Exotic R4

{{Short description|A smooth 4-manifold homeomorphic yet not diffeomorphic to euclidean space}}

In mathematics, an exotic $\R^4$ is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space $\R^4.$ 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.Kirby (1989), p. 95Freedman and Quinn (1990), p. 122 There is a continuum of non-diffeomorphic differentiable structures $\R^4,$ as was shown first by Clifford Taubes.Taubes (1987), Theorem 1.1

Prior to this construction, non-diffeomorphic smooth structures on spheres{{snd}}exotic spheres{{snd}}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 $\R^n;$ in other words, if n ≠ 4 then any smooth manifold homeomorphic to $\R^n$ is diffeomorphic to $\R^n.$ Stallings (1962), in particular Corollary 5.2

Small exotic R<sup>4</sup>s

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

Small exotic $\R^4$ 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.

Large exotic R<sup>4</sup>s

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

Examples of large exotic $\R^4$ 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).

{{harvs|txt| last1=Freedman | first1=Michael Hartley | last2=Taylor | first2=Laurence R. | title=A universal smoothing of four-space | url= http://projecteuclid.org/getRecord?id=euclid.jdg/1214440258 | mr=857376 | year=1986 | journal=Journal of Differential Geometry | issn=0022-040X | volume =24 | issue=1 | pages=69–78}} showed that there is a maximal exotic $\R^4,$ into which all other $\R^4$ can be smoothly embedded as open subsets.

Related exotic structures

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

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.

Notes

References

{{cite book | last1 = Freedman | first1 = Michael H. | authorlink1 = Michael Freedman | last2 = Quinn | first2 = Frank | authorlink2 = Frank Quinn (mathematician) | title = Topology of 4-manifolds | series = Princeton Mathematical Series | volume = 39 | publisher = Princeton University Press | location = Princeton, NJ | year = 1990 | isbn = 0-691-08577-3 | url-access = registration | url = https://archive.org/details/topologyof4manif0000free }}
{{cite journal| last1 = Freedman | first1 = Michael H. | author-link = Michael Freedman | last2 = Taylor | first2 = Laurence R. | title = A universal smoothing of four-space | url = https://projecteuclid.org/euclid.jdg/1214440258 | mr = 857376 | year = 1986 | journal = Journal of Differential Geometry | issn = 0022-040X | volume = 24 | issue = 1 | pages = 69–78 | doi = 10.4310/jdg/1214440258 | doi-access = free | url-access = subscription }}
{{cite book| last = Kirby | first = Robion C. | author-link = Robion Kirby | title = The topology of 4-manifolds | series = Lecture Notes in Mathematics | volume = 1374 | publisher = Springer-Verlag | location = Berlin | year = 1989 | isbn = 3-540-51148-2 }}
{{cite book| last = Scorpan | first = Alexandru | title = The wild world of 4-manifolds | publisher = American Mathematical Society | location = Providence, RI | year = 2005 | isbn = 978-0-8218-3749-8}}
{{cite journal| last = Stallings | first = John | author-link = John R. Stallings | title = The piecewise-linear structure of Euclidean space | url = http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2053736&fulltextType=RA&fileId=S0305004100036756 | journal = Mathematical Proceedings of the Cambridge Philosophical Society | volume = 58 | issue = 3 | year = 1962 | pages = 481–488 | doi=10.1017/s0305004100036756| bibcode = 1962PCPS...58..481S | s2cid = 120418488 | url-access = subscription }} {{MathSciNet|id=0149457}}
{{cite book| last1 = Gompf | first1 = Robert E. | author-link = Robert Gompf | last2 = Stipsicz | first2 = András I. | title = 4-manifolds and Kirby calculus | series = Graduate Studies in Mathematics | volume = 20 | publisher = American Mathematical Society | location = Providence, RI | year = 1999| isbn = 0-8218-0994-6}}
{{cite journal| last = Taubes | first = Clifford Henry | author-link = Clifford Henry Taubes | title = Gauge theory on asymptotically periodic 4-manifolds | url = http://projecteuclid.org/euclid.jdg/1214440981 | journal = Journal of Differential Geometry | volume = 25 | year = 1987 | issue = 3 | pages = 363–430 | doi = 10.4310/jdg/1214440981 | mr = 882829 | id = {{Project Euclid|1214440981}}| doi-access = free }}

Category:4-manifolds

Category:Differential structures