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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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Finiteness properties of general topological spaces
It is known that all metric compact ANR have the homotopy type of finite CW complexes. Which spaces are homotopy equivalent or finitely dominated by CW complexes of finite type?