Euler characteristic of a finite CW complex

Euler 数

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

Other names / variants: Euler示性数

Translation evidence: Literature usage

Meaning in context

For a finite CW complex, the integer obtained by summing the number of cells in each dimension with alternating signs, starting positive in dimension zero; it is independent of the chosen finite cell structure.

对有限 CW 复形，以零维为正号，按维数交错求和各维胞腔数所得的整数；它不依赖所选的有限胞腔结构。

Example

The torus has one zero-cell, two one-cells and one two-cell, so $\chi(T^2)=1-2+1=0$. A sphere has one zero-cell and one two-cell, giving $2$.

Usage and distinctions

This is an integer invariant, not the Euler cohomology class of a vector bundle. An infinite alternating cell count is not automatically defined; the finite hypothesis is essential here.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic TopologyChapter 0, Cell Complexes, Euler-characteristic definition; §2.2, Euler Characteristic, Theorem 2.44, printed p. 146 · Checked 2026-10-14

Translation sources

1. Bao Zhiqiang — Peking University topology syllabus第9周11月9日：Euler 数与定向；主要概念：闭曲面的 Euler 数 · Checked 2026-10-14
2. Peking University — 2022 mathematics teaching plans课程号00137914，第15–16周：Euler示性数；PDF p. 78, printed p. 75 · Checked 2026-10-14

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