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 ...
, 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 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 und ...
. The cup product was introduced in work of
J. W. Alexander,
Eduard ÄŒech
Eduard Čech (; 29 June 1893 – 15 March 1960) was a Czech mathematician. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topo ...
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 integration t ...
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 ...
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 und ...
''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 points ...
''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 th ...
s: if
is a ''p''-cochain and
is a ''q''-cochain, then
:
where σ is a
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
(''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
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 gi ...
of the simplex spanned by S into the
-simplex whose vertices are indexed by
.
Informally,
is the ''p''-th front face and
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 th ...
of the cup product of cochains
and
is given by
:
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,
:
Properties
The cup product operation in cohomology satisfies the identity
:
so that the corresponding multiplication is
graded-commutative.
The cup product is
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
ial, in the following sense: if
:
is a continuous function, and
:
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 "same" ...
in cohomology, then
:
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 preservi ...
.
Interpretation
It is possible to view the cup product
as induced from the following composition:
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 t ...
es of
and
, 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 δΠ...
.
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:
induces a map
but would also induce a map
, which goes the wrong way round to allow us to define a product. This is however of use in defining the
cap product
In algebraic topology the cap product is a method of adjoining a chain of degree ''p'' with a cochain of degree ''q'', such that ''q'' ≤ ''p'', to form a composite chain of degree ''p'' − ''q''. It was introduced by Eduard Čech in 1936, ...
.
Bilinearity follows from this presentation of cup product, i.e.
and
Examples
Cup products may be used to distinguish manifolds from wedges of spaces with identical cohomology groups. The space
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 th ...
s associated to the copies of
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, 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 convert ...
. 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
![Linking Number 1](https://upload.wikimedia.org/wikipedia/commons/5/56/Linking_Number_1.svg)
For oriented manifolds, there is a geometric heuristic that "the cup product is dual to intersections."
Indeed, let
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
. If two submanifolds
of codimension
and
intersect
transversely, then their intersection
is again a submanifold of codimension
. 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
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* L ...
to the cup product, in the sense that taking the Poincaré pairings
then there is the following equality :
.
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
![BorromeanRings](https://upload.wikimedia.org/wikipedia/commons/c/c2/BorromeanRings.svg)
The cup product is a binary (2-ary) operation; one can define a ternary (3-ary) and higher order operation called the
Massey product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist.
Massey triple product
Le ...
, which generalizes the cup product. This is a higher order
cohomology operation In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if ''F'' is a functor defining a cohomology theory, then a coho ...
, 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
In algebraic topology the cap product is a method of adjoining a chain of degree ''p'' with a cochain of degree ''q'', such that ''q'' ≤ ''p'', to form a composite chain of degree ''p'' − ''q''. It was introduced by Eduard Čech in 1936, ...
*
Massey product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist.
Massey triple product
Le ...
*
Torelli group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mot ...
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