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
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. 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.
Definition
In
singular cohomology, the cup product is a construction giving a product on the
graded cohomology ring ''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 poin ...
''X''.
The construction starts with a product of
cochains: 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 and
is the canonical
embedding 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 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
functorial, in the following sense: if
:
is a continuous function, and
:
is the induced
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
in cohomology, then
:
for all classes α, β in ''H''
*(''Y''). In other words, ''f''
* is a (graded)
ring homomorphism.
Interpretation
It is possible to view the cup product
as induced from the following composition:
in terms of the
chain complexes 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
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
cochains 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 is induced by the
wedge product. In other words, the wedge product of
two
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
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
be an oriented
smooth manifold 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 to the cup product, in the sense that taking the Poincaré pairings
then there is the following equality :
.
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.
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
*
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
Glen Eugene Bredon (August 24, 1932 in Fresno, California – May 8, 2000, in North Fork, California) was an American mathematician who worked in the area of topology.
Education and career
Bredon received a bachelor's degree from Stanford Univ ...
, "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