Cup product

In singular cohomology, the cup product is a construction giving a product on the graded cohomology ring H^∗(X) of a topological space X.

The construction starts with a product of cochains: if c^p is a p-cochain and d^q is a q-cochain, then

${\displaystyle (c^{p}\smile d^{q})(\sigma )=c^{p}(\sigma \circ \iota _{0,1,...p})\cdot d^{q}(\sigma \circ \iota _{p,p+1,...,p+q})}$

where σ is a singular (p + q) -simplex and ${\displaystyle \iota _{S},S\subset \{0,1,...,p+q\}}$ is the canonical embedding of the simplex spanned by S into the ${\displaystyle (p+q)}$-simplex whose vertices are indexed by ${\displaystyle \{0,...,p+q\}}$.

Informally, ${\displaystyle \sigma \circ \iota _{0,1,...,p}}$ is the p-th front face and ${\displaystyle \sigma \circ \iota _{p,p+1,...,p+q}}$ is the q-th back face of σ, respectively.

The coboundary of the cup product of cochains c^p and d^q is given by

${\displaystyle \delta (c^{p}\smile d^{q})=\delta {c^{p}}\smile d^{q}+(-1)^{p}(c^{p}\smile \delta {d^{q}}).}$

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,

${\displaystyle H^{p}(X)\times H^{q}(X)\to H^{p+q}(X).}$

The cup product operation in cohomology satisfies the identity

${\displaystyle \alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})}$

so that the corresponding multiplication is graded-commutative.

The cup product is functorial, in the following sense: if

${\displaystyle f\colon X\to Y}$

is a continuous function, and

${\displaystyle f^{*}\colon H^{*}(Y)\to H^{*}(X)}$

is the induced homomorphism in cohomology, then

${\displaystyle f^{*}(\alpha \smile \beta )=f^{*}(\alpha )\smile f^{*}(\beta ),}$

for all classes α, β in H ^*(Y). In other words, f ^* is a (graded) ring homomorphism.

It is possible to view the cup product ${\displaystyle \smile \colon H^{p}(X)\times H^{q}(X)\to H^{p+q}(X)}$ as induced from the following composition:

${\displaystyle \displaystyle C^{\bullet }(X)\times C^{\bullet }(X)\to C^{\bullet }(X\times X){\overset {\Delta ^{*}}{\to }}C^{\bullet }(X)}$

in terms of the chain complexes of ${\displaystyle X}$ and ${\displaystyle X\times X}$, where the first map is the Künneth map and the second is the map induced by the diagonal ${\displaystyle \Delta \colon X\to X\times X}$.

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: ${\displaystyle \Delta \colon X\to X\times X}$ induces a map ${\displaystyle \Delta ^{*}\colon H^{\bullet }(X\times X)\to H^{\bullet }(X)}$ but would also induce a map ${\displaystyle \Delta _{*}\colon H_{\bullet }(X)\to H_{\bullet }(X\times X)}$, 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. ${\displaystyle (u_{1}+u_{2})\smile v=u_{1}\smile v+u_{2}\smile v}$ and ${\displaystyle u\smile (v_{1}+v_{2})=u\smile v_{1}+u\smile v_{2}.}$

Cup products may be used to distinguish manifolds from wedges of spaces with identical cohomology groups. The space ${\displaystyle X:=S^{2}\vee S^{1}\vee S^{1}}$ 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 ${\displaystyle S^{1}}$ 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).

Massey products generalize cup product, allowing one to define "higher order linking numbers", the Milnor invariants.

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).

• Singular homology
• Homology theory
• Cap product
• Massey product
• Torelli group

• James R. Munkres, "Elements of Algebraic Topology", Perseus Publishing, Cambridge Massachusetts (1984) ISBN 0-201-04586-9 (hardcover) ISBN 0-201-62728-0 (paperback)
• Glen E. Bredon, "Topology and Geometry", Springer-Verlag, New York (1993) ISBN 0-387-97926-3
• Allen Hatcher, "Algebraic Topology", Cambridge Publishing Company (2002) ISBN 0-521-79540-0