# Applying Dehn's Lemma/The Loop Theorem to make a surface $\pi_1$-injective

While reading the paper On Scott's Core Theorem [Swarup & Rubinstein: 1990], I was stumped trying to recreate the following, which I feel ought to be very straightforward.

Suppose we have a two-sided surface $S$, which forms a component of the frontier of a 3-manifold $V$, itself lying inside some 3-manifold $M$ (possibly with boundary). Suppose also that the map generated by inclusion: $\pi_1(S) \to \pi_1(M)$, is not injective.

$$S^2 \subset \text{fr}(V^3) \subset V^3 \subset M^3:$$

It is alluded in the paper above that one should be able to apply Dehn's Lemma/The Loop Theorem (perhaps repeatedly) to modify the surface $S$ (either by removing disks from $V$ or adding [fattened] disks to $M\backslash V$) to obtain $\pi_1$-injective surface(s) in its place.

However, I had difficulty in verifying this for myself, as my attempts either failed to satisfy the hypotheses of the theorems above, or obtained too weak a result. Here are some difficulties I faced:

1. For a loop $\gamma$ in $S$ which is nullhomotopic in $\pi_1(M)$, can a nullhomotopy be taken to lie entirely either in $\overline{M\backslash V}$ or $V$?

2. Is it guaranteed that $\ker(\pi_1(S) \to \pi_1(M))$ is finitely generated?

3. Can one guarantee that a generator of this kernel is a simple closed loop, or at least has finitely many self-intersections?

(I include these, as they perhaps highlight a flaw in my naïve approach, which is to induct on the number of generators in a minimal generating set of $\ker(\pi_1(S) \to \pi_1(M))$.)

My question is this: Can one apply Dehn's Lemma/The Loop Theorem as indicated in the third paragraph, without any further assumptions? If so, what approach should one take?

Kneser's lemma. Let $$S$$ be a two-sided properly embeddeded surface in a $$3$$-manifold $$M$$. If the induced map $$\pi_1(S)\to\pi_1(M)$$ is not injective, then there is an embedded disk $$D\subset M$$ such that $$D\cap S=\partial D$$ is transverse and $$[\partial D]$$ is an essential loop in $$S$$. (Or equivalently, via the Loop Theorem: then the inclusion $$\partial\nu(S)\to M-\nu(S)$$ does not induce an injection on $$\pi_1$$, where $$\nu(S)$$ denotes the embedded normal bundle.)
Proof. Since $$S$$ is two-sided, there is a pushoff $$S_+$$ of $$S$$ in its embedded normal bundle. This, too, is not $$\pi_1$$-injective. There is a nullhomotopy $$f:D^2\to M$$ of some non-trivial element of $$\pi_1(S_+)$$. By some small homotopy, we can assume $$f$$ is transverse to $$S$$, so $$f^{-1}(S)$$ is a (possibly empty) finite collection of closed loops; no arcs since we arranged for the boundary of $$D^2$$ to map away from $$S$$. If the intersection is empty, then $$f$$ gives a nullhomotopy of a loop of $$S_+$$ in $$M-\nu(S)$$, to which we may apply the Loop Theorem and get an embedded disk. Otherwise, take a loop in $$f^{-1}(S)$$ that bounds an innermost disk $$A\subset D$$. If $$[f(\partial A)]$$ is essential in $$S$$, then replace $$f$$ with $$f|_A$$, which can be pushed off from $$S$$ to $$S_+$$ and does not intersect $$S$$. On the other hand, if $$[f(\partial A)]$$ is nullhomotopic, there is a nullhomotopy $$A\to S$$ of $$f(\partial A)$$, and with this modify $$f$$ so it does this nullhomotopy along $$A$$. There is a direction to push the nullhomotopy slightly off $$S$$ to reduce the number of loops in $$f^{-1}(S)$$. QED
If $$S$$ is in the frontier of some submanifold $$V\subset M$$ that does not induce an injection $$\pi_1(S)\to\pi_1(M)$$, then Kneser's lemma gives a compression disk $$D$$ on either side. If inside, then $$V-\nu(D)$$ effectively does the compression, and if outside $$V\cup \nu(D)$$ effectively does the compression. If $$S$$ is compact, then since each compression reduces the complexity of $$S$$, the process will eventually terminate.