Timeline for Torsion in homology or fundamental group of subsets of Euclidean 3-space
Current License: CC BY-SA 4.0
8 events
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| Jan 21, 2022 at 2:52 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question has been bumped anyway)
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| Jul 14, 2010 at 0:23 | comment | added | Autumn Kent | Tom, I think the Sphere Theorem (that any 3-manifold containing an essential sphere contains an essential embedded sphere) works for any 3-manifold, so a submanifold of $S^3$ with connected complement would be aspherical and so wouldn't have torsion in $\pi_1$. For the problem at hand, you can of course assume the subset of interest, X say, is connected, but it would seem that the issue is that you're interested in the fundamental group of X, and not the complementary 3-manifold (and you don't have duality at your disposal for $\pi_1$. | |
| Jul 13, 2010 at 23:34 | comment | added | Tom Goodwillie | n>1, I meant of course | |
| Jul 13, 2010 at 18:35 | comment | added | Tom Goodwillie | If a manifold is aspherical (a $K(G,1)$) then its fundamental group cannot have an element of order $n>0$ because then the manifold would have a covering space which is both a manifold and a $K(\mathbb Z/n,1)$. Thus a noncompact $3$-manifold with torsion in $\pi_1$ would have to have $\pi_2\ne 0$. Does that help? | |
| Nov 7, 2009 at 2:09 | comment | added | Autumn Kent | Oh, very good. I knew the argument for compact submanifolds, but was spacing on whether or not you can immediately jump to a compact submanifold. Thanks. | |
| Nov 7, 2009 at 2:04 | comment | added | Ryan Budney | Ah, that's nice. The answer to your question is no. If you had torsion in H_1 of an open 3-dimensional submanifold of R^3, you'd have it in a compact 3-dimensional submanifold of R^3. That doesn't happen -- I didn't supply the argument but it it boils down to what's known as Fox's Re-Embedding Theorem (which is an application of Dehn's Lemma), that such a 3-manifold can be re-embedded to be the complement of disjoint embedded handlebodies, another duality application says H_1 is free. So part of my question is answered. The fundamental group question remains open. | |
| Nov 7, 2009 at 2:02 | history | edited | Autumn Kent | CC BY-SA 2.5 |
added 44 characters in body
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| Nov 7, 2009 at 1:51 | history | answered | Autumn Kent | CC BY-SA 2.5 |