlov | @user51223: I added a detailed description of the map. Note that for the case of sets (i.e., discrete spaces) this is the same map that Galatius—Madsen—Tillmann—Weiss use in the formula (2.7) in their paper. | |
Jul 28, 2015 at 12:20 | history | edited | Dmitri Pavlov | CC BY-SA 3.0 |
added 742 characters in body
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Jul 28, 2015 at 11:20 | comment | added | user51223 | Thank you very much for the answer. However, I still couldn't follow how you get a map from the classifying space to the moduli space/stack or the way around?!? Even, in the case of Madsen-Tillmann map, this does not happen, unless we accept that the moduli space of Riemannian surfaces is that same as $B\Gamma_\infty^+$?! | |
Jul 27, 2015 at 20:07 | history | answered | Dmitri Pavlov | CC BY-SA 3.0 |