Poincaré duality

Poincaré 对偶

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Translation evidence: Literature usage

Meaning in context

For a closed oriented n-manifold with chosen integral fundamental class, cap product with that class is an isomorphism from H^k(M;Z) to H_{n-k}(M;Z).

对选定整数基本类的闭定向 $n$ 维流形，与该基本类取卡积给出 $H^k(M;\mathbb Z)$ 到 $H_{n-k}(M;\mathbb Z)$ 的同构。

Example

On an oriented closed surface, degree-one cohomology is dual by cap product to degree-one homology.

Usage and distinctions

Noncompact manifolds require compact supports or locally finite homology; manifolds with boundary require the relative Lefschetz version.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic Topology§3.3, Theorem 3.30, printed p. 241 · 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：Poincaré 对偶 · Checked 2026-10-14

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