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.