nLab cobordism theory

Idea

The concept of cobordism sits at a subtle connection between differential topology/differential geometry and stable homotopy theory/higher category theory, this is the content of what is often called “cobordism theory”.

The insight goes back to the seminal thesis (Thom 54), which established that the Pontryagin-Thom construction exhibits the cobordism ring, whose elements are cobordism equivalence classes of manifolds, as the homotopy groups of a spectrum, whence called the Thom spectrum.

Via the Brown representability theorem, the full Thom spectrum represents a generalized cohomology theory, whence called cobordism cohomology theory. For every kind of topological structure carried by manifolds this has a variant, such as notably complex cobordism cohomology theory. These Thom spectra and their cobordism cohomology theories play a special role in stable homotopy theory, for instance for the concepts of orientation in generalized cohomology and the concept of genus.

A more recent refinement of this statement is (the proof of) the cobordism hypothesis which identified the (framed) (∞,n)-category of cobordisms as the free symmetric monoidal (∞,n)-category with duals on a single object.

For more introduction see at Introduction to Cobordism and Complex Oriented Cohomology.

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

References

Original articles:

• René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, (1954). 17-86 (doi:10.1007/BF02566923, dml:139072, pdf)

• Michael Atiyah, Thom complexes, Proc. London Math. Soc. (3), 11:291–310, 1961. 10

Textbook accounts:

• Robert Stong, Notes on cobordism theory , Princeton Univ. Press (1968) (pdf)

• Theodor Bröcker, Tammo tom Dieck, Kobordismentheorie, Lecture Notes in Mathematics 178, Springer (1970) [ISBN:9783540053415]

• Sergei Novikov, Section 4.3 in: Topology I – General survey, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (doi:10.1007/978-3-662-10579-5, pdf)

• Stanley Kochman, chapters I and IV of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Yuli Rudyak, On Thom spectra, orientability and cobordism, Springer Monographs in Mathematics, 1998 (pdf)

Lecture notes include

• Tom Weston, An introduction to cobordism theory (pdf)

• John Francis, Topology of manifolds course notes (2010) (web), Lecture 2 Cobordisms (notes by Owen Gwilliam) (pdf), Lecture 3 Thom’s theorem (notes by A. Smith) (pdf)

• Cary Malkiewich, Unoriented cobordism and $M O$, 2011 (pdf)

• Johannes Ebert, A lecture course on Cobordism Theory, 2012 (pdf)

• Introduction to Cobordism and Complex Oriented Cohomology

This one here includes the connection to the (infinity,n)-category of cobordisms

• Dan Freed, Bordism: old and new, 2013 (pdf)

For complex cobordism cohomology see there and see

• Frank Adams, part I, part II of Stable homotopy and generalized homology, Chicago Lectures in mathematics, 1974