In mathematics, the oriented cobordism ring is a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

where elements are oriented

cobordism class In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dim ...

es of

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s, the multiplication is given by the Cartesian product of manifolds and the addition is given as the disjoint union of manifolds. The ring is graded by dimensions of manifolds and is denoted by :

\Omega^_* = \oplus_0^\infty \Omega^_n

where

\Omega^_n

consists of oriented cobordism classes of manifolds of dimension ''n''. One can also define an unoriented cobordism ring, denoted by

\Omega^O_*

. If ''O'' is replaced ''U'', then one gets the

complex cobordism ring In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it ...

, oriented or unoriented. In general, one writes

\Omega^B_*

for the cobordism ring of manifolds with structure ''B''. A theorem of Thom says: :

\Omega^O_n = \pi_(MO)

where ''MO'' is the

Thom spectrum In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact ...

Notes

References

External links

bordism ring in nLabThe unoriented cobordism ring
a blog post by Akhil Mathew {{topology-stub Algebraic topology