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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).

7 votes

Constructible sheaves and dg-modules

This is not a helpful answer to your main question, but merely a negative answer to your "In particular..what happens.." question. But, the general idea may be helpful in figuring out what more preci …
Anatoly Preygel's user avatar