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 unresolved 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 S^2 is the boundary of an embedded D^2. This allows one to determine isotopy classes of embedded curves in arbitrary surfaces.

* Alexander’s theorem in dimension 3 states that an embedded S^2 in S^3 is the boundary of an embedded D^3. 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 D^4 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 S^2 in S^4 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 S^2 in S^4 is the boundary of many distinct smoothly-embedded 3-discs in S^4. By distinct, I mean, up to isotopy leaving the unknotted S^2 fixed.

If we jump back to 3-manifolds, in S^3 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 S^1xD^3 has infinitely-many non-separating properly-embedded 3-discs, up to isotopy. We further prove that the group of diffeomorphisms Diff(S^1xD^3) 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 I will continue do just that, here.

All reducing discs in S^1xD^3 appear as fibres of smooth fibre-bundles S^1xD^3–>S^1. This result was quite surprising, for two reasons. We did not expect there to be non-standard reducing discs in S^1xD^3. Further, our proof that reducing discs in S^1xD^n were fibres of smooth fibre-bundles S^1xD^n–>S^1 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 D^n in S^1xD^n such that the boundary of D^n agrees with {1}xS^(n-1).

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

(1) The isotopy extension theorem.

(2) The classification of tubular neighbourhoods.

(3) Sard’s theorem: transversality and intersection theory.

(4) The inverse and implicit function theorems.

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 S^1xD^n together to produce a new copy of S^1xD^n. 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 S^1xD^n.

I ask the kind reader: is this result about reducing discs being a group trivial? Perhaps the answer depends on whether or not the group contains more than one element.

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 S^1xD^n has the consequence that the reducing-disc group of S^1xD^n is a direct-sum of countably-many copies of Z_2. This requires n to be 6 or larger.

Some care is needed. The result of Hatcher-Wagoner states that the mapping class group of S^1xD^n, provided n is 6 or larger, is isomorphic to a direct sum of the three subgroups:

(1) The mapping-class group of D^(n+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 D^n.

(3) An infinite direct-sum of copies of Z_2.

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 D^n into S^1xD^n 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 Z_2.

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