Topological 4-Manifolds – Independent Evidence for Classification

gn.general-topologygt.geometric-topology

Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? The argument there is extraordinarily complicated and a simpler proof would be desirable.

Is there evidence from any other source that would suggest that topological 4-manifolds are so much simpler than smooth 4-manifolds, or does it all hinge on Freedman's proof that Casson handles are homeomorphic to standard handles?

My question is motivated from a number of points of view:

  1. The classification of topological 4-manifolds is now 30 years old and an easier version of the proof has not emerged. In contrast, Donaldson's invariants have been superseded by more easily computed invariants. This is a very unsatisfactory state of affairs for such a far-reaching topological result, particularly as it is so regularly used in proof-by-contradiction arguments against results in smooth 4-manifold theory.

  2. As the Bing topologists familiar with these arguments retire, the hopes of reproducing the details of the proof are fading, and with it, the insight that such a spectacular proof affords. I am delighted to see that the MPIM, Bonn is running a special semester on this topic next year. Hopefully this will introduce these techniques to a new generation of mathematicians (and save them from having to reinvent them!)

  3. It may be possible to refine the proof to gain more control over the resulting infinite towers – and perhaps get Hoelder maps rather than homeomorphisms, for example. This would require either a better exposition of the fundamental result or some new independent insight, which was the basis of my question.

Best Answer

The answer to this question might have changed since it was first asked nine years ago: a book is now available whose goal it is to give a detailed elaboration on Freedman's work:

The Disc Embedding Theorem, ed. Behrens, Kalmar, Kim, Powell, Ray (Oxford University Press, 2021).

Some excerpts from the Preface:

We choose to follow the proof from [FQ90], using gropes, which differs in many respects from Freedman's original proof using Casson towers [Fre82a]. The infinite construction using gropes, which we call a skyscraper, simplifies several key steps of the proof, and the known extensions of the theory to the non-simply connected case rely on this approach. …

We briefly indicate, for the experts, the salient differences between the proof given in this book and that given in [FQ90]. First, there is a slight change in the definition of towers (and therefore of skyscrapers). …

Additionally, the statement of the disc embedding theorem in [FQ90] asserts that immersed discs, under certain conditions including the existence of framed, algebraically transverse spheres, may be replaced by flat embedded discs with the same boundary and geometrically transverse spheres. The proofs given in [Fre82a, FQ90] produce the embedded discs but not the geometrically transverse spheres. We remedy this omission by modifying the start of the proof given in [FQ90], as in [PRT20]. … Besides these points, the proof of the disc embedding theorem given in this book only differs from that in [FQ90] in the increased amount of detail and number of illustrations.

In Section 1.5 they mention some things that are not covered in the book. In addition to bypassing the part of Freedman's original proof that "consisted of embedding uncountably many compactified Casson handles within the original Casson handle and then applying techniques of decomposition space theory and Kirby calculus," they say:

Note that the ambient manifold is required to be smooth in the statement of the disc embedding theorem. There exists a category preserving version of the theorem, where ‘immersed’ discs in a topological manifold are promoted to embedded ones. However, the proof requires the notion of topological transversality and smoothing away from a point (see Section 1.6). These facts, established by Quinn, in turn depend upon the disc embedding theorem in a smooth 4-manifold stated above. The fully topological version of the disc embedding theorem is beyond the scope of this book, since we will not discuss Quinn's proof of transversality.

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