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Geometry-Topology Seminar

Thursday, February 23, 2023 - 5:15pm

Paul Melvin

Bryn Mawr College

Location

University of Pennsylvania to participat

DRL 4C8

The Geomeetry-Topology Seminar is running in "hybrid style" this semester, with the talks in DRL 4C8 but also Zoomed to reach a wider audience. If you would like to participate by Zoom, just e-mail gluck@math.upenn.edu for the Zoom link.

 

 

This talk will begin with a review of some standard notions of stabilization of 4-manifolds, two celebrated results of Wall from the 1960s, and Freedman's topological classification of closed, oriented simply-connected 4-manifolds from the 1980s.  We will then introduce what it means for a topological 4-manifold to be fertile – meaning (roughly) it has loads of smooth structures – and for it to be homologically fertile – meaning all its (primitive, ordinary) 2-dimensional homology classes can be represented (in some smooth structure) by loads of smoothly embedded spheres.  Finally, we'll indicate how gauge theory and symplectic topology show that many smoothable 4-manifolds are fertile, and explain how it follows that their stabilizations are homologically fertile.