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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
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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.
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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
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  • 201