genus of a closed orientable surface

亏格

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

Translation evidence: Literature usage

Meaning in context

For a connected closed orientable surface, the nonnegative integer g determined by its Euler characteristic through chi=2-2g; equivalently, the number of torus summands in its classification, with the sphere having genus zero.

对连通闭可定向曲面，由 Euler 数 χ=2−2g\chi=2-2g 确定的非负整数 gg；等价地，它是分类中环面连通和项的个数，球面的亏格为零。

Example

A connected sum of two tori has χ=−2\chi=-2, so g=(2−(−2))/2=2g=(2-(-2))/2=2. The sphere has χ=2\chi=2 and genus zero.

Usage and distinctions

The scope is closed orientable connected surfaces. Nonorientable genus counts crosscaps and obeys a different Euler formula; algebraic-curve genus is not defined by this unqualified surface formula.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§4.2, Definition 4.2.39 and Remark 4.2.40 · Checked 2026-10-14
  2. Allen Hatcher — Algebraic Topology§2.2, Example 2.36, closed orientable surfaces and their cellular homology · Checked 2026-10-14

Translation sources

  1. Bao Zhiqiang — Peking University topology syllabus第8周10月31日，主要概念：曲面的可定向性及亏格；第9周11月9日，常用结论：gT² 的 Euler 数为 2-2g · Checked 2026-10-14
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