cohomology ring

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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 poi ...
''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 viewed ...
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 commuta ...
serving as the ring multiplication. Here 'cohomology' is usually understood as
singular 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 viewed ...
, but the ring structure is also present in other theories such as de Rham cohomology. It is also
functorial In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
: for a
continuous mapping In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
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 commuta ...
, which takes the form :$H^k\left(X;R\right) \times H^\ell\left(X;R\right) \to H^\left(X; R\right).$ The cup product gives a multiplication on the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of the cohomology groups :$H^\bullet\left(X;R\right) = \bigoplus_ H^k\left(X; R\right).$ This multiplication turns ''H''(''X'';''R'') into a ring. In fact, it is naturally an N-
graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
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 :$\left(\alpha^k \smile \beta^\ell\right) = \left(-1\right)^\left(\beta^\ell \smile \alpha^k\right).$ 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 In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of ...
has cup-length equal to its
complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a ...
.

# Examples

*$\operatorname^*\left(\mathbbP^n; \mathbb_2\right) = \mathbb_2$
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , wh ...
(\alpha^) where $, \alpha, =1$. *$\operatorname^*\left(\mathbbP^\infty; \mathbb_2\right) = \mathbb_2$
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , wh ...
/math> where $, \alpha, =1$. *$\operatorname^*\left(\mathbbP^n; \mathbb\right) = \mathbb$
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , wh ...
(\alpha^) where $, \alpha, =2$. *$\operatorname^*\left(\mathbbP^\infty; \mathbb\right) = \mathbb$
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , wh ...
/math> where $, \alpha, =2$. *$\operatorname^*\left(\mathbbP^n; \mathbb\right) = \mathbb$
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , wh ...
(\alpha^) where $, \alpha, =4$. *$\operatorname^*\left(\mathbbP^\infty; \mathbb\right) = \mathbb$
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , wh ...
/math> where $, \alpha, =4$. *By the Künneth formula, the mod 2 cohomology ring of the cartesian product of ''n'' copies of $\mathbbP^\infty$ is a polynomial ring in ''n'' variables with coefficients in $\mathbb_2$. *The reduced cohomology ring of wedge sums is the direct product of their reduced cohomology rings. *The cohomology ring of suspensions vanishes except for the degree 0 part.

# See also

* Quantum cohomology

# References

* * {{Hatcher AT Homology theory