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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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Equivalence of knotted spheres in $S^4$

Say we have two smoothly embedded spheres $K, K' \subset S^4$ that are equivalent in the sense that there is a diffeomorphism of pairs $(S^4, K)$ and $(S^4, K')$. Does it follow that they are equivale …