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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
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Lifting homology classes to the unit tangent bundle, a la Johnson
Let $M$ be a oriented smooth closed 2-manifold, and let $\gamma$ be an oriented smooth simple closed curve in $M$.
In Spin structures and quadratic forms on surfaces, Johnson definines a standard way …