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 $B$.

Related facts

Corollaries

• If the base space $B$ is a space with zero Euler characteristic, then the covering space $E$ (assuming finite degree covering) is also a space with zero Euler characteristic.
• If the covering space $E$ is a space with zero Euler characteristic, then the base space $B$ (assuming finite degree covering) is also a space with zero Euler characteristic.
• For a finite degree covering, the sign of the Euler characteristic of the base space and covering space is the same.
• Classifying space of nontrivial finite group cannot have finitely generated homology: A space with Euler characteristic one (and hence, in particular, a contractible space, weakly contractible space, or acyclic space) cannot arise as a covering space of finite degree greater than one for any other space with finitely generated homology. In particular, this implies that the classifying space of any nontrivial finite group (viewed as a discrete group) cannot be a space with finitely generated homology.

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 $\mathbb {R} \mathbb {P} ^{n}$	link	$1+x^{n}$	0	sphere $S^{n}$	link	$1+x^{n}$	0
2	Even nonnegative integer $n$	real projective space $\mathbb {R} \mathbb {P} ^{n}$	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