Mathematical Sciences: Dynamical Systems in 3-dimensional Topology

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The study of 3-dimensional topology is often enriched by imposing extra structure on 3-manifolds. A recent construction due to Gabai shows that a 3-manifold supports many interesting dynamical systems, namely pseudo-Anosov flows. The first two parts of this project aim towards showing that dynamical systems on a 3-manifold can be used to give complete information about an important topological invariant, the Thurston norm on homology. The main work of the first part will be to show that Gabai's flows are well-behaved enough to determine completely the Thurston norm. The second part will focus on explicitly constructing these flows, so as to have an efficient algorithm for computing the norm. A bonus will be a method for efficient computation of complete homeomorphism invariants of 3-manifolds which fiber over the circle. The third and final part of this project addresses a long range goal, to investigate the conjecture that every 3-manifold which is irreducible and contains a non-separating surface has a finite cover which fibers over the circle. Dynamical systems are potentially useful for this investigation, due to work of Fried. Mosher will attempt to prove the conjecture for some simple examples which are "as close as possible" to fibering without actually fibering. If successful, this may yield suggestions for an attack on the conjecture in general.