Definition:Euler-Poincaré Characteristic

Definition

Let $K$ be a simplical complex.

The Euler-Poincaré characteristic of $K$ is defined and denoted:

$\map \chi K = \ds \sum_{n \mathop \ge 0} \paren {-1}^n \alpha_n$

where $\alpha_n$ denotes the number of $n$-simplexes of $K$.

Also see

• Definition:Euler Characteristic of Surface

• Results about the Euler-Poincaré characteristic can be found here.

Source of Name

This entry was named for Leonhard Paul Euler and Henri Poincaré.

Historical Note

The concept of the Euler-Poincaré characteristic originated with Leonhard Paul Euler, who first noted that $\map \chi K = 2$ when $K$ is a regular polyhedron in Cartesian $3$-space.

His original definition was extended by Augustin Louis Cauchy in $1813$ and then Henri Poincaré in $1895$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler-Poincaré characteristic
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler-Poincaré characteristic