exotic R4’s

If n4 then the smooth manifoldsMathworldPlanetmath 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 n.

By contrast one may choose uncountably many open sets in 4, which are all homeomorphic to 4, but which are pairwise non-diffeomorphic.

A smooth manifold homeomorphic to 4, but not diffeomorphic to it is called an exotic 4.

Given an exotic 4, E, we have a diffeomorphism E×5. (As there is only one smooth manifold homeomorphic to 5). Hence exotic 4’s may be identified with closed submanifoldsMathworldPlanetmath of 5. In particular this means the cardinality of the set of exotic 4’s is precisely continuumPlanetmathPlanetmath.

Historically, Donaldson’s theorem led to the discovery of the Donaldson Freedman exotic 4.

Title exotic R4’s
Canonical name ExoticR4s
Date of creation 2013-03-22 15:37:33
Last modified on 2013-03-22 15:37:33
Owner whm22 (2009)
Last modified by whm22 (2009)
Numerical id 21
Author whm22 (2009)
Entry type Definition
Classification msc 57R12
Classification msc 14J80
Related topic DonaldsonsTheorem
Related topic DonaldsonFreedmanExoticR4