Cup product

Builds on Singular cohomology Commutative ring

Definition

Let $X$ be a topological space and let $R$ be a commutative ring with $1$ . For a singular simplex $\sigma \colon \Delta^{n} \to X$ with vertices $v_0, \ldots, v_n$ , write $\sigma|_{[v_i, \ldots, v_j]}$ for its restriction to the face spanned by $v_i, \ldots, v_j$ . For cochains $\varphi \in C^k(X; R)$ and $\psi \in C^l(X; R)$ , the cup product $\varphi \smile \psi \in C^{k+l}(X; R)$ is $(\varphi \smile \psi)(\sigma) = \varphi\big(\sigma|_{[v_0, \ldots, v_k]}\big) \cdot \psi\big(\sigma|_{[v_k, \ldots, v_{k+l}]}\big)$ for every singular simplex $\sigma \colon \Delta^{k+l} \to X$ . It satisfies $\delta(\varphi \smile \psi) = \delta\varphi \smile \psi + (-1)^k \varphi \smile \delta\psi$ . Hence it induces a bilinear map $\smile \colon H^k(X; R) \times H^l(X; R) \to H^{k+l}(X; R)$ on singular cohomology, also called the cup product.

Remarks

The cup product on cohomology is associative, the class of the constant $0$ -cochain $1$ is a unit, and $a \smile b = (-1)^{kl}\, b \smile a$ for $a \in H^k$ and $b \in H^l$ . Graded commutativity holds on cohomology but not on cochains. It is natural: $f^*(a \smile b) = f^*(a) \smile f^*(b)$ for a continuous map $f$ . The same formula gives relative products such as $H^k(X; R) \times H^l(X, A; R) \to H^{k+l}(X, A; R)$ .

Example: for the torus $T^2 = S^1 \times S^1$ , $H^1(T^2; \Z) \cong \Z^2$ with generators $\alpha$ and $\beta$ dual to the two circles. Then $\alpha \smile \beta$ generates $H^2(T^2; \Z) \cong \Z$ , and $\alpha \smile \alpha = 0$ .

Non-example: the pointwise product $(\varphi\psi)(\sigma) = \varphi(\sigma)\psi(\sigma)$ only makes sense when $\varphi$ , $\psi$ and $\sigma$ all have the same dimension. So it does not define a product $C^k \times C^l \to C^{k+l}$ . The cup product avoids this by splitting each simplex into a front face and a back face.

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Commutative ring concept
Singular cohomology concept
Cup product definition

Tag	`W10ERK1ADE`
URI	`m2://W10ERK1ADE`
Kind	definition
Writing	W0 stub
MSC	55N45
Defines	Cup product
Uses	Singular cohomology, Commutative ring
Changed	2 November 2026

Cup product

Definition

Remarks

Reading path

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