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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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"Bad" valid edge contractions
you can take a triangle, that is, three vertices connected by three edges. If you contract one of the edges, then you would obtain two vertices, connected by two edges, which is not allowed in simplic …