# Fake versus Exotic

Without recourse to the Disc Theorem (or its progeny), is it true that all known examples of exotic differentiable structures on 4-manifolds would be fake rather than exotic?

Terminology (perhaps non-standard):

(1) By "Disc Theorem" I mean the statement, contained in [1], that kinky handles are not only homotopic to standard 2-handles (as proved by Casson [2]), but homeomorphic to standard 2-handles,

(2) The "progeny" is all extensions of the Disc Theorem to non-simply connected settings (including gropes, capped gropes, etc),

(3) A smooth 4-manifold is "exotic" if it is homeomorphically equivalent to, but smoothly inequivalent to a standard 4-manifold,

(4) A smooth manifold is "fake" if it is homotopically equivalent to, but smoothly inequivalent to a standard 4-manifold.

The motivation for this question is the same as that of the earlier Question, namely:

1. The classification [2] of topological 4-manifolds, which follows from the Disc Theorem, is now 30 years old and an easier version of the proof has not emerged. In contrast, Donaldson's invariants have been followed by more easily computed invariants. This asymmetry is an 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.

3. It may be possible to refine the proof under the assumption of more regularity to gain more control over the resulting infinite towers - and perhaps get Hoelder maps rather than homeomorphisms, for example.

References:

[1] Freedman, M. H. (1982), "The topology of four-dimensional manifolds", Journal of Differential Geometry 17 (3), 357-453.

[2] 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.

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As far as I know, Freedman's result is the only method to tell that two 4-manifolds are homeomorphic, without explicitly demonstrating a diffeomorphism (by e.g. Kirby Calculus). –  Ian Agol Oct 2 '12 at 16:04
I try to make a little "reassuring" comment: do these proofs-by-contradiction have really such an enormous impact? Does Freedman's theorem (roughly, two homotopy equivalent closed 4-manifolds with some condition on their pi_1 are homeomorphic) really have such an enormous impact, aside from writing many times "homeomorphic" instead of "homotopy equivalent"? –  Bruno Martelli Oct 4 '12 at 10:28
@Bruno Agreed. The content of this question is essentially: for the known exotic 4-manifolds does it all hinge upon the stiffening of homotopy to homeomorphism afforded by the Disc Theorem? You're saying, that this is the only concrete impact (yielding exotic rather than just fake 4-manifolds). On the other hand, perhaps we are so far from proving the last remaining Poincare conjecture (smooth 4-d) because the proof of the topological 4-d conjecture is so poorly understood. –  Brendan Guilfoyle Oct 7 '12 at 12:57
Here is some further progeny, namely the outcome (so far) of the Freedman part of the Bonn semester: mypage.iu.edu/~macp/Freedman2013.pdf –  Dave Aug 28 at 5:29