# 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 – user39253 Aug 28 '13 at 5:29
The lecture notes linked by user39253 have moved (for now) to renyi.hu/~sbehrens/freedman.html – j.c. Sep 24 '15 at 15:13

The answer appears to be "Yes" - without recourse to the Disc Theorem, all known examples of exotic differentiable structures on 4-manifolds would be fake rather than exotic.

Casson's result is that his infinite kinky handle construction produces objects that are homotopic to standard 2-handles and so, a priori, leads only to fake 4-manifolds. Since fakeness is not such an unusual phenomenon (homotopies are weak creatures), one could interpret the result as something of a failure - perhaps this explains why the result languished unpublished for almost a decade.

On the other hand, the existence of exotic 4-manifolds, with all of their strangeness, follows solely from the central claim of the disc theorem: that Casson handles are homeomorphic to standard 2-handles. This introduces extraordinary dichotomies in 4-dimensions, as it contradicts many different subsequent smooth results. For example, it leads to:

1. Small exotic structures: disc theorem versus smooth h-cobordism theorem
2. Large exotic structures: disc theorem versus smooth connected-sum-splitting
3. Topologically slice knots: disc theorem versus smooth knot theory

All of this is deeply unsettling.

As an attempt at an alternative geometrization of these paradigms, I have just posted a preprint to the arxiv (joint work with Wilhelm klingenberg) utilizing a neutral Kaehler structure on 4-manifolds. While the main result is the extension of a differential geometry result of 1846, this is essentially a byproduct of our consideration of the disc theorem. The topological motivation is more fully explained in the video:

A global version of a classical result of Joachimsthal and the slice problem for knots

In the near future I will post further details of an alternative construction of Casson handles with more geometric control of the boundary in order to explore these questions directly.

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The formula near the end of the paper looks reminiscent of a formula for geodesic torsion (but missing a $\cos(\phi_i)$ factor). Since the geodesic torsion of a curve of intersection of surfaces intersecting at constant angle is the same, I think this reproves your formula (maybe with a $\cos(\phi_i)$ factor or using $\sin(2\phi_i)$ instead)? encyclopediaofmath.org/index.php/Geodesic_torsion – Ian Agol Jul 22 '14 at 19:42