Timeline for Do most manifolds have symmetries? or not?
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| Jul 28, 2021 at 10:59 | comment | added | Nick L | Furthermore, the question does a generic simply connected, smooth $6$-manifold admit no finite group actions makes perfect sense, because such manifolds are uniquely determined by their cohomolgy ring and characteristic classes (Wall-Jupp-Jubr) so the set of diffeomorphism classes sits in a space suitable for such quantitative questions. Puppe proved several striking statements in the case when the manifolds are spin (a substantial subset), for example that the set of manifolds having a cohomologically non-trivial finite group action has density 0. See Theorem 3 of the paper mentioned above. | |
| Jul 28, 2021 at 10:45 | comment | added | Nick L | For a specific reference for the simply connected case of "counter-question" is Theorem 4 of arxiv.org/pdf/math/0606714.pdf. Puppe constructed a simply connected $6$-manifold on which no finite group can act effectively by orientation preserving transformations. | |
| Jul 28, 2021 at 10:10 | history | edited | archipelago |
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| Jul 28, 2021 at 4:44 | comment | added | Ryan Budney | The answer to your question is going to be sensitive to dimension. There aren't any compact simply-connected 1-manifolds. In dimension 2 there's only the one manifold $S^2$, so your answer will be negative. In dimension 3, again you run into the problem of not having interesting simply-connected manifolds. I think your question is somewhat bogged-down in the issue of what "most" should mean. The question still makes sense for manifolds with fundamental groups. In that case, you start seeing symmetryless manifolds in dimension 3. | |
| Jul 27, 2021 at 20:43 | comment | added | markvs | Did you discuss it with D.Fuchs? | |
| Jul 27, 2021 at 20:29 | comment | added | Denis T | There's a paper by Volker Puppe with a title very reminiscent of the question, and there he proves that "most" 3-manifolds have no finite order diffeomorphisms. | |
| Jul 27, 2021 at 20:07 | comment | added | mme | "...Also we better restrict to connected manifolds, and maybe also to simply connected manifolds." I missed this sentence. I think there is likely to be substantial difference between the s.c. case and the general case: one reason to believe your Statement is that one might believe that most manifolds satisfy Borel's hypotheses. | |
| Jul 27, 2021 at 20:06 | comment | added | archipelago | Any finite order diffeomorphism of a closed manifold that fixes a point and the tangent space at that point is the identity (choose an invariant Riemannian metric and shoot geodesics away from that point). In particular $Diff_\partial(D^d)$ is torsion-free (extend the diffeomorphism to the sphere and use the above). | |
| Jul 27, 2021 at 19:57 | comment | added | mme | Counter-question. Yes, I believe the first example is due to Borel ("On periodic maps of certain K(pi,1)"), which proves that any aspherical manifold with centerless fundamental group and torsion-free outer automorphism group supports no finite order diffeomorphisms. For hyperbolic manifolds I believe every finite group action is conjugate to an action by hyperbolic isometries, so you're just asking (by Mostow) for hyperbolic manifolds with Out(pi_1) tors-free, just like in Borel's statement. | |
| Jul 27, 2021 at 19:45 | history | asked | Chris Schommer-Pries | CC BY-SA 4.0 |