# Euler characteristic of covering space is product of degree of covering and Euler characteristic of base

## Statement

Suppose $E$ and $B$ are both topological spaces, with $B$ a path-connected space, and $p:E \to B$ is a covering map (surjective) having finite degree. If $E$ and $B$ both have finitely generated homology, then we have the following relation:

$\chi(E) = \operatorname{deg}(p) \cdot \chi(B)$

where:

• $\chi(E)$ is the Euler characteristic (?) of $E$.
• $\operatorname{deg}(p)$ is the degree of the covering map $p$, i.e., the size of any of the fibers. The fibers all have the same size because the base space is path-connected and the map is a covering map.
• $\chi(B)$ is the Euler characteristic (?) of $B$.

## Particular cases

The Poincare polynomial of a topological space is the ordinary generating function of the Betti numbers, and evaluating this polynomial at $-1$ gives the Euler characteristic.

Degree of covering Parameter Base space Homology information Poincare polynomial Euler characteristic Covering space Homology information Poincare polynomial Euler characteristic
2 Odd nonnegative integer $n$ real projective space $\R\mathbb{P}^n$ link $1 + x^n$ 0 sphere $S^n$ link $1 + x^n$ 0
2 Even nonnegative integer $n$ real projective space $\R\mathbb{P}^n$ link $1$ 1 sphere $S^n$ link $1 + x^n$ 2
2 -- Klein bottle link $1 + x$ 0 2-torus $S^1 \times S^1$ link $1 + 2x + x^2 = (1 + x)^2$ 0
$m$ -- circle $S^1$ link $1 + x$ 0 circle $S^1$ (via $m^{th}$ power map in $\mathbb{C}$) link $1 + x$ 0