---
tag: 8TPW8B9G48
kind: definition
name: Cap product
msc: [55N45]
defines: N7SXBQ7WK0
uses: [6N7BC70CX0, M0Q5CFJ7ZH, JX47MS1R1H, 0310REJJY8]
sources:
  - url: https://en.wikipedia.org/wiki/Cap_product
---

## Definition

Let $X$ be a topological space and let $R$ be a [[commutative ring|0310REJJY8]] with $1$.
For a singular simplex $\sigma \colon \Delta^n \to X$ with vertices $v_0, \ldots, v_n$ and a cochain $\varphi \in C^k(X; R)$ with $k \le n$, the *cap product* is the chain
$$\sigma \frown \varphi = \varphi\big(\sigma|_{[v_0, \ldots, v_k]}\big) \, \sigma|_{[v_k, \ldots, v_n]} \in C_{n-k}(X; R).$$
It extends linearly to a bilinear map $C_n(X; R) \times C^k(X; R) \to C_{n-k}(X; R)$.
It satisfies $\partial(\sigma \frown \varphi) = (-1)^k \big(\partial\sigma \frown \varphi - \sigma \frown \delta\varphi\big)$.
Hence it induces a bilinear map $\frown \colon H_n(X; R) \times H^k(X; R) \to H_{n-k}(X; R)$ between [[singular homology|M0Q5CFJ7ZH]] and [[singular cohomology|6N7BC70CX0]], also called the cap product.

## Remarks

Cap and [[cup product|JX47MS1R1H]] are related by $\psi(\sigma \frown \varphi) = (\varphi \smile \psi)(\sigma)$ for $\varphi \in C^k$ and $\psi \in C^{n-k}$.
On homology, $(c \frown \varphi) \frown \psi = c \frown (\varphi \smile \psi)$, so $H_\bullet(X; R)$ is a module over $H^\bullet(X; R)$.
It is natural in the form $f_*(c \frown f^*\varphi) = f_*(c) \frown \varphi$ for a continuous map $f$.

Example: for $k = n$, $\sigma \frown \varphi = \varphi(\sigma)\, \sigma(v_n)$ is a $0$-chain.
Its image under the augmentation $C_0(X; R) \to R$ is $\varphi(\sigma)$, so the cap product in top degree followed by augmentation is the Kronecker pairing of cohomology with homology.

Non-example: for $k > n$ the formula does not apply, since an $n$-simplex has no face of dimension $k$; the product $H_n \times H^k \to H_{n-k}$ is only defined for $k \le n$.