# Extension of homeomorphism of boundaries to a homeomorphism of a cobordism

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.

-
What sort of conditions do you have in mind? Are you looking for an h-cobordism-type theorem for $3$-manifolds cobounding surfaces? The h-cobordism does hold in this case, which follows from the Poincare conjecture (=theorem). –  Jim Conant Feb 18 '12 at 19:15
The question is whether the s-cobordism theorem holds for surfaces –  Fernando Muro Feb 19 '12 at 0:07
Your question is quite vague. You've already listed one condition -- that $f_{\pm}$ extends to a homeomorphism. Why isn't that an answer? –  Ryan Budney Feb 19 '12 at 0:39
You've been asking quite a few similar questions recently. I suspect your low response rate is due to the vague nature of your questions. See: mathoverflow.net/questions/88609/… and mathoverflow.net/questions/88334/isomorphism-of-cobordisms –  Ryan Budney Feb 19 '12 at 0:43
s-cobordism theorem is close to what I wanted to know but I don't know how to fit my questiono into it yet. @Ryan, thank you for your advise. Yes it is ture my question is vague. By the way, the statement "$f_{\pm}$ extends..." is not a condition but the conclusion. I want some condision on which this statement holds. –  knot Feb 19 '12 at 6:08