manifolds and cobordisms

cobordism theory, Introduction

Definitions

• locally Euclidean space

  • coordinate chart, coordinate transformation

  • atlas,

  • smooth structure

• manifold

  • topological manifold

  • differentiable manifold, ,smooth manifold

• infinite dimensional manifold

  • Banach manifold, Hilbert manifold, ILH manifold, Frechet manifold, convenient manifold

• tangent bundle

• normal bundle

• G-structure, torsion of a G-structure

  • orientation, spin structure, string structure, fivebrane structure

• Cartan geometry:

  • Riemannian manifold

  • complex manifold

  • symplectic manifold

• cobordism

  • B-bordism

  • extended cobordism

  • cobordism category

  • (∞,n)-category of cobordisms

    • functorial quantum field theory

  • Thom spectrum

    • cobordism ring

    • genus

Genera and invariants

• signature genus, Kervaire invariant

• A-hat genus, Witten genus

Classification

• 2-manifolds/surfaces

  • genus of a surface

• 3-manifolds

  • Kirby calculus

• 4-manifolds

  • Dehn surgery

  • exotic smooth structure

Theorems

• Whitney embedding theorem

• Thom's transversality theorem

• Pontrjagin-Thom construction

• Galatius-Tillmann-Madsen-Weiss theorem

• geometrization conjecture,

  • Poincaré conjecture

  • elliptization conjecture

• cobordism hypothesis-theorem