Timeline for Homotopy type of set of self homotopy-equivalences of a surface
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Mar 19, 2010 at 16:24 | comment | added | Dan Ramras | I think Allen's answer is pretty much along the lines of the second half of my comment above. Should have read his answer first... | |
| Mar 19, 2010 at 14:31 | comment | added | Andy Putman | I decided to switch the accepted answer to Hatcher's answer, since I suspect that people who are interested in the topology of surfaces will have an easier time understanding it. | |
| Mar 19, 2010 at 4:58 | comment | added | Dan Ramras | You might try looking at Bill Dwyer's notes from the Haynes Miller birthday conference. I don't quite see this in there (there's a lot to look through, though) but he definitely discusses similar things. See math.ku.dk/~jg/homotopical2008 Also, Theorem 2.6 in Atiyah-Bott's paper on Yang-Mills theory might be relevant here. The result says that maps from a finite complex into an E-M space is a product of E-M spaces, and is easy to prove by induction over skeleta. (Although maybe that result only applies to Map(X, K(G, n)) with n>1? I'm a little confused at the moment.) | |
| Mar 19, 2010 at 2:52 | comment | added | Anatoly Preygel | I don't know if this helps, but you can interpret Reid's answer as a combination of (a refinement of) classifying spaces + covering space theory: Say $X$ is a pointed connected space. Then, $Map(X, K(G,1))$ turns out to be homotopy equivalent to the geometric realization of the category of $G$-bundles on $X$ (the $\pi_0$ statement is the usual classifying space story). And, covering space theory tells us this is equivalent to the category whose objects are homomorphisms $\pi_1 X \to G$, and whose morphisms are given by conjugation on $G$. Now plug in $X = K(G,1)$ to this computation. | |
| Mar 19, 2010 at 2:07 | vote | accept | Andy Putman | ||
| Mar 19, 2010 at 14:30 | |||||
| Mar 19, 2010 at 2:07 | comment | added | Andy Putman | That's a bit too fancy for a geometric topologist like me to understand! I'm ashamed to admit that my knowledge of algebraic topology extends only to the late 60's or so. If anyone knows a more down-to-earth explanation or a reference, please share. | |
| Mar 19, 2010 at 1:59 | comment | added | Reid Barton | Oh. I don't have a reference offhand, though I'm sure someone could provide one. The way I would do the mapping space computation is to check that we can do it in groupoids, where it simply comes down to the definition of functor and natural transformation. To do this I think you just need the fact that groupoids = 1-truncated spaces is a reflexive sub-$(\infty,1)$-category of spaces. | |
| Mar 19, 2010 at 1:52 | comment | added | Andy Putman | No, I can do the group theory <grin>. I mean the statement about the mapping space that you quoted. | |
| Mar 19, 2010 at 1:52 | comment | added | Reid Barton | Which, the statement about the center? No, it just seems overwhelmingly likely due to the results 1-3, and I imagined it was a "simple matter of algebra", but didn't think about whether it's actually easy to prove. | |
| Mar 19, 2010 at 1:47 | comment | added | Andy Putman | Very nice! The center is indeed trivial for g at least 2. Do you know a nice reference for this result? | |
| Mar 19, 2010 at 1:41 | history | answered | Reid Barton | CC BY-SA 2.5 |