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.
The creator of the cobordism theory is considered to be René Thom (1954)although some crucial ideas were already anticipated by Lev Pontryagin (1950 and before).
A cobordism between two manifolds
is a manifold
for their edge
are then called unoriented cobordant.
More commonly, however, oriented cobordism is used. Two oriented manifolds
are called oriented cobordant if there is an oriented manifold
orientation, whereby the orientation towards
the orientation of
induced orientation on the edge and
with the opposite orientation.
According to a theorem of Thom, two manifolds are oriented cobordant if and only if all their Pontrjagin numbers and Stiefel-Whitney numbers agree.
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.
- John Milnor: A survey of cobordism theory. Enseignement Math. (2) 8 1962 16-23. online (pdf)
- Steimle, op.cit.
- Thom, Quelques propriétés globales des variétés différentiables, Comm.Math.Helvetici, vol. 28, 1954, pp. 17-86, digitalisat