Exotic R⁴

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In mathematics, an exotic ${\displaystyle \mathbb {R} ^{4}}$ is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space ${\displaystyle \mathbb {R} ^{4}.}$ 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 ${\displaystyle \mathbb {R} ^{4},}$ 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 ${\displaystyle \mathbb {R} ^{n};}$ in other words, if n ≠ 4 then any smooth manifold homeomorphic to ${\displaystyle \mathbb {R} ^{n}}$ is diffeomorphic to ${\displaystyle \mathbb {R} ^{n}.}$^[4]

An exotic ${\displaystyle \mathbb {R} ^{4}}$ is called small if it can be smoothly embedded as an open subset of the standard ${\displaystyle \mathbb {R} ^{4}.}$

Small exotic ${\displaystyle \mathbb {R} ^{4}}$ 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.

An exotic ${\displaystyle \mathbb {R} ^{4}}$ is called large if it cannot be smoothly embedded as an open subset of the standard ${\displaystyle \mathbb {R} ^{4}.}$

Examples of large exotic ${\displaystyle \mathbb {R} ^{4}}$ 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 ${\displaystyle \mathbb {R} ^{4},}$ into which all other ${\displaystyle \mathbb {R} ^{4}}$ can be smoothly embedded as open subsets.

Casson handles are homeomorphic to ${\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}}$ by Freedman's theorem (where ${\displaystyle \mathbb {D} ^{2}}$ is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to ${\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}.}$ In other words, some Casson handles are exotic ${\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}.}$

It is not known (as of 2022) 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.

• Akbulut cork - tool used to construct exotic ${\displaystyle \mathbb {R} ^{4}}$'s from classes in ${\displaystyle H^{3}(S^{3},\mathbb {R} )}$^[5]
• Atlas (topology)

• Freedman, Michael H.; Quinn, Frank (1990). Topology of 4-manifolds. Princeton Mathematical Series. Vol. 39. Princeton, NJ: Princeton University Press. ISBN 0-691-08577-3.
• 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. MR 0857376.
• Kirby, Robion C. (1989). The topology of 4-manifolds. Lecture Notes in Mathematics. Vol. 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". Proc. Cambridge Philos. Soc. 58 (3): 481–488. Bibcode:1962PCPS...58..481S. doi:10.1017/s0305004100036756. S2CID 120418488. MR0149457
• Gompf, Robert E.; Stipsicz, András I. (1999). 4-manifolds and Kirby calculus. Graduate Studies in Mathematics. Vol. 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. MR 0882829. PE 1214440981.