Generalised Dehn twists and Torelli groups of simply-connected 4-manifolds

Daniel Galvin (University of Glasgow)

Monday 19th February 16:00-17:00 Maths 311B

Abstract

By the work of Freedman and Perron-Quinn, the topological mapping class group of a (smooth) simply-connected closed 4-manifold is entirely determined by the group of automorphisms of its intersection form, defined on its second homology.  Hence its Torelli group, the subgroup of the mapping class group consisting of homeomorphisms that induce the trivial map on homology, is trivial.  If we started with a smooth manifold, this means that all elements of the Torelli group are isotopic to diffeomorphisms, i.e. are smoothable.  However, once we allow the boundary to be non-empty, then the work of Saeki and Orson-Powell tells us that this Torelli group can be non-trivial.  In this talk, I will show that there exist simply-connected smooth 4-manifolds with boundary whose Torelli group contains non-smoothable diffeomorphisms, and show that the obvious way to produce smoothable elements of the Torelli group, by defining so-called 'generalised Dehn twists', does not always give all smoothable elements of the Torelli group.  This is joint work with Roberto Ladu.

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