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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, specifically in
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
, the cup product is a method of adjoining two
cocycle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 ringIn mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring (mathematics), ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomol ...
. 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, immersion (mathematics), immersions, characteristic classes, and ...
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 Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph ...
in 1944.

# Definition

In
singular cohomology Singular may refer to: * Singular, the grammatical number In linguistics, grammatical number is a grammatical category of nouns, pronouns, adjectives, and verb agreement (linguistics), agreement that expresses count distinctions (such as "one", ...
, the cup product is a construction giving a product on the graded
cohomology ringIn mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring (mathematics), ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomol ...
''H''(''X'') of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''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 module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
s: if $\alpha^p$ is a ''p''-cochain and $\beta^q$ is a ''q''-cochain, then :$\left(\alpha^p \smile \beta^q\right)\left(\sigma\right) = \alpha^p\left(\sigma \circ \iota_\right) \cdot \beta^q\left(\sigma \circ \iota_\right)$ where σ is a Singular homology, singular (''p'' + ''q'') -simplex and $\iota_S , S \subset \$ is the canonical embedding of the simplex spanned by S into the $\left(p+q\right)$-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 of the cup product of cochains $\alpha^p$ and $\beta^q$ is given by :$\delta\left(\alpha^p \smile \beta^q\right) = \delta \smile \beta^q + \left(-1\right)^p\left(\alpha^p \smile \delta\right).$ 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, : $H^p\left(X\right) \times H^q\left(X\right) \to H^\left(X\right).$

# Properties

The cup product operation in cohomology satisfies the identity :$\alpha^p \smile \beta^q = \left(-1\right)^\left(\beta^q \smile \alpha^p\right)$ so that the corresponding multiplication is supercommutative, graded-commutative. The cup product is functorial, in the following sense: if :$f\colon X\to Y$ is a continuous function, and :$f^*\colon H^*\left(Y\right)\to H^*\left(X\right)$ is the induced homomorphism in cohomology, then :$f^*\left(\alpha \smile \beta\right) =f^*\left(\alpha\right) \smile f^*\left(\beta\right),$ 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 $\smile \colon H^p\left(X\right) \times H^q\left(X\right) \to H^\left(X\right)$ as induced from the following composition:
$\displaystyle C^\bullet\left(X\right) \times C^\bullet\left(X\right) \to C^\bullet\left(X \times X\right) \overset C^\bullet\left(X\right)$
in terms of the chain complexes of $X$ and $X \times X$, where the first map is the Künneth formula, Künneth map and the second is the map induced by the diagonal functor, diagonal $\Delta \colon X \to X \times X$. 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 $\Delta^* \colon H^\bullet\left(X \times X\right) \to H^\bullet\left(X\right)$ but would also induce a map $\Delta_* \colon H_\bullet\left(X\right) \to H_\bullet\left(X \times X\right)$, 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. $\left(u_1 + u_2\right) \smile v = u_1 \smile v + u_2 \smile v$ and $u \smile \left(v_1 + v_2\right) = u \smile v_1 + u \smile v_2.$

# 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 cochains 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, the cup product of differential forms is induced by the wedge product. In other words, the wedge product of two Closed and exact differential forms, 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 of dimension $n$. If two submanifolds $A,B$ of codimension $i$ and $j$ intersect Transversality (mathematics), 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 duality, Poincaré dual to the cup product, in the sense that taking the Poincaré pairings $\left[A\right]^*, \left[B\right]^* \in H^,H^$ then there is the following equality : $\left[A\right]^* \smile \left[B\right]^*=\left[A \cap B\right]^* \in H^\left(X, \mathbb Z\right)$. 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).