nLab B-bordism

Idea

A $B$-bordism is a cobordism $W$ equipped with extra “topological structure” with respect to (B,f)-structure, i.e. in the form of a lift of the classifying map $W \to B O$ of its tangent bundle/stable normal bundle? through some fibration $B \to B O$ over the classifying space of the orthogonal group.

Commonly considered are lifts through the Whitehead tower of the orthogonal group), corresponding, in this order, to cobordisms with

• spin structure - spin bordism

• string structure - string bordism

• fivebrane structure - fivebrane bordism?

etc.

Examples

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

References

• Manifold Atlas, B-Bordism