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Probably the simplest statement of the Loop Theorem in 3-manifolds is as follows: Let $M$ be a 3-manifold and let $D$ be a 2-disk. If there is a map $$(D, \partial D) \rightarrow (M, \partial M)$$ with $f| \partial D \rightarrow \partial M$ not nullhomotopic, then there is an embedding with the same property.

Can anyone suggest an intuitive way to look at this result? It would also be helpful to see an actual manifold with a disc mapped into as described.


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up vote 2 down vote accepted

I like to think about the Loop theorem as a kind of "regularization" result: you have a map $f:(D,\partial D) \to (M, \partial M)$ which is just continuous and such that $\gamma = f (\partial D)$ is not nullhomotopic in $\partial M$ (it is the "loop" in "Loop theorem"), and the theorem allows you to promote $f$ to a map $g:(D,\partial D) \to (M, \partial M)$ which is also injective.

To get the intuitive feeling you are looking for, you should probably have a look at Dehn's Lemma, which is actually a corollary of the Loop Theorem. It says that if you have a map $f:(D,\partial D) \to (M, \partial M)$ which is injective near $\partial D$ then $f|_{ \partial D}$ can be extended to a continuous embedding $f:(D,\partial D) \to (M, \partial M)$ (i.e. with $f|_{ \partial D}=g|_{ \partial D}$). In other words, if an embedded loop in $\partial M$ is nullhomotopic in $M$ then it is the boundary of an embedded disk in $M$.

For a reference, have a look at chapter 4 of Hempel's 3-Manifolds, or at chapter 3 in Hatcher.

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