Mathematical Sciences: Laminations, Representations and the Topology of 3-Manifolds

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The goal of this project, devoted to manifolds and cell complexes, is to explore the interrelationships between the topology of 3-manifolds and the types of laminations and representations that 3-manifolds possess. The main questions to be investigated are the following: (1) Given a compact oriented laminated 3-manifold N and an irreducible 3-manifold M homotopy equivalent to N, is M homeomorphic to N? If a homeomorphism from M to itself is homotopic to the identity, is it isotopic to the identity? (2) What 3-manifolds are laminated? (3) Calculate the degenerate invariant of a knot in the 3-sphere. Given an oriented link in the 3-sphere, find an effective way to decide if it is fibred and if so complete its monodromy. Can one obtain a counterexample to the Poincare conjecture by surgery on a knot? (4) Calculate the topological type of the representation spaces of the Brieskorn homology spheres. (5) Define and calculate equivariant instanton homology. (6) Calculate the relative Floer index of two flat connections on homology spheres obtained by surgery on knots. (7) Show that a link of two components in the 4-sphere is nullhomotopic.