cup product

杯积

Geometry & Topology · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: 上积

Translation evidence: Literature usage

Meaning in context

The product on singular cohomology with coefficients in a commutative ring, induced by evaluating a p-cochain on the first p-face and a q-cochain on the last q-face of an ordered (p+q)-simplex and multiplying.

以交换环为系数时，在有序 $(p+q)$ 单形的前 $p$ 面与后 $q$ 面分别计算余链并相乘，由此诱导的上同调乘积。

Example

On $T^2$, degree-one generators a,b can be chosen so $a\smile b$ generates $H^2(T^2;\mathbb Z)$.

Usage and distinctions

Graded commutativity holds on cohomology; the usual cochain formula is not literally graded commutative.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic Topology§3.2, Cup Product, printed pp. 206–208 · Checked 2026-10-14

Translation sources

1. Li Xiaobin — Southwest Jiaotong University, Gromov–Witten study guide§1.2 基础拓扑学，核心内容：杯积 · Checked 2026-10-14
2. Peking University — 2022 mathematics teaching plans00137914 course plan, 第9–10周：上同调的环结构（上积）简介，PDF p. 78 (printed p. 75) · Checked 2026-10-14

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