In ^{âˆ—}(''X''), called the

^{âˆ—}(''X'') of a

^{*}(''Y''). In other words, ''f'' ^{*} is a (graded)

Algebraic Topology

, Cambridge Publishing Company (2002) {{ISBN, 0-521-79540-0 Homology theory Algebraic topology Binary operations

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 in 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 u ...

, the cup product is a method of adjoining two cocycle
In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous ...

s of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space ''X'' into a graded ring, ''H''cohomology ring 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 u ...

. The cup product was introduced in work of J. W. Alexander, Eduard ÄŒech and Hassler Whitney
Hassler Whitney (March 23, 1907 â€“ May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integratio ...

from 1935â€“1938, and, in full generality, by Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 â€“ January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.
Early life and education
He was born in Warsaw, Kingdom of Poland to a ...

in 1944.
Definition

Insingular 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 ...

, the cup product is a construction giving a product on the graded cohomology ring 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 u ...

''H''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 point ...

''X''.
The construction starts with a product of cochain
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...

s: if $\backslash alpha^p$ is a ''p''-cochain and
$\backslash beta^q$ is a ''q''-cochain, then
:$(\backslash alpha^p\; \backslash smile\; \backslash beta^q)(\backslash sigma)\; =\; \backslash alpha^p(\backslash sigma\; \backslash circ\; \backslash iota\_)\; \backslash cdot\; \backslash beta^q(\backslash sigma\; \backslash circ\; \backslash iota\_)$
where Ïƒ is a singular (''p'' + ''q'') -simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

and $\backslash iota\_S\; ,\; S\; \backslash subset\; \backslash $
is the canonical embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...

of the simplex spanned by S into the $(p+q)$-simplex whose vertices are indexed by $\backslash $.
Informally, $\backslash sigma\; \backslash circ\; \backslash iota\_$ is the ''p''-th front face and $\backslash sigma\; \backslash circ\; \backslash iota\_$ is the ''q''-th back face of Ïƒ, respectively.
The coboundary
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...

of the cup product of cochains $\backslash alpha^p$ and $\backslash beta^q$ is given by
:$\backslash delta(\backslash alpha^p\; \backslash smile\; \backslash beta^q)\; =\; \backslash delta\; \backslash smile\; \backslash beta^q\; +\; (-1)^p(\backslash alpha^p\; \backslash smile\; \backslash delta).$
The cup product of two cocycles is again a cocycle, and the product of a coboundary with a cocycle (in either order) is a coboundary. The cup product operation induces a bilinear operation on cohomology,
: $H^p(X)\; \backslash times\; H^q(X)\; \backslash to\; H^(X).$
Properties

The cup product operation in cohomology satisfies the identity :$\backslash alpha^p\; \backslash smile\; \backslash beta^q\; =\; (-1)^(\backslash beta^q\; \backslash smile\; \backslash alpha^p)$ so that the corresponding multiplication is graded-commutative. The cup product isfunctor
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 ...

ial, in the following sense: if
:$f\backslash colon\; X\backslash to\; Y$
is a continuous function, and
:$f^*\backslash colon\; H^*(Y)\backslash to\; H^*(X)$
is the induced homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sam ...

in cohomology, then
:$f^*(\backslash alpha\; \backslash smile\; \backslash beta)\; =f^*(\backslash alpha)\; \backslash smile\; f^*(\backslash beta),$
for all classes Î±, Î² in ''H'' ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preserv ...

.
Interpretation

It is possible to view the cup product $\backslash smile\; \backslash colon\; H^p(X)\; \backslash times\; H^q(X)\; \backslash to\; H^(X)$ as induced from the following composition:$\backslash displaystyle\; C^\backslash bullet(X)\; \backslash times\; C^\backslash bullet(X)\; \backslash to\; C^\backslash bullet(X\; \backslash times\; X)\; \backslash overset\; C^\backslash bullet(X)$in terms of the

chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...

es of $X$ and $X\; \backslash times\; X$, where the first map is the KÃ¼nneth map and the second is the map induced by the diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek Î´Î ...

$\backslash Delta\; \backslash colon\; X\; \backslash to\; X\; \backslash times\; X$.
This composition passes to the quotient to give a well-defined map in terms of cohomology, this is the cup product. This approach explains the existence of a cup product for cohomology but not for homology: $\backslash Delta\; \backslash colon\; X\; \backslash to\; X\; \backslash times\; X$ induces a map $\backslash Delta^*\; \backslash colon\; H^\backslash bullet(X\; \backslash times\; X)\; \backslash to\; H^\backslash bullet(X)$ but would also induce a map $\backslash Delta\_*\; \backslash colon\; H\_\backslash bullet(X)\; \backslash to\; H\_\backslash bullet(X\; \backslash times\; X)$, which goes the wrong way round to allow us to define a product. This is however of use in defining the cap product.
Bilinearity follows from this presentation of cup product, i.e. $(u\_1\; +\; u\_2)\; \backslash smile\; v\; =\; u\_1\; \backslash smile\; v\; +\; u\_2\; \backslash smile\; v$ and $u\; \backslash smile\; (v\_1\; +\; v\_2)\; =\; u\; \backslash smile\; v\_1\; +\; u\; \backslash smile\; v\_2.$
Examples

Cup products may be used to distinguish manifolds from wedges of spaces with identical cohomology groups. The space $X:=\; S^2\backslash vee\; S^1\backslash vee\; S^1$ has the same cohomology groups as the torus ''T'', but with a different cup product. In the case of ''X'' the multiplication of thecochain
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...

s associated to the copies of $S^1$ is degenerate, whereas in ''T'' multiplication in the first cohomology group can be used to decompose the torus as a 2-cell diagram, thus having product equal to Z (more generally ''M'' where this is the base module).
Other definitions

Cup product and differential forms

Inde 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 adapt ...

, the cup product of differential forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Ã‰lie Cartan. It has many applications ...

is induced by the wedge product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...

. In other words, the wedge product of
two closed differential forms belongs to the de Rham class of the cup product of the two original de Rham classes.
Cup product and geometric intersections

For oriented manifolds, there is a geometric heuristic that "the cup product is dual to intersections." Indeed, let $M$ be an orientedsmooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

of dimension $n$. If two submanifolds $A,B$ of codimension $i$ and $j$ intersect transversely, then their intersection $A\; \backslash cap\; B$ is again a submanifold of codimension $i+j$. By taking the images of the fundamental homology classes of these manifolds under inclusion, one can obtain a bilinear product on homology. This product is PoincarÃ© dual to the cup product, in the sense that taking the PoincarÃ© pairings $;\; href="/html/ALL/s/.html"\; ;"title="">$ then there is the following equality :
$;\; href="/html/ALL/s/.html"\; ;"title="">$.
Similarly, the linking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In E ...

can be defined in terms of intersections, shifting dimensions by 1, or alternatively in terms of a non-vanishing cup product on the complement of a link.
Massey products

The cup product is a binary (2-ary) operation; one can define a ternary (3-ary) and higher order operation called the Massey product, which generalizes the cup product. This is a higher order cohomology operation, which is only partly defined (only defined for some triples).See also

*Singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...

* Homology theory
* Cap product
* Massey product
* Torelli group
References

* James R. Munkres, "Elements of Algebraic Topology", Perseus Publishing, Cambridge Massachusetts (1984) (hardcover) (paperback) * Glen E. Bredon, "Topology and Geometry", Springer-Verlag, New York (1993) * Allen Hatcher,Algebraic Topology

, Cambridge Publishing Company (2002) {{ISBN, 0-521-79540-0 Homology theory Algebraic topology Binary operations