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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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The topological analog of flatness?
An immediate comment is that, in the case of smooth manifolds, the notion that you suggest needs to take into account more than just the differentiable structure of $X$ and $Y$: any two smooth, connec …