singular cohomology

奇异上同调群

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

Other names / variants: singular cohomology group

Translation evidence: Literature usage

Meaning in context

For an abelian coefficient group A, the cohomology of Hom(C_*(X;Z),A), with coboundary given by precomposing a cochain with the singular boundary.

给定阿贝尔系数群 AA，对奇异链复形取 Hom⁡(C∗(X;Z),A)\operatorname{Hom}(C_*(X;\mathbb Z),A)，余边界由预复合链边界给出，其上同调群即所定义的群。

Example

H1(S1;Z)≅ZH^1(S^1;\mathbb Z)\cong\mathbb Z.

Usage and distinctions

Induced maps reverse direction; with integral coefficients this is not always simply the dual of homology because Ext terms can occur.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Allen Hatcher — Algebraic Topology§3.1, Cohomology of Spaces, printed pp. 197–199 · Checked 2026-10-14

Translation sources

  1. Peking University — 2022 mathematics teaching plans00137914 course plan, 第5–6周：奇异同调群及奇异上同调群的定义，PDF p. 78 (printed p. 75) · Checked 2026-10-14
Edit source / propose a PRReport a correction

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.