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Oriented Cobordism Ring
In mathematics, the oriented cobordism ring is a ring (mathematics), ring where elements are oriented cobordism classes 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 ring, 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, 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. Notes References * External links bordism ring in nLab [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
