manifolds and cobordisms
manifold
coordinate chart, atlas
orientation, spin structure, string structure, fivebrane structure
smooth manifold, Riemannian manifold, complex manifold
cobordism
B-bordism
extended cobordism
cobordism category
(∞,n)-category of cobordisms
Thom spectrum
cobordism ring
genus
infinite dimensional manifold
signature genus, Kervaire invariant
A-hat genus, Witten genus
Thom's transversality theorem
Pontrjagin-Thom construction
Galatius-Tillmann-Madsen-Weiss theorem
geometrization conjecture,
Poincaré conjecture
elliptization conjecture
cobordism hypothesis-theorem
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A surface is a space of dimension 2.
In differential geometry this means a 2-dimensional smooth manifold.