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 or something thereby parametrized?.