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The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
2
votes
Accepted
Smooth (locally trivial) fibration and submanifold
Let f:R^2 → R be the projection on the first factor and let L be the closed submanifold given by {(x,y)| x>0,xy=1 } union {(x,y)| y=-1 } . I think this is a counterexample.
2
votes
Accepted
Relative equivariant Thom transversality
See Prop 2.2 of ON THE GROUPS JO(G), Chung-Nim Lee and Arthur Wasserman, Memoirs of the American Mathematical Society Number 159