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, MSpin^c, MSpin^h MString, MFivebrane, M2-Orient, M2-Spin, MNinebrane (see also pin⁻ bordism, pin⁺ bordism, pinᶜ bordism, spin bordism, spinᶜ bordism, spinʰ bordism, string bordism, fivebrane bordism, 2-oriented bordism, 2-spin bordism, ninebrane bordism)

• MU, MSU, MΩΩSU(n)

• MP, MR

• MSp

• MTO, MTSO

• 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