Isotopy in dimension 4

Four-dimensional manifold theory is remarkable for a variety of reasons.  It has the only outstanding generalized smooth Poincare conjecture.  It is the only dimension where vector spaces have more than one smooth structure.  The only dimension with an outstanding generalized Shoenflies problem.  The list goes on.  One issue that is perhaps not discussed enough is the paucity of theorems about smooth isotopy.    In dimensions 2 and 3, the Schoenflies and Alexander theorems are the backbone of all theorems about isotopy, allowing one to work from the ground-up.

  • The Schoenflies theorem states that an embedded circle in S2 is the boundary of an embedded D2.  This allows one to determine isotopy classes of embedded curves in arbitrary surfaces.
  • Alexander's theorem in dimension 3 states that an embedded S2 in S3 is the boundary of an embedded D3.    Alexander's theorem and Dehn's Lemma together allow a standard innermost circle isotopy argument, which allows one to readily determine the isotopy relation among incompressible surfaces.  Moreover, this is one of the key theorems in latticework that created the framework of "sufficiently large" or Haken manifolds, leading to geometrization. 

In dimension 5 and up, the analogue of the Schoenflies/Alexander theorems are true, but the proof has a rather different form where one proves the theorem in the tame topological category (Mazur) then applies h-cobordism. 

In dimension 4 one still has Mazur's theorem, but the question of if D4 admits an exotic smooth structure is open, so the Schoenflies problem remains open.  Other basic isotopy questions remain open as well.   For example, it remains an open question as to whether or not an embedded S2 in S4 is unknotted if and only if the exterior has infinite-cyclic fundamental group.   

A theorem David Gabai and I recently proved is closely related to that latter problem.  We have shown that an unknotted S2 in S4 is the boundary of many distinct smoothly-embedded 3-discs in S4.  By distinct, I mean, up to isotopy leaving the unknotted S2 fixed. 

If we jump back to 3-manifolds, in S3 the spanning 2-disc for an unknot is an incompressible surface and is unique up to isotopy leaving the boundary circle fixed.  That proof involves Schoenflies and Dehn's Lemma.  Thus, in dimension 4, if there will eventually be anything analogous to the theory of incompressible surfaces, it will be quite different from the 3-dimensional variety. 

Another way to state our theorem is that the 4-manifold S1xD3 has infinitely-many non-separating properly-embedded 3-discs, up to isotopy.  We further prove that the group of diffeomorphisms Diff(S1xD3) acts transitively on these discs. We call these discs "reducing balls" in the paper.  David prefers the terminology "ball" to "disc", so we use ball.  I prefer calling them "reducing discs," so in this blog, I will do just that. 

All reducing discs in S1xD3 appear as fibres of smooth fibre-bundles S1xD3-->S1.  This result was quite surprising, for two reasons.  We did not expect there to be non-standard reducing discs in S1xD3. Further, our proof that reducing discs in S1xDn were fibres of smooth fibre-bundles S1xDn-->S1 was far more general than we expected -- the proof works for all n.  Let me say that again -- there is no adaptation for low dimensions, or high dimensions.  The proof in dimension 4 is the same as the high-dimensional proof, and the low-dimensional proof. In this context, a reducing disc is a smoothly embedded Dn in S1xDn such that the boundary of Dn agrees with {1}xSn-1

I know a very short list of non-trivial theorems about manifolds whose proofs are independent of the dimension of the manifold:

  • The isotopy extension theorem. 
  • The classification of tubular neighbourhoods.
  • Sard's theorem: transversality and intersection theory.

Our proof that reducing discs are fibres of fibre bundles has a further consequence.  Reducing discs have a concatenation operation -- think of stacking two copies of S1xDn together to produce a new copy of S1xDn.  This stacking operation turns isotopy-classes of reducing discs into a monoid, and our proof shows there are inverses.  So we might as well call this the reducing-disc group of S1xDn

When n=1, it is classical that the reducing-disc group is the integers.  When n=2, it is a consequence of the Schoenflies theorem that the reducing-disc group is trivial.  When n=3, we do not compute the reducing-disc group, but our paper proves it contains a free-abelian group of infiniite rank.

The only other result I know of concerning the reducing-disc group comes from the Hatcher-Wagoner book.  Although they did not explicitly write it as so, their theorem concerning the structure of the mapping-class group of S1xDhas the consequence that the reducing-disc group of S1xDis a direct-sum of countably-many copies of Z2.  This requires n to be 6 or larger.   

Some care is needed.  The result of Hatcher-Wagoner states that the mapping class group of S1xDn, provided n is 6 or larger, is isomorphic to a direct sum of the three subgroups:

  1. The mapping-class group of Dn+1.  i.e. isotopy-classes of diffeomorphisms of the (n+1)-disc that are the identity on the boundary.
  2. The mapping-class group of Dn
  3. An infinite direct-sum of copies of Z2

To go between their result and ours, one needs to observe that the reducing-disc group is the path-components of the space of embeddings of Dn into S^1xDn that agree with the standard inclusion on the boundary, modulo parametrization. i.e. I am thinking of the elements of the reducing-disc group as not being equipped with a parametrization.  Thus items (1) and (2) are trivial in the reducing-disc group, and we are left with only the infinite direct-sum of Z2