Exotic R4

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In mathematics, an exotic is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space 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 of 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 still remains open as of 2019). For any positive integer n other than 4, there are no exotic smooth structures on in other words, if n ≠ 4 then any smooth manifold homeomorphic to is diffeomorphic to [4]

Small exotic R4s[edit]

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

Small exotic 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[edit]

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

Examples of large exotic 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 into which all other can be smoothly embedded as open subsets.

Related exotic structures[edit]

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

It is not known (as of 2017) 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[edit]

Notes[edit]

  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

References[edit]

  • 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 0857376.
  • 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 0882829. PE 1214440981.