exotic R4’s

If $n\eq 4$ then the smooth manifolds homeomorphic to a given topological $n$ - manifold, $M$ , are parameterized by some discrete algebraic invariant of $M$ . In particular there is a unique smooth manifold homeomorphic to $\\mathbb{R}^{n}$ .

By contrast one may choose uncountably many open sets in $\\mathbb{R}^{4}$ , which are all homeomorphic to $\\mathbb{R}^{4}$ , but which are pairwise non-diffeomorphic.

A smooth manifold homeomorphic to $\\mathbb{R}^{4}$ , but not diffeomorphic to it is called an exotic $\\mathbb{R}^{4}$ .

Given an exotic $\\mathbb{R}^{4}$ , $E$ , we have a diffeomorphism $E\\times\\mathbb{R}\\to\\mathbb{R}^{5}$ . (As there is only one smooth manifold homeomorphic to $\\mathbb{R}^{5}$ ). Hence exotic $\\mathbb{R}^{4}$ ’s may be identified with closed submanifolds of $\\mathbb{R}^{5}$ . In particular this means the cardinality of the set of exotic $\\mathbb{R}^{4}$ ’s is precisely continuum.

Historically, Donaldson’s theorem led to the discovery of the Donaldson Freedman exotic $\\mathbb{R}^{4}$ .

单词	ExoticR4s
释义	exotic R4’s If $n\eq 4$ then the smooth manifolds homeomorphic to a given topological $n$ - manifold, $M$ , are parameterized by some discrete algebraic invariant of $M$ . In particular there is a unique smooth manifold homeomorphic to $\\mathbb{R}^{n}$ . By contrast one may choose uncountably many open sets in $\\mathbb{R}^{4}$ , which are all homeomorphic to $\\mathbb{R}^{4}$ , but which are pairwise non-diffeomorphic. A smooth manifold homeomorphic to $\\mathbb{R}^{4}$ , but not diffeomorphic to it is called an exotic $\\mathbb{R}^{4}$ . Given an exotic $\\mathbb{R}^{4}$ , $E$ , we have a diffeomorphism $E\\times\\mathbb{R}\\to\\mathbb{R}^{5}$ . (As there is only one smooth manifold homeomorphic to $\\mathbb{R}^{5}$ ). Hence exotic $\\mathbb{R}^{4}$ ’s may be identified with closed submanifolds of $\\mathbb{R}^{5}$ . In particular this means the cardinality of the set of exotic $\\mathbb{R}^{4}$ ’s is precisely continuum. Historically, Donaldson’s theorem led to the discovery of the Donaldson Freedman exotic $\\mathbb{R}^{4}$ .
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