Timeline for Injectivity of the Dehn-Nielsen-Baer map?
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18 events
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| Jul 7, 2021 at 8:07 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Mar 9, 2021 at 8:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Feb 7, 2021 at 7:38 | answer | added | mathreader | timeline score: 1 | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| May 28, 2015 at 17:08 | comment | added | Dylan Thurston | They address it, but don't really go into the proof (as is sensible for that point in the book). | |
| May 28, 2015 at 16:57 | comment | added | Ian Agol | I guess they address the issue of homotopy/homotopy equivalence vs. diffeomorphism/isotopy in Chapter 1. | |
| May 27, 2015 at 21:43 | comment | added | Ian Agol | The mapping class group is usually defined as diffeos. modulo isotopies. To show equivalence with $Out(\pi_1(M))$, first one may show equivalence with self-homotopy equivalences up to homotopy, and then show that homotopy equivalences up to homotopy are equivalent to diffeomorphisms up to isotopy. | |
| May 27, 2015 at 20:48 | comment | added | Dylan Thurston | Ian, can you be more explicit about exactly which statement you are arguing for? Those statements are both specific to surfaces, right? | |
| May 27, 2015 at 20:45 | comment | added | Ian Agol | As for homotopy equivalence implies homeomorphism, one may prove this by induction on a hierarchy, a la Waldhausen. Similarly for homotopy equivalent homeos. are isotopic. | |
| May 27, 2015 at 18:32 | comment | added | Dylan Thurston | Thanks, everyone. I wish authors would spell this out a bit more. | |
| May 27, 2015 at 18:29 | comment | added | mme | $[X,K(G,n)] = H^n(X;G) = \text{Hom}(H_n(X;\Bbb Z);G) = \text{Hom}(\pi_n(X);G)$. For a more down-to-earth example of a map $X \to K(\pi,n)$ that's not null-homotopic but kills $\pi_n$ we could take the fibration $S^1 \to \Bbb{RP}^\infty \to \Bbb{CP}^\infty$ obtained by considering $\Bbb{RP}^\infty$ as a quotient of the infinite "odd-dimensional sphere" and extending this to a quotient by $S^1$. If this map was null-homotopic so would be the identity map of $\Bbb{RP}^\infty$. This should correspond to the Bockstein $H^1(-,\Bbb Z/2) \to H^2(-,\Bbb Z)$. | |
| May 27, 2015 at 18:27 | comment | added | mme | I think you need some connectivity conditions on the domain - it seems to me there is a nontrivial map $K(\Bbb Z_2,1) \to K(\Bbb Z_2, 2)$ given by the cup square. (It is a theorem that there is a bijection between homotopy classes of maps $K(\pi, n) \to K(\pi', m)$ and natural transformations between the Set-valued functors $H^n(-, \pi) \to H^m(-,\pi')$. The cup square represents a nontrivial natural transformation because it does so on $\Bbb{RP}^2$, say.) If you assume the domain is $(n-1)$-connected you can prove what you want by exploiting the isomorphisms... | |
| May 27, 2015 at 18:13 | comment | added | Ryan Budney | Correct. Things get complicated when your spaces have two or more non-trivial stages in their Postnikov towers. | |
| May 27, 2015 at 18:00 | comment | added | Dylan Thurston | Thanks! I was looking in completely the wrong section of Hatcher. I guess the generally false belief is true more generally as long as the domain is a CW complex and the target is a $K(\pi, n)$, right? | |
| May 27, 2015 at 17:51 | comment | added | mme | @DylanThurston: The generally false belief is true for based homotopy classes of maps $K(\pi, 1) \to K(\pi',1)$, which are in bijection with homomorphisms $\pi \to \pi'$. The proof Ryan alluded to is Hatcher's Theorem 1B.9. This theorem immediately implies that the homomorphisms induced by homotopy equivalences are precisely the automorphisms, and we pass to $\text{Out}(\pi,1)$ when we forget about the basepoint. | |
| May 27, 2015 at 17:35 | comment | added | Dylan Thurston | Ryan, you're starting to sketch an argument for surjectivity of the map to endomorphisms of $\pi_1$. What about injectivity? Is, for instance, the false belief that I mentioned true for self-maps of $K(\pi,1)$ spaces, or for homotopy equivalences? This does fit within obstruction theory, and I looked a little in Hatcher's book before posting, but I couldn't find anything directly relevant (in terms of finding homotopies). I'd appreciate a more detailed pointer. | |
| May 27, 2015 at 17:19 | comment | added | Ryan Budney | The link between continuous maps of a $K(\pi,1)$ and homomorphisms of $\pi_1$ up to conjugation comes from obstruction theory. Given a homomorphism of $\pi_1$ you define a map between the $K(\pi,1)$ and itself, inducing the map on $\pi_1$, cellularly. You can extend the map to the entire space since it has no higher homotopy groups. I think there is a semi-detailed write up in Hatcher's algebraic topology book. The same kind of argument shows you any continuous map between $K(\pi,1)$ spaces is induced via this kind of construction. | |
| May 27, 2015 at 17:10 | history | asked | Dylan Thurston | CC BY-SA 3.0 |