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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 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''(''X''), called the
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 Čech and
Hassler Whitney Hassler Whitney (March 23, 1907 – 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 1935–1938, and, in full generality, by
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – 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

In
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 ...
, 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''(''X'') 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 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 \alpha^p is a ''p''-cochain and \beta^q is a ''q''-cochain, then :(\alpha^p \smile \beta^q)(\sigma) = \alpha^p(\sigma \circ \iota_) \cdot \beta^q(\sigma \circ \iota_) where σ 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 \iota_S , S \subset \ 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 \. Informally, \sigma \circ \iota_ is the ''p''-th front face and \sigma \circ \iota_ is the ''q''-th back face of σ, 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 \alpha^p and \beta^q is given by :\delta(\alpha^p \smile \beta^q) = \delta \smile \beta^q + (-1)^p(\alpha^p \smile \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) \times H^q(X) \to H^(X).


Properties

The cup product operation in cohomology satisfies the identity :\alpha^p \smile \beta^q = (-1)^(\beta^q \smile \alpha^p) so that the corresponding multiplication is graded-commutative. The cup product is
functor 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\colon X\to Y is a continuous function, and :f^*\colon H^*(Y)\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^*(\alpha \smile \beta) =f^*(\alpha) \smile f^*(\beta), for all classes α, β in ''H'' *(''Y''). In other words, ''f'' * is a (graded)
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 \smile \colon H^p(X) \times H^q(X) \to H^(X) as induced from the following composition:
\displaystyle C^\bullet(X) \times C^\bullet(X) \to C^\bullet(X \times X) \overset C^\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 \times X, where the first map is the Künneth 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 δ ...
\Delta \colon X \to X \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: \Delta \colon X \to X \times X induces a map \Delta^* \colon H^\bullet(X \times X) \to H^\bullet(X) but would also induce a map \Delta_* \colon H_\bullet(X) \to H_\bullet(X \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) \smile v = u_1 \smile v + u_2 \smile v and u \smile (v_1 + v_2) = u \smile v_1 + u \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\vee S^1\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 the
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 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

In
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 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 Élie 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 oriented
smooth 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 \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 Poincaré dual to the cup product, in the sense that taking the Poincaré pairings *, * \in H^,H^ then there is the following equality : * \smile *= \cap B* \in H^(X, \mathbb Z). 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