If then the smooth manifolds homeomorphic to a given topological - manifold, , are parameterized by some discrete algebraic invariant of . In particular there is a unique smooth manifold homeomorphic to .
By contrast one may choose uncountably many open sets in , which are all homeomorphic to , but which are pairwise non-diffeomorphic.
A smooth manifold homeomorphic to , but not diffeomorphic to it is called an exotic .
Given an exotic , , we have a diffeomorphism . (As there is only one smooth manifold homeomorphic to ). Hence exotic ’s may be identified with closed submanifolds of . In particular this means the cardinality of the set of exotic ’s is precisely continuum.
Historically, Donaldson’s theorem led to the discovery of the Donaldson Freedman exotic .
|Date of creation||2013-03-22 15:37:33|
|Last modified on||2013-03-22 15:37:33|
|Last modified by||whm22 (2009)|