Cup product
Encyclopedia
In mathematics
, specifically in algebraic topology
, the cup product is a method of adjoining two cocycle
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
. 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∗(X) of a topological space
X.
The construction starts with a product of cochains: if cp is a p-cochain and
dq is a q-cochain, then
where σ is a (p + q) -singular simplex
and
is the canonical embedding
of the simplex spanned by S into the
-standard simplex.
Informally,
is the p-th front face and 
is the q-th back face of σ, respectively.
The coboundary of the cup product of cocycles cp and dq 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. Thus, the cup product operation passes to cohomology, defining a bilinear operation
so that the corresponding multiplication is graded-commutative.
The cup product is functor
ial, in the following sense: if
is a continuous function, and
is the induced homomorphism
in cohomology, then
for all classes α, β in H *(Y). In other words, f * is a (graded) ring homomorphism
.
as induced from the following composition:
in terms of the chain complex
es of
and 
, where the first map is the Künneth map and the second is the map induced by the diagonal 
.
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
.
Bilinearity follows from this presentation of cup product, i.e.
and 
T have identical cohomology groups in all dimensions, but the multiplication of the cup product distinguishes the associated cohomology rings. In the former case the multiplication of the cochains associated to the loops is degenerate, whereas in the latter case 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).
, the cup product of differential forms is also known as the wedge product, and in this sense is a special case of Grassmann
's exterior product.
When two submanifolds of a smooth manifold intersect transversely, their intersection is again a submanifold. By taking the fundamental homology class of these manifolds, this yields a bilinear product on homology. This product is dual to the cup product, i.e. the homology class of the intersection of two submanifolds is the Poincaré dual of the cup product of their Poincaré duals.
Similarly, the linking number
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.
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).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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.Although algebraic topology...
, the cup product is a method of adjoining two cocycle
Cocycle
A cocycle refers to any one of the following:*A closed cochain in algebraic topology is called a cocycle.*A particular type of map in an autonomous dynamical system; see Oseledec theorem....
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
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 understood as singular cohomology, but the ring structure is...
. The cup product was introduced in work of J. W. Alexander
James Waddell Alexander II
James Waddell Alexander II was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others...
, Eduard Čech
Eduard Cech
Eduard Čech was a Czech mathematician born in Stračov, Bohemia . His research interests included projective differential geometry and topology. In 1921–1922 he collaborated with Guido Fubini in Turin...
and Hassler Whitney
Hassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
from 1935–1938, and, in full generality, by Samuel Eilenberg
Samuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
in 1944.
Definition
In singular cohomology, the cup product is a construction giving a product on the graded cohomology ringCohomology 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 understood as singular cohomology, but the ring structure is...
H∗(X) of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X.
The construction starts with a product of cochains: if cp is a p-cochain and
dq is a q-cochain, then
where σ is a (p + q) -singular simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
and
is the canonical embedding
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
of the simplex spanned by S into the
-standard simplex.Informally,
is the p-th front face and is the q-th back face of σ, respectively.The coboundary of the cup product of cocycles cp and dq 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. Thus, the cup product operation passes to cohomology, defining a bilinear operation
Properties
The cup product operation in cohomology satisfies the identityso that the corresponding multiplication is graded-commutative.
The cup product is functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
ial, in the following sense: if
is a continuous function, and
is the induced homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
in cohomology, then
for all classes α, β in H *(Y). In other words, f * is a (graded) ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
.
Interpretation
It is possible to view the cup product as induced from the following composition:in terms of the chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
es of
and , where the first map is the Künneth map and the second is the map induced by the diagonal .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 productCap 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, and independently by Hassler Whitney in 1938.-Definition:Let X be a topological...
.
Bilinearity follows from this presentation of cup product, i.e.
and Examples
As singular spaces, the 2-sphere S2 with two disjoint 1-dimensional loops attached by their endpoints to the surface and the torusTorus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
T have identical cohomology groups in all dimensions, but the multiplication of the cup product distinguishes the associated cohomology rings. In the former case the multiplication of the cochains associated to the loops is degenerate, whereas in the latter case 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).
Cup product and differential forms
In de Rham cohomologyDe Rham cohomology
In mathematics, de Rham cohomology 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 adapted to computation and the concrete representation of cohomology classes...
, the cup product of differential forms is also known as the wedge product, and in this sense is a special case of Grassmann
Hermann Grassmann
Hermann Günther Grassmann was a German polymath, renowned in his day as a linguist and now also admired as a mathematician. He was also a physicist, neohumanist, general scholar, and publisher...
's exterior product.
Cup product and geometric intersections
Similarly, the linking number
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...
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
Massey product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product.-Massey triple product:...
, which generalizes the cup product. This is a higher order cohomology operation
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 cohomology operation should be a natural transformation from...
, which is only partly defined (only defined for some triples).