Editing Ring (mathematics) (section)

=== Cohomology ring of a topological space ===
To any [[topological space]] {{mvar|X}} one can associate its integral [[cohomology ring]]
:<math>H^*(X,\Z ) = \bigoplus_{i=0}^{\infty} H^i(X,\Z ),</math>
a [[graded ring]]. There are also [[homology group]]s <math>H_i(X,\Z )</math> of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the [[sphere]]s and [[torus|tori]], for which the methods of [[point-set topology]] are not well-suited. [[Cohomology group]]s were later defined in terms of homology groups in a way which is roughly analogous to the dual of a [[vector space]]. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the [[universal coefficient theorem]]. However, the advantage of the cohomology groups is that there is a [[cup product|natural product]], which is analogous to the observation that one can multiply pointwise a {{mvar|k}}-[[multilinear form]] and an {{mvar|l}}-multilinear form to get a ({{math|''k'' + ''l''}})-multilinear form.

The ring structure in cohomology provides the foundation for [[characteristic class]]es of [[fiber bundle]]s, intersection theory on manifolds and [[algebraic variety|algebraic varieties]], [[Schubert calculus]] and much more.