1
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

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.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.