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

$$


M


{displaystyle M}

and

$$


N


{displaystyle N}

is a manifold

$$


W


{displaystyle W}

for their edge

$$


∂
W


{displaystyle partial W}

valid

$$


∂
W
=
M
∪
N


{displaystyle partial W=Mcup N}

.

$$


M


{displaystyle M}

and

$$


N


{displaystyle N}

are then called unoriented cobordant.

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

$$


M


{displaystyle M}

and

$$


N


{displaystyle N}

are called oriented cobordant if there is an oriented manifold

$$


W


{displaystyle W}

with

$$


∂
W
=
M
∪


N
¯




{displaystyle partial W=Mcup {overline {N}}

orientation, whereby the orientation towards

$$


∂
W


{displaystyle partial W}

the orientation of

$$


W


{displaystyle W}

induced orientation on the edge and

$$




N
¯




{displaystyle {overline {N}}

the variety

$$


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)

Web links

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

Individual references

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