nLab cobordism theory -- contents

cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory

• equivariant cobordism theory

Concepts of cobordism theory

• manifold, differentiable manifold, smooth manifold

• tangential structure

• cobordism, cobordism class

  cobordism ring

• submanifold,

  normal bundle

• Pontrjagin's theorem (equivariant, twisted):

  $\phantom{\leftrightarrow}$ Cohomotopy

  $\leftrightarrow$ cobordism classes of normally framed submanifolds

• Thom's theorem:

  $\phantom{\leftrightarrow}$ homotopy classes of maps to Thom space MO

  $\leftrightarrow$ cobordism classes of normally oriented submanifolds

• vector bundle

  Thom space

  Thom isomorphism

• Thom spectrum

  Pontryagin-Thom collapse construction

• cobordism cohomology theory

  complex cobordism cohomology theory

  orientation in generalized cohomology

  genus

• (∞,n)-category of cobordisms

  cobordism hypothesis

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism