manifolds and cobordisms
coordinate chart, atlas
orientation, spin structure, string structure, fivebrane structure
smooth manifold, Riemannian manifold, complex manifold
(∞,n)-category of cobordisms
infinite dimensional manifold
signature genus, Kervaire invariant
A-hat genus, Witten genus
Thom's transversality theorem
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A BB-bordism is a cobordism WW equipped with extra “topological structure” in the form of a lift of the classifying map W→BOW \to B O of its tangent bundle through some fibration B→BOB \to B O over the classifying space of the orthogonal group.
Commonly considered are lifts through the Whitehead tower of the orthogonal group), corresponding, in this order, to cobordisms with
spin structure - spin bordism
string structure - string bordism
fivebrane structure - fivebrane bordism?