Homologically-trivial codimension-2 submanifolds must bound codimension-1 submanifolds

Question

I'm not sure about the exact statement, but I've heard that a homologically-trivial codimension-2 submanifolds must bound codimension-1 submanifolds. Is there a reference a proof for this statement? Thanks in advance.

Answer 1

In the setting of smooth manifolds (judging by the tags this is the case you are interested in), see Theorem 3 on page 50 of

Kirby, Robion C., The topology of 4-manifolds, Lecture Notes in Mathematics, 1374. Berlin etc.: Springer-Verlag. vi, 108 p. DM 25.00 (1989). ZBL0668.57001.

I am not sure about the topological category (PL should work the same). I remember, Mike Miller wrote a more detailed account of this proof (and mentioned it in one of the MSE questions), I just forgot where it was. You may want to ask Mike directly, he is at Columbia U.

Edit. In the answer I was assuming that you are using homology with integer coefficients and your codimension 2 submanifold $M\subset N^n$ is closed, connected and oriented, and $N$ is also oriented (not sure if this assumption is essential here, but it is used in the proof). Then the fundamental class $[M]$ of $M$ is well-defined and the condition that $[M]=0\in H_{n-2}(N)$ is well-posed.

Here is the proof taken from Kirby's book: