For an embedded compact 3-manifold , implies .

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# Drawable 3-manifolds

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For an embedded compact 3-manifold , implies .

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Notice that the boundary of is a disjoint union of compact, orientable surfaces. Call them and thus . It suffices to show that: iff contains only 2-spheres iff .

Let us prove that if the boundary does not contain only 2-spheres then the homology, and thus also the fundamental group, are non-trivial. For this look at embedded in . Call to the complement of in and enlarge to an open neighborhood that retracts to , call this neighborhood . Thus is homotopically equivalent to . Now using the Alexander duality with coefficients in and the universal coefficients theorem we deduce .

Doing Mayer-Vietoris with and we get an exact sequence: (The zero comes form the fact that the union of and is , which has trivial second homology). And thus if contains some surface of positive genus cannot be trivial (because is isomorphic to ).

For the other implication it suffices to show that if all the components of the boundary are 2-spheres then both and are trivial. For this we use a strong result that says that a *smoothly* embedded 2-sphere in the 3-sphere divides the 3-sphere in two 3-balls (an instance of the generalized Schoenflies theorem). Now by induction on the number of connected components of the boundary of we can “fill” each 2-sphere to a 3-ball, which is just adjoining a 3-cell, and thus the and the remain invariant untill the end, where we get the 3-sphere. This means that both the first homology and the fundamental group were trivial.

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