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

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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

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