Cobordism ring

In mathematics, the oriented cobordism ring is a ring where elements are oriented cobordism classes[1] of manifolds, 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

${\displaystyle \Omega _{*}^{SO}=\oplus _{0}^{\infty }\Omega _{n}^{SO}}$

where ${\displaystyle \Omega _{n}^{SO}}$ consists of oriented cobordism classes of manifolds of dimension n. One can also define an unoriented cobordism ring, denoted by ${\displaystyle \Omega _{*}^{O}}$. If O is replaced U, then one gets the complex cobordism ring, oriented or unoriented.

In general, one writes ${\displaystyle \Omega _{*}^{B}}$ for the cobordism ring of manifolds with structure B.

A theorem of Thom[2] says:

${\displaystyle \Omega _{n}^{O}=\pi _{n}(MO)}$

where MO is the Thom spectrum.

Notes

1. Two compact oriented manifolds M, N are oriented cobordant if there is a compact manifold with boundary such that the boundary is diffeomorphic to the disjoint union of M with the given orientation and N with the reversed orientation.
2. http://math.northwestern.edu/~jnkf/classes/mflds/3thom.pdf

References

• Milnor, John Willard; Stasheff, James D. (1974), Characteristic classes, Annals of Mathematics Studies, 76, Princeton University Press; University of Tokyo Press, ISBN 978-0-691-08122-9

External links

• http://ncatlab.org:8080/nlab/show/cobordism+ring
• https://amathew.wordpress.com/2012/05/23/the-unoriented-cobordism-ring/, a blog post by Akhil Mathew