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.

(Danny Calegari's notes have Kneser's lemma and the Scott Core Theorem.)


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