In mathematics, specifically in algebraic topology, 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. 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, the Oseledec theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. ... 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. ... J. W. Alexander James Waddell Alexander II (September 19, 1888 – September 23, 1971) was an important topologist of the pre-WWII era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ... Wikipedia does not have an article with this exact name. ... Hassler Whitney (23 March 1907 – 10 May 1989) was an American mathematician who was one of the founders of singularity theory, PhB, Yale University, 1928; MusB, 1929; ScD (Honorary), 1947; PhD, Harvard University, under G.D. Birkhoff, 1932. ... Samuel Eilenberg (September 30, 1913-January 30, 1998) was a Polish mathematician. ...

Definition

In singular cohomology, the cup product is a construction giving a product on the graded cohomology ring H^∗(X) of a topological space X. In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... 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. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

The construction starts with a product of cochains: if c^p is a p-cochain and d^q is a q-cochain, then In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...

where σ is a (p + q) -simplex and (d₀, …, d_p) and (d_p, …, d_{p + q}) are the natural injections into σ, sometimes called the p-th front face and the q-th back face, respectively. In geometry, a simplex (plural: simplices) or n-simplex is an n-dimensional analogue of a triangle. ... In mathematics, inclusion is a partial order on sets. ...

The coboundary of the cup product of cocycles c^p and d^q is given by In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ...

Thus, the cup product operation passes to cohomology, defining a bilinear operation

Properties

The cup product operation in cohomology satisfies the identity

so that the corresponding multiplication is graded-commutative. A supercommutative algebra is a Z2 graded algebra such that for any two pure elements x,y of the algebra, yx=(-1)xyxy Equivalently, it is an algebra where the supercommutator [x,y)≡xy-(-1)|x||y|yx always vanishes. ...

The cup product is functorial, in the following sense: if Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

is a continuous function, and

is the induced homomorphism in cohomology, then In abstract algebra, a homomorphism is a structure-preserving map. ...

for all classes α, β in H ^*(Y). In other words, f ^* is a (graded) ring homomorphism. In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

Examples

As singular spaces, the 2-sphere S² with two disjoint 1-dimensional loops attached by their endpoints to the surface and the torus 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). In geometry, a torus (pl. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...

Other definitions

Cup product and differential forms

In De Rham cohomology, 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. 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. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... Hermann Günther Grassmann (April 15, 1809, Stettin – September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...

Cup product and geometric intersections

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. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Transversality in mathematics is a notion that describes how spaces can intersect; transversality can be seen as the opposite of tangency, and plays a role in general position. ...

See also

• cohomology ring
• singular homology
• homology theory
• cap product