Cohomology ring

Last updated October 23, 2019

In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product 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. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

Algebraic topology branch of mathematics

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 up to homotopy equivalence.

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

In ring theory, a branch of abstract algebra, 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 noncommutative rings where multiplication is not required to be commutative.

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 graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H^∗(X), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944.

H^{k}(X;R)\times H^{\ell }(X;R)\to H^{k+\ell }(X;R).

The cup product gives a multiplication on the direct sum of the cohomology groups

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.

H^{\bullet }(X;R)=\bigoplus _{k\in \mathbb {N} }H^{k}(X;R).

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.

In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups $such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading .$

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

(\alpha ^{k}\smile \beta ^{\ell })=(-1)^{k\ell }(\beta ^{\ell }\smile \alpha ^{k}).

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.

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 a complex projective space label the complex lines through the origin of a complex Euclidean space. Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(Cⁿ⁺¹), P_n(C) or CPⁿ. When n = 1, the complex projective space CP¹ is the Riemann sphere, and when n = 2, CP² is the complex projective plane.

In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or a complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d. That is, the smooth manifold M has dimension 2d; and away from any singular point V will also be a smooth manifold of dimension 2d.

Examples

$\operatorname {H} ^{*}(\mathbb {R} P^{n};\mathbb {F} _{2})=\mathbb {F} _{2}[\alpha ]/(\alpha ^{n+1})$ where $|\alpha |=1$ .
$\operatorname {H} ^{*}(\mathbb {R} P^{\infty };\mathbb {F} _{2})=\mathbb {F} _{2}[\alpha ]$ where $|\alpha |=1$ .
By the Künneth formula, the mod 2 cohomology ring of the cartesian product of n copies of $\mathbb {R} P^{\infty }$ is a polynomial ring in n variables with coefficients in $\mathbb {F} _{2}$ .

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In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by $, is the set of all prime ideals of R . It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme .$

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number. Thus a formal power series differs from a polynomial in that it may have infinitely many terms, and differs from a power series, whose variables can take on numerical values. One way to view a formal power series is as an infinite ordered sequence of numbers. In this case, the powers of the variable are used only to indicate the order of the coefficients, so that the coefficient of $is the fifth term in the sequence. In combinatorics, formal power series provide representations of numerical sequences and of multisets, and for instance allow concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. More generally, formal power series can include series with any finite number of variables, and with coefficients in an arbitrary ring. Formal power series can be created from Taylor polynomials using formal moduli.$

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, the adele ring is defined in class field theory, a branch of algebraic number theory. It allows one to elegantly describe the Artin reciprocity law. The adele ring is a self-dual topological ring, which is built on a global field. It is the restricted product of all the completions of the global field and therefore contains all the completions of the global field.

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi-Yau manifolds, string theory, Chern-Simons theory, knot theory, Gromov-Witten invariants, topological quantum field theory, the Chern theorem etc.

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a 'motif' is the 'cohomology essence' of a variety.

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext¹ classifies extensions of one module by another.

In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor₁ and the torsion subgroup of an abelian group.

In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.

In abstract algebra, Hilbert's Theorem 90 is an important result on cyclic extensions of fields that leads to Kummer theory. In its most basic form, it states that if L/K is a cyclic extension of fields with Galois group G = Gal(L/K) generated by an element $and if is an element of L of relative norm 1, then there exists in L such that$

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices.

In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products, but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.

In algebraic geometry, the Chow groups of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map $is surjective. An element of that restricts to the canonical generator of the reduced theory is called a complex orientation . The notion is central to Quillen's work relating cohomology to formal group laws.$

References

Novikov, S. P. (1996). Topology I, General Survey. Springer-Verlag. ISBN 7-03-016673-6.
Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0 .

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Cohomology ring

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