3-Manifold Geometry and Topology

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Under this grant, the PI will work on problems related to surfaces in 3-manifolds. The virtual Haken conjecture asks whether any aspherical closed 3-manifold has a finite-sheeted cover which contains a closed surface which injects on the fundamental group. The PI proposes to work on this conjecture making use of recent results of Kahn-Markovic and Wise. This would have ramifications for related conjectures, such as the virtual fibering conjecture, and would greatly elucidate the structure of 3-manifolds and their invariants, such as twisted Alexander polynomials. The PI also proposes to study the Thurston norm, and the relation between various surfaces realizing the Thurston norm in a minimal homology class via sutured manifold hierarchies. He hopes to use these relations to study various questions about the Thurston norm and its relation to other invariants of the 3-manifold.The mathematical study of 3-dimensional spaces goes back to the work of Poincare. Because classical physics describes our universe to be a 3-dimensional space, the classification of 3-dimensional spaces is an important mathematical endeavor, since it may have ramifications for the global structure of our universe. This project will pursue various aspects of the classification of 3-dimensional spaces. The principal and most ambitious project will study 2-dimensional objects inside of 3-dimensional spaces, and how one may resolve these in related "covering spaces". Studying these 2-dimensional objects has implications for the global structure of finite 3-dimensional spaces. The techniques of the project will have relations to other areas of mathematics such as group theory, geometry, and the study of 4-dimensional spaces.