Let $W: \emptyset \Rightarrow M$ be a smooth cobordism from $\emptyset$ to a smooth closed $n$-manifold $M$. Are there reasonably simple conditions on $W$ which guarantee the existence of another cobordism $W': M \Rightarrow \emptyset$ such that $W' \circ W$ is diffeomorphic to the $(n+1)$-sphere?

I am mostly interested in the higher dimensional case where surgery theory works. There are some obvious necessary conditions, such as the triviality of the tangent bundle of $W$ and the surjectivity of $H_k(M) \to H_k(W)$ for $0 \leq k \leq n$. It seems plausible that if one could find the right configuration of conditions then a surgery-theoretic argument would yield the result, but I was not able to find any such claim in the literature.

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    $\begingroup$ Equivalently: which (n+1)-manifolds with boundary embed into $\mathbb R^{n+1}$. $\endgroup$ – nsrt Jun 11 '14 at 10:32
  • $\begingroup$ Yes, smoothly embed, that's another way to look at it. But it seems much less tractable in this formulation (although this might be due to my ignorance of embedding theory). $\endgroup$ – Yonatan Harpaz Jun 11 '14 at 12:27

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