3 – Coverings of a Topological Space
285
as their union the set p
−1 (D) of points of X projecting onto D. The maps
p : D k −→ D and f k : D −→ D k being continuous and mutually inverse, p is
a homeomorphism from D k onto D, so that the D k are connected by arcs like
D. Finally, D k is open in X since it is the set of (z, ζ) ∈ X satisfying z ∈ D
and ζ ∈ W k , where W k is an open neighbourhood of α k . As a result, the
D k are the connected components of p
−1 (D) and p is a local homeomorphism
from X onto B.
On the other hand, for all k, the couple (D k , p) is clearly a local chart
of X; we thus obtain an atlas for X, a priori C
0 . It is in fact holomorphic.
Indeed, let (a, α), (b, β) be two points of X and f, g be the local uniform
branches at a and b such that f (a) = α, g(b) = β. They are defined on the
discs U and V centered at a and b and map them homeomorphically onto
open neighbourhoods f (U ) and g(V ) of (a, α) and (b, β). If f (U ) ∩ g(V ) = ∅,
f and g are equal at least at one point of U ∩ V , hence on all of U ∩ V
(uniqueness of uniform branches on a connected open set), and the change of
charts taking the local chart (f (U ), p) to the local chart (g(V ), p) transforms
the coordinate p(z, ζ) = z of a point of the first chart into its coordinate
p(z, ζ) = z in the second. The change of coordinates is, therefore, the map
z → z, which is as holomorphic as possible. The conclusion that follows from
these arguments is that there a complex analytic structure on X turning X
into a Riemann surface. It is not yet compact, but it is a start.
To go from here to the existence of global uniform branches on every
simply connected open set contained in B and to complete X to obtain a
compact Riemann surface ˆ
X associated to P , it is helpful to develop some
aspects of general topology that can also be useful elsewhere.
3 – Coverings of a Topological Space
As this theory requires quite a few explanations, I will break it down into
several parts and confine myself to the essential minimum.
8 In particular, I
will not mention the notion of a fundamental group of a space as it is not
needed to construct Riemann surfaces of algebraic functions.
(i) Definition of a covering. The notion of a covering space of a topological
space (separated, i.e. satisfying Hausdorff’s axiom) generalizes the situation
encountered at the end of the previous n
◦ . Take two separated spaces X and
B and a continuous and surjective map p : X −→ B ; although this is not
always necessary, we will suppose that the “ base ” B is connected
9 and locally
8 This section follows quite closely Chapter XVI.28 in Dieudonn´ e’s El´ ements
d’analyse, which follows even more closely what N. Bourbaki has written on the
subject when it was on its agenda in the 1950s. As at the time, many homotopy
experts belonged to the group, to start with, Samuel Eilenberg and Jean-Pierre
Serre, and other people who had seriously thought about the subject, it is unlikely that a anything better could be achieved.
9 In all of this n
◦ and in the rest of this §, “ connected ” will mean arc-connected
285
as their union the set p
−1 (D) of points of X projecting onto D. The maps
p : D k −→ D and f k : D −→ D k being continuous and mutually inverse, p is
a homeomorphism from D k onto D, so that the D k are connected by arcs like
D. Finally, D k is open in X since it is the set of (z, ζ) ∈ X satisfying z ∈ D
and ζ ∈ W k , where W k is an open neighbourhood of α k . As a result, the
D k are the connected components of p
−1 (D) and p is a local homeomorphism
from X onto B.
On the other hand, for all k, the couple (D k , p) is clearly a local chart
of X; we thus obtain an atlas for X, a priori C
0 . It is in fact holomorphic.
Indeed, let (a, α), (b, β) be two points of X and f, g be the local uniform
branches at a and b such that f (a) = α, g(b) = β. They are defined on the
discs U and V centered at a and b and map them homeomorphically onto
open neighbourhoods f (U ) and g(V ) of (a, α) and (b, β). If f (U ) ∩ g(V ) = ∅,
f and g are equal at least at one point of U ∩ V , hence on all of U ∩ V
(uniqueness of uniform branches on a connected open set), and the change of
charts taking the local chart (f (U ), p) to the local chart (g(V ), p) transforms
the coordinate p(z, ζ) = z of a point of the first chart into its coordinate
p(z, ζ) = z in the second. The change of coordinates is, therefore, the map
z → z, which is as holomorphic as possible. The conclusion that follows from
these arguments is that there a complex analytic structure on X turning X
into a Riemann surface. It is not yet compact, but it is a start.
To go from here to the existence of global uniform branches on every
simply connected open set contained in B and to complete X to obtain a
compact Riemann surface ˆ
X associated to P , it is helpful to develop some
aspects of general topology that can also be useful elsewhere.
3 – Coverings of a Topological Space
As this theory requires quite a few explanations, I will break it down into
several parts and confine myself to the essential minimum.
8 In particular, I
will not mention the notion of a fundamental group of a space as it is not
needed to construct Riemann surfaces of algebraic functions.
(i) Definition of a covering. The notion of a covering space of a topological
space (separated, i.e. satisfying Hausdorff’s axiom) generalizes the situation
encountered at the end of the previous n
◦ . Take two separated spaces X and
B and a continuous and surjective map p : X −→ B ; although this is not
always necessary, we will suppose that the “ base ” B is connected
9 and locally
8 This section follows quite closely Chapter XVI.28 in Dieudonn´ e’s El´ ements
d’analyse, which follows even more closely what N. Bourbaki has written on the
subject when it was on its agenda in the 1950s. As at the time, many homotopy
experts belonged to the group, to start with, Samuel Eilenberg and Jean-Pierre
Serre, and other people who had seriously thought about the subject, it is unlikely that a anything better could be achieved.
9 In all of this n
◦ and in the rest of this §, “ connected ” will mean arc-connected