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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 In contrast, Donaldson's invariants have been superseded by more easily computed invariants.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 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.

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.

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?

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 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.

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 it would be nice to have multiple arguments.

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?

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?