COMPLEX GEOMETRY Course notes

2.6 Covering spaces
Recall that an étalé space (or étale map) over X is a continuous map p : ˜X −→ X such that p is a local
homeomorphism: that is, for every x ∈ ˜X, there is an open set U in ˜X containing x such that the image
p(U) is open in X and the restriction of p to U is a homeomorphism p| U : U −→ p(U). A connected covering
space p : ˜X −→ X is a universal cover if ˜X is simply connected. The name universal cover comes from
the following important property: if the map p : ˜X −→ X is a universal cover of the space X and the map
p ′ : X ′ −→ X is any cover of the space X where the covering space X ′ is connected, then there exists a
covering map f : X ′ −→ ˜X such that p ◦ f = p ′ . Any manifold X has a universal cover ˜X with étale covering
map f : ˜X −→ X and π1 (X) acts on ˜X discretely and freely with quotient f. Moreover, there exists a
bijection
étale
π 1 (S) ⊇ H ↦→ { ˜X/H −→ X}
between the set of subgroups of π 1 (X) up to conjugation and the set of étale coverings from a connected
manifold.
G/H for H normal ⇐⇒ Galois (regular) covering.
Recall that a covering map p : ˜X −→ X is said to be Galois if for every x ∈ X and ˜x ∈ p −1 (x), the subgroup
p ∗ π 1 ( ˜X, ˜x) is normal in π 1 (X, x).
If X is a Riemann surface, then the complex charts on X lifts to any covering space.
Lemma 2.6.1. Let f : Z 1 −→ Z be a finite étale covering corresponding to H ↩→ π 1 (Z). Then there exists
a finite regular covering h : Z 2 −→ Z and a finite étale covering g : Z 2 −→ Z 1 such that f ◦ g = h.
Z 1 Z 2
étale covering
Z
∃regular
Proof: H has a finite number of conjugates in π 1 (Z), such a number equals [π 1 (Z) : N(H)] and these
intersection is then of finite index.
Corollary 2.6.1. For every n, there exists a unique étale n-sheeted covering Z −→ D ∗ and it is isomorphic
to D ∗ −→ D ∗ (z ↦→ z n ).
Example 2.6.1. If f : Z 1 −→ Z 2 is finite between Riemann surfaces, then there is a finite étale covering
where ∆ = f(suppR f ) is the branching locus.
Z 1 − f −1 (∆) −→ Z 2 − ∆
18