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Yesterday and today Marty Scharlemann gave two talks on the Schoenflies Conjecture, one of the great open problems in low dimensional topology. These talks were very clear and inspiring, and I thought it would be useful to summarize what Marty said in a blog post, just for my own benefit.

The story starts with the following classical theorem, usually called the Jordan curve theorem, or Jordan-Schoenflies theorem:

Theorem (Jordan-Schoenflies): Let P be a simple closed curve in the plane. Then its complement has a unique bounded component, whose closure is homeomorphic to the disk in such a way that P becomes the boundary of the disk.

In order to make the relationship between the two complementary components more symmetric, one could express this theorem by saying that a simple closed curve P in the 2-sphere separates the 2-sphere into two components X and Y, each of which has closure homeomorphic to a disk with P as the boundary.

Based on this simple but powerful fact in dimension 2, Schoenflies asked: is it true for every n that every n-sphere P in the (n+1)-sphere splits the (n+1)-sphere into two standard (n+1)-balls?

I was at UC Riverside this past weekend, attending the regional meeting, and giving a talk in a special session on knot theory in memory of the late Xiao-Song Lin. After lunch, I joined in a conversation between Rob Kirby and Mike Freedman on the recent flurry of activity this summer, in which Selman Akbulut showed (and his work was further extended by Bob Gompf ) that certain infinite families of Cappell-Shaneson manifolds — smooth $4$-manifolds known since Freedman’s work to be homeomorphic to $S^4$ — are in fact diffeomorphic to the standard smooth $S^4$ (actually, Cappell-Shaneson’s manifolds have the additional feature that they admit a free $\mathbb{Z}/2\mathbb{Z}$ action, giving rise to fake $\mathbb{RP}^4$‘s, which was actually their original interest). (Note: an earlier version of this post falsely implied that Gompf’s work was done independently of Akbulut’s, whereas in fact it came later, as Gompf readily acknowledges).

Apparently these constructions had somewhat altered the experts’s (i.e. Freedman and Kirby) feelings about whether the smooth $4$-dimensional Poincaré conjecture is likely to be true. The Cappell-Shaneson manifolds are constructed by doing surgery on certain torus bundles over a circle — those with monodromy chosen so that the resulting torus bundles have the homology of a $S^1 \times S^3$. A suitable surgery, killing the $S^1$ factor makes the manifold into homology $S^4$‘s, and also kills the subgroup of the fundamental group normally generated by the $S^1$ factor. On the other hand, everything else in the fundamental group “comes from” the torus, whose fundamental group is abelian, and therefore the resulting manifold is simply-connected. Since it is a homology $4$-sphere, is it therefore a homotopy $4$-sphere, and consequently (by Freedman), a topological $4$-sphere.

Gompf shows these $4$-spheres are standard by showing that a certain move — which simplifies the monodromy of the $T^3$ fiber — can be realized by a diffeomorphism. The move is an example of what is known to $4$-manifold topologists as a “log transform” (and to $3$-manifold topologists as “Dehn surgery times $S^1$”). A log transform takes as input a smooth embedded torus. A tubular neighborhood of this torus is a product $T^2 \times D^2$ whose boundary is a $3$-torus $T^3$. This tubular neighborhood is removed, and reglued by an automorphism of the $T^3$ factor. Usually a log transform will change the topology of the manifold, or at least the smooth structure. But in this case, the surgered torus is contained in a $T^3$ fiber, and the log transform can be shown to be isotopic to the identity, by using the monodromy of the fibration (technically, the monodromy of the fibration produces a once-punctured torus in the $T^3$ bundle with boundary on the curve along which the log transform “twists”, but after doing surgery to produce the homology $S^4$‘s, this once-punctured torus is “capped” to become a smooth disk).

The point of this blog post is to show how to construct many, many other smooth $4$-manifolds which are topological $4$-spheres, and for which Gompf’s method of showing they are standard does not work. Are these manifolds counterexamples to the smooth $4$-dimensional Poincaré conjecture? I am really not the person to ask.

The construction takes as input a fibered knot — i.e. a knot $K$ in the $3$-sphere $S^3$ whose complement fibers over a circle. In other words, there is a fibration $S \to S^3 - K \to S^1$, where $S$ is a (minimal genus) Seifert surface for the knot $K$. The fibration of spaces gives rise to a short exact sequence of fundamental groups (in general, one gets a long exact sequence of homotopy groups, but the spaces $S, S^3-K,S^1$ are all $K(\pi,1)$‘s — i.e. their homotopy groups in dimension other than $1$ all vanish). Since $S$ has boundary, the fundamental group of $S$ is free and finitely generated of rank $2g$, where $g$ is the genus. The fundamental group of $S^1$ is $\mathbb{Z}$. So one exhibits $\pi_1(S^3 - K)$ as an HNN extension of a free group, where the meridian $m$ acts by conjugation on the free group $\pi_1(S)$ by some automorphism $\phi:\pi_1(S) \to \pi_1(S)$.

