Timeline for Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
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8 events
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| Dec 6, 2011 at 3:30 | comment | added | Lucas Culler | You can also use an embedding in S^6 (a bit easier to construct). If nM = normal bundle, then H^2(S^6-nM) = Z by Mayer Vietoris. A geometric representative of the generating class will be a 4-manifold X with dX = M. More precisely, dX will be cobordant to a nonvanishing section of nM. | |
| Apr 29, 2011 at 23:37 | comment | added | Dylan Thurston | @Greg: It is related, though in a different direction; see my answer. @Daniel: I disagree that Whitney immersion is hard, and Hirsch's theorem is not necessary, as Bruno sketches. | |
| Apr 29, 2011 at 16:47 | comment | added | Daniel Moskovich | This is beautifully intuitive! But it's definitely massive overkill- each step is in itself more difficult that the fact being proven. Statement (1) is the easiest statement among: 1) Omega_3=0 2)Whitney immersion theorem 3)Hirsch's theorem 4) Alexander duality in this context. This is highlighted by Rourke's short elementary proof (which isn't half as conceptually satisfying for me, though). | |
| Apr 29, 2011 at 16:34 | comment | added | Bruno Martelli | You can avoid Hirsch's theorem by using Whitney's immersion theorem of $M^n$ in $\mathbb R^{2n-1}$, which implies that every 3-manifold $M$ immerses in $\mathbb R^5$. Such an immersion can be perturbed so that self-intersections have dimension 3+3-5 = 1, i.e. are circles. You can then surger the manifold around the circles to get an embedding, and such a surgery can easily be realized by a cobordism. Then you use Alexander duality. This was Rohlin's original argument. | |
| Apr 29, 2011 at 13:52 | comment | added | Autumn Kent | It's funny, I sorta thought this was how everybody thought about it. I guess having Cameron Gordon as an advisor colors your world view. | |
| Apr 29, 2011 at 13:24 | comment | added | Greg Kuperberg | That's quite interesting! And potentially related to the paper of Costantino and Thurston on complexity of 4-manifolds bounded by a 3-manifold. | |
| Apr 29, 2011 at 13:01 | history | edited | Autumn Kent | CC BY-SA 3.0 |
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| Apr 29, 2011 at 12:36 | history | answered | Autumn Kent | CC BY-SA 3.0 |