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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”.
6
votes
Cohomology of foliations and closed forms along the leaves
If $F$ is a regular foliation then we have a Lie algebroid on $M$ given as the (integrable) sub-bundle $L\subset T_M$ of vectors that are tangent to the leaves.
There is a Chevalley-Eilenberg complex …
2
votes
A quantity associated with a smooth groupoid
Here are a few more examples and thoughts.
first of all, it seems to me that it does not necessarily vanish for the particular case of a Lie group $G$, unless $G$ is assumed to be connected. Indeed …