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K-State Today

September 15, 2016

Mathematics Colloquium lecture Sept. 15

By Reta McDermott

Paul Melvin, Bryn Mawr College, will present "4-Dimensional Exoticity" as part of the Mathematics Department Colloquium Lecture series at 2:30 p.m. Thursday, Sept. 15, in 122 Cardwell Hall.

The abstract for the lecture: The existence of exotic smooth structures on high-dimensional manifolds — precluded in low dimensions by classical work of Rado and Moise — was first realized by Milnor in 1956. He noticed that certain sphere bundles over spheres were homeomorphic but not diffeomorphic to the 7-sphere, and furthermore, that these examples could be obtained from the standard 7-sphere by removing a small 7-ball and regluing it by an automorphism of its boundary. This sort of ball-twisting, which can be performed on other manifolds and in other dimensions to produce exotic smooth structures, is in some sense the source of all such exoticity... except in dimension 4.

For 4-manifolds, ball-twisting has no effect, by work of Cerf in the early 60s. But there is a variant called cork-twisting, introduced by Akbulut in 1991, in which one replaces the ball by a contractible manifold. This operation is now known to be responsible for all exotic smooth structures on simply-connected 4-manifolds.

This presentation will describe cork twisting in some detail, and explore its versatility and applications.