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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).

4 votes

What should be taught in a 1st course on smooth manifolds?

This is in agreement with Igor's comment on Anton's answer, but became too long. I'd say whatever approach you ultimately take, for a first-year grad course it surely has to be done 'properly', i.e. …
Spencer's user avatar
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2 votes
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When is the group of homeomorphisms of a compact space locally compact?

When is the group of homeomorphisms of a compact space locally compact? I am interested in finding out when the group of homeomorphisms of a compact topological space $X$ (with appropriate topo …