In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic 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. There is a continuum of non-diffeomorphic differentiable structures of R4, as was shown first by Clifford Taubes.
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 still remains open as of 2014). For any positive integer n other than 4, there are no exotic smooth structures on Rn; in other words, if n ≠ 4 then any smooth manifold homeomorphic to Rn is diffeomorphic to Rn.
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 R4s 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 R4s 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 R4s 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×R2s.
It is not known (as of 2009) 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.
- Kirby (1989), p. 95
- Freedman and Quinn (1990), p. 122
- Taubes (1987), Theorem 1.1
- Stallings (1962), in particular Corollary 5.2
- Freedman, Michael H.; Quinn, Frank (1990). Topology of 4-manifolds. Princeton Mathematical Series. 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. ISSN 0022-040X. MR 857376.
- 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". Proc. Cambridge Philos. Soc. 58 (3): 481–488. doi:10.1017/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. MR 882829. PE 1214440981.