Assume $\Sigma\_1$ and $\Sigma\_2$ are two embedded compact surfaces 
(say orientable) in an orientable 3-manifold $M$. Assume $\Sigma\_1$ and $\Sigma\_2$ 
are homotopic in $M$. Then are they isotopic? 



Isotopy in 3-manifolds
No, generally they're not. 


For example, there's only one homotopy class $S^2 \to \mathbb R^3$ but there's two isotopy classes of embeddings (given via how the embedding orients the compact 3-manifold it bounds). 


edit: I think if your 3-manifold is irreducible and if your maps $S^2 \to M$ are not null homotopic then the answer is likely yes. But if your 3-manifold is say a connect sum of lens spaces then I suspect it's false but I haven't come up with a nice example yet. As Allen points out in the comments below, a connect-sum of lens spaces won't work, at least not when your surface is a sphere. 


edit2: As Misha Kapovich points out, for irreducible 3-manifolds and incompressible surfaces homotopy implies isotopy. This is an old theorem of Waldhausen's. "On Irreducible 3-manifolds which are sufficiently large" Ann. of Math (2) 87 (1968) 56--88. 


