Cup product

Properties

The cup product operation in cohomology satisfies the identity

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

so that the corresponding multiplication is graded-commutative.

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

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

is a continuous function, and

${\displaystyle f^{\ast }:H^{\ast }(Y)\to H^{\ast }(X)}$

is the induced homomorphism in cohomology, then

${\displaystyle f^{\ast }(\alpha ⌣\beta )=f^{\ast }(\alpha )⌣f^{\ast }(\beta ),}$

for all classes ${\displaystyle \alpha ,\beta }$ in ${\displaystyle H^{\ast }(Y)}$. In other words, ${\displaystyle f^{\ast }}$ is a (graded) ring homomorphism.

Interpretation

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

${\displaystyle {\displaystyle C^{\bullet }(X)\times C^{\bullet }(X)\to C^{\bullet }(X\times X)\stackrel{{\mathrm{\Delta }}^{\ast }}{\to }{C}^{\bullet }(X)}}$

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 ${\displaystyle \mathrm{\Delta }:X\to X\times X}$.

This composition passes to the quotient to give a well-defined map in terms of cohomology, and this is the cup product. This approach explains the existence of a cup product for cohomology but not for homology: ${\displaystyle \mathrm{\Delta }:X\to X\times X}$ induces a map ${\displaystyle {\mathrm{\Delta }}^{\ast }:H^{\bullet }(X\times X)\to H^{\bullet }(X)}$ but would also induce a map ${\displaystyle {\mathrm{\Delta }}_{\ast }:H_{\bullet }(X)\to H_{\bullet }(X\times X)}$, 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. ${\displaystyle (u_1+u_2)⌣v=u_1⌣v+u_2⌣v}$ and ${\displaystyle u⌣(v_1+v_2)=u⌣v_1+u⌣v_2.}$

Examples

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).

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 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 ${\displaystyle M}$ be an oriented smooth manifold of dimension ${\displaystyle n}$. If two submanifolds ${\displaystyle A,B}$ of codimension ${\displaystyle i}$ and ${\displaystyle j}$ intersect transversely, then their intersection ${\displaystyle A\cap B}$ is again a submanifold of codimension ${\displaystyle i+j}$. 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 dual to the cup product, in the sense that taking the Poincaré pairings ${\displaystyle [A{]}^{\ast },[B{]}^{\ast }\in H^i,H^j}$ then there is the following equality:

${\displaystyle [A{]}^{\ast }⌣[B{]}^{\ast }=[A\cap B{]}^{\ast }\in H^{i+j}(X,\mathbb{Z})}$.

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).