Article

# 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]

${displaystyle W}$

W

{displaystyle W}

is a cobordism between

${displaystyle M}$

M

{displaystyle M}

and

${displaystyle N}$

N

{displaystyle N}

.

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

${displaystyle M}$

M

{displaystyle M}

and

${displaystyle N}$

N

{displaystyle N}

is a manifold

${displaystyle W}$

W

{displaystyle W}

for their edge

${displaystyle partial W}$

W

{displaystyle partial W}

valid

The circle is oriented cobordant to the union of two circles.

${displaystyle M}$

M

{displaystyle M}

and

${displaystyle N}$

N

{displaystyle N}

are then called unoriented cobordant.

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

${displaystyle M}$

M

{displaystyle M}

and

${displaystyle N}$

N

{displaystyle N}

are called oriented cobordant if there is an oriented manifold

${displaystyle W}$

W

{displaystyle W}

with

orientation, whereby the orientation towards

${displaystyle partial W}$

W

{displaystyle partial W}

the orientation of

${displaystyle W}$

W

{displaystyle W}

induced orientation on the edge and

${displaystyle {overline {N}}}$

N
¯

{displaystyle {overline {N}}

the variety

${displaystyle N}$

N

{displaystyle N}

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)