Suppos we have a cobordism $(M, \partial_{-}M, \partial_{+}M)$, where $M$ is a oriented compact (topological) 3-manifold.

Assume we have orientation preserving homeomorphism $f_{\pm}: \Sigma \to \partial_{\pm}M$ respectively, where $\Sigma$ is a oriented compact 2-surface. (I have $\Sigma=S^1 \times S^1$ in my mind.)

**Question;** When $f_{\pm}$ extends to a homeomorphim from $(\Sigma \times \[0,1\], \Sigma \times 0, \Sigma \times 1)$ to $(M, \partial_{-}M, \partial_{+}M)$?

In other words, under what condisions is there a homeomorphism from $\Sigma \times \[0,1\]$ to $M$ such that the restriction of it on boundaries are $f_{\pm}$ respectively.