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It was mentioned in Topological gauge theories and group cohomology, Robbert Dijkgraaf, Edward Witten, Comm. Math. Phys. 129(2): 393-429 (1990).

Any 3 dimensional manifold can be realized as the boundary of a 4 dimensional manifold. (In page 2 (labeled 394)).

It seems that there are some unstated assumptions given here. The paper may mean that the 3-manifold is orientable. My questions:

  1. What are assumptions?

  • orientable or non-orientable?

  • compact or non-compact?

  • smooth differentiable or not?

  1. How to prove: Any 3 dimensional manifold (with what assumptions) can be realized as the boundary of a 4 dimensional manifold.

How does it apply to for example $\mathbf{RP}^2 \times S^1$?

I find some related questions, but I still hope someone can provide new and easier answers than these:

$RP^2$ as a boundary of a 3-manifold

https://mathoverflow.net/questions/63373/elegant-proof-that-any-closed-oriented-3-manifold-is-the-boundary-of-some-orien

Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

annie marie cœur
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    There's an awful lot of information in this link you included. Did you look carefully at any of the proofs mentioned there to see if they might satisfy your requirements for an "easy answer"? Part of the reason I ask is that the proofs mentioned in that link include some of the most modern proofs. – Lee Mosher May 07 '21 at 18:48
  • I skim through it. What I concern is whether the statement in Comm. Math. Phys. 129(2): 393-429 (1990) paper is more general than the closed orientable assumption used in other MO post. :) I hope someone can confirm this MUST be the assumption in the witten paper – annie marie cœur May 07 '21 at 18:51
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    The question is what you mean by "is a boundary". If you mean "there is a 4-manifold whose boundary is the given 3-manifold $M$", then the answer is trivial by just taking $M\times[0,1)$. This is why one usually talks about the non-trivial question of whether a given manifold is the boundary of a compact manifold, but this of course only is a meaningful question if the manifold we started with is compact. Usually, these problems are stated and solved in the smooth category, but for 3-manifolds, this is a non-assumption by Moise's theorem. – Thorgott May 07 '21 at 21:46
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    Regarding orientability vs. non-orientability, is is both true that a compact (not necessarily oriented) 3-manifold is the boundary of some compact 4-manifold and that an oriented, compact 3-manifold is the oriented boundary of some oriented compact 4-manifold (which is a stronger statement). The relevant term to look up here is "(oriented) cobordism". – Thorgott May 07 '21 at 21:47
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    $\bf RP^2\times D_2$ – Thomas May 08 '21 at 08:10

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