Geometry and Topology Seminar
Moser's classification of Dehn surgeries on torus knots (1971) inspired a now fifty-years-old project to classify "exceptional" Dehn surgeries on knots in the three-sphere. A prominent component of this project seeks to classify which knots admit surgeries to the "simplest" non-trivial 3-manifolds--lens spaces. By combining data from Floer homology and the theory of integer lattices into the notion of a changemaker lattice, Greene (2010) solved the lens space realization problem: every lens space which may be realized as surgery on a knot in the three-sphere may be realized by a knot already known to surger to that lens space (i.e. a torus knot or a Berge knot). In this talk, we present a survey of techniques in Dehn surgery and their applications, introduce a generalization of Greene's changemaker lattices, and discuss applications to surgeries on knots in the Poincaré homology sphere.