nLab
cobordism theory

Contents

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

Concepts of cobordism theory

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

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

relative bordism theories:

equivariant bordism theory:

global equivariant bordism theory:

algebraic:

References

Original articles:

Textbook accounts:

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 February 3, 2021 at 07:47:59. See the history of this page for a list of all contributions to it.