nLab
cobordism theory

Context

Manifolds and cobordisms

Stable Homotopy theory

Contents

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

References

Original articles include

  • René Thom, Quelques propriétés globales des variétés différentiables Comment. Math. Helv. 28, (1954). 17-86

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

Textbook accounts include

Lecture notes include

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

For complex cobordism cohomology see there and see

Last revised on April 22, 2017 at 04:17:42. See the history of this page for a list of all contributions to it.