# Cobordism

In mathematics, the notion of **cobordism** (also: **bordism**) is most important in topology and its applications, as well as in topological quantum field theory. It is considered to be the most “computable” relation among manifolds, which is geometrically interesting.^{[1]}

The creator of the **cobordism theory** is considered to be René Thom (1954)^{[2]}although some crucial ideas were already anticipated by Lev Pontryagin (1950 and before).

## Definition

A *cobordism* between two manifolds

and

$$is a manifold

$$for their edge

$$valid

$$

and

$$ are then called *unoriented cobordant*.

More commonly, however, *oriented cobordism* is used. Two oriented manifolds

and

$$ are called *oriented cobordant* if there is an oriented manifold

with

orientation, whereby the orientation towards

$$the orientation of

$$induced orientation on the edge and

$$the variety

$$with the opposite orientation.

## Predictability

According to a theorem of Thom, two manifolds are oriented cobordant if and only if all their Pontrjagin numbers and Stiefel-Whitney numbers agree.

## Applications

Cobordism (oriented or unoriented) defines an equivalence relation, the equivalence classes can be conceived as a group with the disjoint union.

René Thom’s calculation (of the torsion-free part) of the (oriented) cobordism group has numerous applications in algebraic topology and beyond. From it followed directly Hirzebruch’s signature theorem, and the original proof of the Atiyah-Singer index theorem was also built on it

Within topology, the notion was fundamental to the development of surgery theory. Furthermore, the oriented cobordism groups are an example of a generalized cohomology theory.

Topological quantum field theory also builds on the notion of cobordism, see cobordism conjecture.

## Variations of terms

Different variants of the notion of cobordism are of importance, in particular framed cobordism (Pontryagin-Thom construction) and h-cobordism.

## Literature

- John Milnor:
*A survey of cobordism theory*. Enseignement Math. (2) 8 1962 16-23. online (pdf)

## Web links

- Steimle: What is cobordism?
- Anosov , Woizechowski : Bordism (Encyclopedia of Mathematics)

## Individual references

- ↑ Steimle, op.cit.
- ↑ Thom, Quelques propriétés globales des variétés différentiables, Comm.Math.Helvetici, vol. 28, 1954, pp. 17-86, digitalisat

- Differential Topology
- Algebraic Topology