Let $M$ be a compact, orientable and irreducible 3-manifold with with boundary consisting of two incompressible components $N\_0,N\_1$, with $N\_i \stackrel{f\_i}{\cong} S\_g$ for some diffeomorphism $f\_i: S\_g \to N\_i$, where $S\_g$ is the closed orientable surface of genus $g $. I wonder if the following is true: 



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> If $f\_0 \simeq f\_1$, i.e, if the $f\_i $ are homotopic as maps from $S\_g$ to $M$, then $M$ is homeomorphic to $ S\_g \times [0,1]$.
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I have the following idea for a proof: Since $f\_0 \simeq f\_1$, there is a map $f: S\_g \times [0,1] \to M$ with $f(x,i) = f\_i(x)$ for $i=0,1$. In particular, we have $f\_0 = f \circ i\_0$ with $i\_0: S\_g \to S\_g \times [0,1]$ the natural inclusion of $S\_g$. Since $f\_0$ is $\pi\_1$-injective and $i\_0$ is a homotopy equivalence, $f$ must be $\pi\_1$ injective. I would be done once I've shown that $f$ is also $\pi\_1$-surjective, since this implies that $f$ is indeed homotopic to a homeomorphism. 


Intuitively, this *should* be the case, but i cannot come up with a formal proof of this. Does anybody have an idea, or a counterexample ?



Is the following 3-manifold always a trivial I-bundle over a surface?
This follows, fairly easily, from the hypothesis of irreducibility and from the "annulus theorem" (see page 130 of Jaco-Shalen's book "Seifert Fibered Spaces in 3-Manifolds"). You can remove the hypothesis of irreducibility if you are willing to use the Poincaré conjecture. 


