Euler characteristic

Builds on Betti number Singular homology CW complex Euler's polyhedron formula

Definition

Let $X$ be a topological space whose singular homology groups $H_k(X)$ are finitely generated for all $k$ and zero for all but finitely many $k$ . The Euler characteristic of $X$ is $\chi(X) = \sum_{k \ge 0} (-1)^k b_k(X),$ where $b_k(X)$ is the $k$ -th Betti number of $X$ .

Remarks

If $X$ is a finite CW complex with $c_k$ cells of dimension $k$ , then $\chi(X) = \sum_k (-1)^k c_k$ . For the boundary of a convex polyhedron this is Euler's polyhedron formula $V - E + F = 2$ . For any field $F$ , $\chi(X) = \sum_k (-1)^k \dim_F H_k(X; F)$ under the same hypothesis.

Example: the torus has $b_0 = 1$ , $b_1 = 2$ , $b_2 = 1$ , so $\chi(T^2) = 1 - 2 + 1 = 0$ . Its usual CW structure has one $0$ -cell, two $1$ -cells and one $2$ -cell, and $1 - 2 + 1 = 0$ again.

Non-example: the wedge of countably many circles has $H_1$ free abelian of infinite rank, so it fails the finiteness hypothesis and has no Euler characteristic in this sense.

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Tag	`SYP3FVM71M`
URI	`m2://SYP3FVM71M`
Kind	definition
Writing	W0 stub
MSC	55N10, 57N65
Defines	Euler characteristic
Uses	Betti number, Singular homology, CW complex, Euler's polyhedron formula
Changed	2 November 2026

Euler characteristic

Definition

Remarks

Reading path

Further reading

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