Since $K$ is a knot in $S^3$, the homology of $S^3-K$ is equal to $\mathbb{Z}$ in dimension $1$. Moreover, since putting $K$ back in recovers $S^3$, it follows that the fundamental group $\pi_1(S^3-K)$ is normally generated by the meridian (which also generates the $\mathbb{Z}$ in $H_1$). For the moment everything is $3$-dimensional, but there is a trick to promote this to $4$ dimensions. In place of the surface $S$, consider the $3$-manifold $M_{2g} = \#_{i=1}^{2g} S^2 \times S^1$. In other words, $M_{2g}$ is obtained by doubling a handlebody of genus $2g$. The fundamental group $\pi_1(M_{2g})$ is free of rank $2g$. Now one builds a $M$ bundle over $S^1$ with monodromy $\phi$; call this $4$-manifold $W_\phi$. The existence of such a manifold depends on being able to realize any automorphism of a free group by a homeomorphism of a doubled handlebody; one way to see this is to observe that $\text{Aut}(F)$ is generated by Nielsen moves — interchanging generators, replacing generators by their inverses, and replacing generators $x,y$ by $xy, y$. These moves are all realizable by homeomorphisms of doubled handlebodies, the last by a “handle slide”.

Now, observe that $\pi_1(W_\phi) = \pi_1(S^3 - K)$, and is normally generated by a loop $\gamma \in W_\phi$ representing the circle direction. Moreover, $H_1(W_\phi) = H_1(S^3-K) = \mathbb{Z}$. To compute $H_2$, observe that $H_2(M) = (H_1(M))^*$ by Poincaré duality. If the action of $\phi$ on $H_1(M)$ (a free abelian group of rank $2g$) is represented by a matrix $A$, then the action on $H_2(M)$ is represented by the transpose $A^T$. The fact that $H_1(W_\phi)=\mathbb{Z}$ is equivalent to the fact that the first homology of $M$ dies in the bundle; i.e. $\det(A - \text{Id})=\pm 1$; hence $\det(A^T - \text{Id}) = \pm 1$, and for the same reason, the second homology of $M$ dies in the bundle, and $H_2(W_\phi)=0$. By ($4$-dimensional) Poincaré duality, $H_3(W_\phi) = \mathbb{Z}$, and we see that $W_\phi$ is a homology $S^1 \times S^3$.

A tubular neighborhood of the loop $\gamma$ is a product $S^1 \times D^3$, since $W_\phi$ is orientable. The boundary of this is $S^1 \times S^2$. So we drill out $\gamma$ and glue in a product $D^2 \times S^2$ to produce $W_\phi'$. Drilling out $\gamma$ does not affect the fundamental group, by Seifert van-Kampen, and the fact that the inclusion $S^1 \times S^2 \to S^1 \times D^3$ is an isomorphism on $\pi_1$. On the other hand, filling in a $D^2 \times S^2$ has the effect of killing the meridian $m$, and therefore (by the discussion above), killing $\pi_1$ completely; i.e. $W_\phi'$ is simply-connected. Hence $H_1(W_\phi') = H_3(W_\phi')=0$. Drilling out a circle and gluing back a sphere increases Euler characteristic by $2$; since the rank of $H_1$ and $H_3$ have both gone down by $1$, it follows that the rank of $H_2(W_\phi')$ is still $0$, and since the fundamental group is trivial, $H_2(W_\phi')=0$. So $W_\phi'$ is a smooth, simply-connected homology sphere, which is to say, a smooth $4$-manifold which is topologically $S^4$.

Back in June, Freedman-Gompf-Morrison-Walker described a way to use Rasmussen’s $s$-invariant to detect exotic $S^4$‘s, and proposed trying this invariant out on the Cappell-Shaneson examples (see Scott Morrison’s post about that here). Is it feasible to compute the invariants on these new examples?

(Corrected Update 11/10:) Some ways of doing this construction give standard $S^4$‘s, some give $S^4$‘s that are not obviously standard. And there are other variations on this construction arising from “non-geometric” automorphisms of free groups that are also not obviously standard. These examples are also not obviously the same as other known potential counterexamples to the smooth Poincare conjecture. So the conclusion seems to be that they deserve further study. (Added 11/15:) This paper by Aitchison-Silver discusses a closely related construction.