Complexes in low-dimensional topology

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).In "3-manifolds as viewed from the curve complex" John Hempel illustrated the utility of the curve complex as a tool for understanding 3-manifolds.His work introduced the techniques used in geometric group theory into the study of low-dimensional manifolds. The Gromov hyperbolicity of the curve complex, established by Howard Masur and Yair Minsky, provides deep insights into the structure of the complex and the information it encodes. The fanstastic work of Hossein Namazi, Saul Schleimer and Juan Souto deserves particular mention in this context. In recent years, the P.I. has studied several complexes, in particular the width complex (defined by the P.I.) and the Kakimizu complex. These complexes shed light on many of the P.I.'s pet problems on genus, rank and stabilization of Heegaard splittings.The P.I. pursues research in the area of low-dimensional topology, specifically 3-manifolds. The project builds on the cross-fertilization of low-dimensional topology and geometric group theory. The P.I. maintains several national and international collaborations with mathematicians in the United States, Germany and Israel. The P.I. aims to involve graduate and undergraduate students in some aspects of the project.