Timeline for Mapping Class Group (MCG) of connected sum of 3-torus and $S^2\times S^1$
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 16, 2018 at 14:29 | comment | added | mme | @Qiaochu This is due to Waldhausen, who proved that the mapping class group of a closed aspherical oriented Haken 3-manifold is identified with $\text{Out}(\pi_1)$; in this case that is $GL_3 \Bbb Z$. There is a proof of Waldhausen's theorem in Hempel's book on 3-manifolds. The proof is an extension-of-ideas from the surface case. | |
| Jul 29, 2016 at 13:35 | answer | added | Allen Hatcher | timeline score: 16 | |
| Jul 29, 2016 at 5:12 | comment | added | Qiaochu Yuan | I do not see what in that post gives the result you want for $T^3$. In any case some further argument beyond what you wrote down is needed. | |
| Jul 29, 2016 at 5:11 | comment | added | SKShukla | yes, but it generalizes to 3-manifold in this specific case. A discussion can be found here: math.stackexchange.com/questions/35702/… | |
| Jul 29, 2016 at 5:10 | comment | added | Qiaochu Yuan | Those are two different statements; the Dehn-Nielsen-Baer theorem is a claim about the mapping class groups of surfaces, whereas your second claim is about homotopy classes of homotopy equivalences. Neither of them imply the desired result about the mapping class group of $T^3$. | |
| Jul 29, 2016 at 5:05 | comment | added | SKShukla | I am not sure I understand your comment. I was just using the fact that Dehn-Nielsen-Baer theorem applies to space $K(\pi,1)$, that is, $\pi_0HomotopyEquivalences(M)$ is isomorphic to $Out(\pi)$. | |
| Jul 29, 2016 at 4:47 | comment | added | Qiaochu Yuan | The fact that $T^n$ is an Eilenberg-MacLane space tells you that the group of homotopy classes of homotopy equivalences is $GL_n(\mathbb{Z})$; it doesn't imply the same fact about mapping class groups, both because 1) a priori a homotopy equivalence may not be representable by a homeomorphism, and 2) a priori a homotopy between homotopy equivalences, even ones that can be represented by homeomorphisms, may not be representable by an isotopy. For the actual mapping class groups, at least when $n \ge 5$, see en.wikipedia.org/wiki/Mapping_class_group#Torus. | |
| Jul 29, 2016 at 4:41 | history | asked | SKShukla | CC BY-SA 3.0 |