Exotic R4

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Short description: Smooth 4-manifold homeomorphic yet not diffeomorphic to Euclidean space

In mathematics, an exotic ℝ4 is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space ℝ4. The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.[1][2] There is a continuum of non-diffeomorphic differentiable structures ℝ4, as was shown first by Clifford Taubes.[3]

Prior to this construction, non-diffeomorphic smooth structures on spheres - exotic spheres - were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and remains open as of 2024). For any positive integer n other than 4, there are no exotic smooth structures ℝn; in other words, if n ≠ 4 then any smooth manifold homeomorphic to ℝn is diffeomorphic to ℝn.[4]

## Contents

* 1 Small exotic R4s
* 2 Large exotic R4s
* 3 Related exotic structures
* 4 See also
* 5 Notes
* 6 References

## Small exotic R4s

An exotic ℝ4 is called small if it can be smoothly embedded as an open subset of the standard ℝ4.

Small exotic ℝ4 can be constructed by starting with a non-trivial smooth 5-dimensional h-cobordism (which exists by Donaldson's proof that the h-cobordism theorem fails in this dimension) and using Freedman's theorem that the topological h-cobordism theorem holds in this dimension.

## Large exotic R4s

An exotic ℝ4 is called large if it cannot be smoothly embedded as an open subset of the standard ℝ4.

Examples of large exotic ℝ4 can be constructed using the fact that compact 4-manifolds can often be split as a topological sum (by Freedman's work), but cannot be split as a smooth sum (by Donaldson's work).

Michael Hartley Freedman and Laurence R. Taylor (1986) showed that there is a maximal exotic ℝ4, into which all other ℝ4 can be smoothly embedded as open subsets.

## Related exotic structures

Casson handles are homeomorphic to 𝔻2×ℝ2 by Freedman's theorem (where 𝔻2 is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to 𝔻2×ℝ2. In other words, some Casson handles are exotic 𝔻2×ℝ2.

It is not known (as of 2024) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincaré conjecture in dimension 4. Some plausible candidates are given by Gluck twists.

## See also

* Akbulut cork \- tool used to construct exotic ℝ4's from classes in H3(S3,ℝ)[5]
* Atlas (topology)

## Notes

1. ↑ Kirby (1989), p. 95
2. ↑ Freedman and Quinn (1990), p. 122
3. ↑ Taubes (1987), Theorem 1.1
4. ↑ Stallings (1962), in particular Corollary 5.2
5. ↑ Asselmeyer-Maluga, Torsten; Król, Jerzy (2014-08-28). "Abelian gerbes, generalized geometries and foliations of small exotic R^4". arXiv:0904.1276 [hep-th].

## References

* Freedman, Michael H.; Quinn, Frank (1990). Topology of 4-manifolds. Princeton Mathematical Series. 39. Princeton, NJ: Princeton University Press. ISBN 0-691-08577-3. https://archive.org/details/topologyof4manif0000free.
* Freedman, Michael H.; Taylor, Laurence R. (1986). "A universal smoothing of four-space". Journal of Differential Geometry 24 (1): 69–78. doi:10.4310/jdg/1214440258. ISSN 0022-040X. https://projecteuclid.org/euclid.jdg/1214440258.
* Kirby, Robion C. (1989). The topology of 4-manifolds. Lecture Notes in Mathematics. 1374. Berlin: Springer-Verlag. ISBN 3-540-51148-2.
* Scorpan, Alexandru (2005). The wild world of 4-manifolds. Providence, RI: American Mathematical Society. ISBN 978-0-8218-3749-8.
* Stallings, John (1962). "The piecewise-linear structure of Euclidean space". Mathematical Proceedings of the Cambridge Philosophical Society 58 (3): 481–488. doi:10.1017/s0305004100036756. Bibcode: 1962PCPS...58..481S. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2053736&fulltextType=RA&fileId=S0305004100036756. MR0149457
* Gompf, Robert E.; Stipsicz, András I. (1999). 4-manifolds and Kirby calculus. Graduate Studies in Mathematics. 20. Providence, RI: American Mathematical Society. ISBN 0-8218-0994-6.
* Taubes, Clifford Henry (1987). "Gauge theory on asymptotically periodic 4-manifolds". Journal of Differential Geometry 25 (3): 363–430. doi:10.4310/jdg/1214440981. Template:Project Euclid. http://projecteuclid.org/euclid.jdg/1214440981.

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