The Freedman Dichotomies

Question

Definition: A categorical dichotomy is said to be a “Freedman Dichotomy” if the sole evidence for its non-vacuity is the Disk Theorem (Theorem 1.1 of (F82)). This claims that a Casson handle is homeomorphic to a standard 2-handle (rather than just homotopic to a standard 2-handle, as originally proven by Casson (C86)).

Question: What are all of the known Freedman Dichotomies?

Let me illustrate by presenting examples I have found.

Definition: A smooth manifold is called “exotic” if it is homeomorphic to some standard smooth manifold, but not diffeomorphic to it. Examples of exotic spheres in dimension 7 and above were famously constructed by Milnor in 1959 (M59).

No examples of exotic manifolds of lower dimension are known, other than in dimension 4, where examples arise due to Freedman’s work.

Answer 1: 4-manifold exoticness is a Freedman Dichotomy.

In a previous MO question I sought, without success, to find any evidence for the Disk Theorem other than that contained in (F82). A follow-up MO question sought to quarantine the implications of the Disk Theorem with regard to exotic 4-manifolds. The result of these questions is that, without the Disk Theorem there would be no evidence for exoticness in 4 dimensions.

In fact, this can be improved:

Answer 2: The distinction between “small” and “large” exotic 4-manifolds is also a Freedman (sub)Dichotomy.

Small exotic structures arise from the contradiction between the Disk Theorem and the smooth h-cobordism theorem, while large exotic arise from the contradiction between the Disk Theorem and smooth connected-sum-splitting. A good overview of this is contained in (Sc05).

Moving on, we have the following.

Definition: A knot in the 3-sphere (the boundary of a 4-ball) is “smoothly slice” if it bounds a proper smoothly embedded disk in the 4-ball. A knot is “topologically slice” if it bounds a proper continuously embedded disk $D$ which can be extended to a continuous embedding of $D\times D$ into a normal neighbourhood of the disk.

Answer 3: The distinction between topological sliceness and smooth sliceness for knots in dimension 3 is a Freedman Dichotomy.

The reason is that the existence of a topological slice disk for knots with Alexander polynomial equal to 1 follows solely from the Disk Theorem, while the non-existence of smooth slice disks comes from smooth knot theory – often allied to gauge theory or smooth invariants of knots.

The E8 lattice is a symmetric $8\times 8$ matrix of integers that appears in the study of Lie groups. As it is unimodular, the question arises as to whether E8 is the intersection form of a closed simply connected 4-manifold (which must be unimodular).

Since the E8 lattice has signature 8, a theorem of Rohlin says that it cannot be the intersection form of a smooth closed simply connected 4-manifold (which must have signature divisible by 16).

Answer 4: The existence/non-existence of a (non-smooth) topological 4-manifold with E8 intersection form is a Freedman Dichotomy.

Such a 4-manifold exists by the classification of topological 4-manifolds, which once again, follows from the Disk Theorem.

Definition: A “triangulation” of a topological manifold is a homeomorphism to a locally finite simplicial complex.

Topological manifolds of dimension greater than 4 which do not admit a triangulation have been known for decades (KS77).

Answer 5: The triangulability/non-triangulability of 4-manifolds is a Freedman Dichotomy.

Unpublished work of Casson relates his invariant to the triangulability or otherwise of topological 4-manifolds, whose existence is guaranteed by their classification - see the exposition (AMcC14).

So what other Freedman Dichotomies exist? And, of course, any corrections or comments on the above five examples are welcome.

(AMcC14) Akbulut, Selman, and John D. McCarthy. Casson's Invariant for Oriented Homology Three-Spheres: An Exposition.(MN-36). Princeton University Press, 2014.

(C86) Casson, A. J. (1986), "Three lectures on new infinite constructions in 4-dimensional manifolds", A la recherche de la topologie perdue, Progr. Math., 62, Boston, MA: Birkhauser Boston, 201-244.

(F82) Freedman, Michael Hartley. "The topology of four-dimensional manifolds." J. Differential Geom 17.3 (1982): 357-453.

(KS77) Kirby, Robion C., and Laurence Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations. No. 88. Princeton University Press, 1977.

(M59) Milnor, John. "Differentiable structures on spheres." American Journal of Mathematics 81.4 (1959): 962-972.

(Sc05) Scorpan, Alexandru. The wild world of 4-manifolds. American Mathematical Soc., 2005.

Answer 1

I would rather make this a comment, since I am only suggesting a hitherto apparently not well known dichotomy in the literature, not in mathematics, but it won't allow me to post comments. Anyway.

Here https://eudml.org/doc/109989 is an account by Siebenmann of Freedman's Casson handle proof. It contains a detailed construction of the partial parametrisation of a Casson handle that Freedman called the design, and another exposition of the sphere to sphere theorem. Mitosis is not proven there, but this was independently described by Gompf and Singh (see Durham conference proceedings 1984).

Also, with regards to recent discussion above, Jim Conant was talking about Kirby's "Topology of 4-manifolds" book, not the Freedman-Quinn book of the same name, because Kirby also sketched Freedman's proof in his book.

Answer 2

One of Freedman's results is that a homology $3$-sphere admits a tame topological embedding into $\mathbb R^4$. So here is an odd fact:

Let $M$ be the Poincare dodecahedral space. There is an open subset of $\mathbb R^4$ homeomorphic to $M \times \mathbb R$. But there is no open subset of $\mathbb R^4$ diffeomorphic to $M \times \mathbb R$.

The obstruction to the smooth embedding of $M$ comes from the Rochlin invariant of the smooth $4$ manifold $M$ bounds in $\mathbb R^4$.

One additional twist to this is that $M \setminus \{*\}$, i.e. the once-punctured Poincare Dodecahedral Space admits a beautiful smooth embedding in $\mathbb R^4$. The problem with this embedding with respect to the above is that the sphere surrounding the puncture point is knotted, i.e. the punctured Poincare dodecahedral space is a Seifert Surface for a non-trivially embedded $S^2$ in $\mathbb R^4$. So there's no clear way of how to modify this embedding to get an embedded $M$.

These embeddings of Freedman's use the standardness of Casson 2-handles. I have not seen a construction of any such embedding by any other means.

edit: I suppose one reason the above could be deemed hard to accept is you would think you could `smooth' a topological embedding $M \to \mathbb R^4$. Smooth maps are dense, so every topological embedding can be approximated by a smooth map, but that smooth map (I presume) would have degeneracies. I think it would be useful to complete this picture to describe the degeneracies of such smooth approximations, in some combinatorial language.