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4-manifolds and their boundaries

Banff International Research Station for Mathematical Innovation and Discovery

I plan to discuss two results related to 4-manifolds with boundary. The first, joint with Dave Auckly, Hee Jung Kim, and Paul Melvin, is a construction of diffeomorphisms of finite order on the boundary of certain contractible manifolds that change their smooth structure relative to the boundary. Tange has recently announced a similar result. We show in fact that for any finite group G acting on the 3-sphere, there is a G-action on the boundary of a contractible manifold, such that every element changes the smooth structure relative to the boundary. Our construction initially produces reducible boundaries, and then we show how to make these hyperbolic. The second set of results, joint with Arunima Ray, is concerned with two analogues of Dehn's lemma for 4-manifolds. We give examples of a reducible 3-manifold Y bounding a 4-manifold W that does not split smoothly as a boundary-connected sum, even though the reducing sphere in Y is null-homotopic in W. By a different construction, we find a contractible 4-manifold W with boundary a 3-manifold Y containing an essential torus that doesn't bound (smoothly, in one version; topologically in another version) a solid torus in W.

Mathematics; Manifolds and cell complexes; Differential geometry; Low dimensional topology

video/mp4

Author affiliation: Brandeis University

2016-08-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58989

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/