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 surface is a space of dimension 2.
In differential geometry this means a 2-dimensional smooth manifold or something thereby parametrized?.
In complex analytic geometry this usually means a complex manifold of complex dimension 2.
Similarly and more generally, in algebraic geometry an algebraic surface is a variety of algebraic dimension 2.
analog for dimension 1: curve, algebraic curve
genus of a surface