Exotic R4

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 2020). 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 R4sEdit

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 R4sEdit

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 structuresEdit

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 alsoEdit

  • Akbulut cork - tool used to construct exotic  's from classes in  [5]
  • Atlas (topology)

NotesEdit

  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 [gr-qc, physics:hep-th, physics:math-ph].

ReferencesEdit