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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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Action of the permutation group on the set of topologies on a continuum

Let $X$ be a set of continuum cardinality. The group of permutations of $X$ acts on the set of topologies on $X$. What can be said about the fixed points of this action? Are manifolds fixed points? Wh …
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