In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the cohomology ring of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'' is a
ring formed from the
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
groups of ''X'' together with the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
serving as the ring multiplication. Here 'cohomology' is usually understood as
singular cohomology, but the ring structure is also present in other theories such as
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
. It is also
functorial: for a
continuous mapping of spaces one obtains a
ring homomorphism on cohomology rings, which is contravariant.
Specifically, given a sequence of cohomology groups ''H''
''k''(''X'';''R'') on ''X'' with coefficients in a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''R'' (typically ''R'' is Z
''n'', Z, Q, R, or C) one can define the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
, which takes the form
:
The cup product gives a multiplication on the
direct sum of the cohomology groups
:
This multiplication turns ''H''
•(''X'';''R'') into a ring. In fact, it is naturally an N-
graded ring with the nonnegative integer ''k'' serving as the degree. The cup product respects this grading.
The cohomology ring is
graded-commutative in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree ''k'' and ℓ; we have
:
A numerical invariant derived from the cohomology ring is the cup-length, which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example a
complex projective space has cup-length equal to its
complex dimension.
Examples
*
where
.
*