292
X – The Riemann Surface of an Algebraic Function
Indeed, the map e : C −→ C
∗ transforms C into a connected and simply
connected covering space of C
∗ [(i), Example 1]. A lifting of γ is a path μ
in C such that e[μ(t)] = γ(t) for all t, in other words, is a unform branch
of Log z along γ in the sense of Chap. IV, § 4, up to the factor 2πi. As γ is
closed, μ(1) = μ(0) + n for some n ∈ Z, so that μ is fixed-endpoint homotopic
to the rectilinear path
t −→ (1 − t)μ(0) + tμ(1) = (1 − t)α + tβ .
As a result, γ is fixed-endpoint homotopic to the path
t −→ e [(1 − t)α + tβ] = γ(0)e(nt) ,
i.e. to nu. The end of the proof can be left to the reader.
The corollary shows that γ is homotopic to a point in C
∗ if and only if
the variation of Log z or, equivalently, of Arg z over γ is zero. If the variation
of the argument is 2πn, then γ is homotopic to a circle centered at 0 traveled
n times.
In the previous statement, C
∗ could be replace by a pointed open disc
centered at 0, for example 0 < |z| < 1 ; it suffices to argue in the Poincare
half-plane rather than in C.
Exercise 2. Extract from the preceding arguments a direct elementary
proof of corollary 2.
(iv) Coverings of a simply connected space. We are now ready to prove
one the main results of the theory:
Theorem 3. Every covering (X, B, p) of a simply connected and locally
connected space is trivial.
It all amounts to showing the existence of a global section with given value
α ∈ X at a given point a ∈ B. To define it at an arbitrary z ∈ B, connect a
to z by a path γ and consider the lifting μ of γ with initial point α. If γ is
replaced by another path γ
connecting a to z, the terminal point μ(1) does
not change because, B being simply connected, γ and γ
are fixed-endpoint
homotopic, and hence so are their liftings. A map f : B −→ X can, therefore,
be defined without any ambiguity by setting f (z) = μ(1). So p[f (z)] = z for
all z ∈ B. For good reasons, this is very much like the construction of a
primitive of a holomorphic function on a simply connected open set.
To show that f is a section, it suffices to prove that it is continuous at
every z ∈ B. But let D be a connected open neighbourhood of z over which X
is trivial. To calculate f (z
) for z
∈ D, choose once for all a path γ connecting
a to z. Let it be followed by a path γ
connecting z to z
in D. To lift the new
path to X, lift γ onto a path μ with initial point α and terminal point f (z)
by definition, then lift γ
onto a path with initial point f (z) ; its terminal
point is f (z
). But f (z) belongs to one of the connected components D i of
p
−1 (D), hence so does the lifting of γ
since its image is a connected subset
of p
−1 (D). As a result, f (z
) is the point of D i projecting onto z
, qed.
X – The Riemann Surface of an Algebraic Function
Indeed, the map e : C −→ C
∗ transforms C into a connected and simply
connected covering space of C
∗ [(i), Example 1]. A lifting of γ is a path μ
in C such that e[μ(t)] = γ(t) for all t, in other words, is a unform branch
of Log z along γ in the sense of Chap. IV, § 4, up to the factor 2πi. As γ is
closed, μ(1) = μ(0) + n for some n ∈ Z, so that μ is fixed-endpoint homotopic
to the rectilinear path
t −→ (1 − t)μ(0) + tμ(1) = (1 − t)α + tβ .
As a result, γ is fixed-endpoint homotopic to the path
t −→ e [(1 − t)α + tβ] = γ(0)e(nt) ,
i.e. to nu. The end of the proof can be left to the reader.
The corollary shows that γ is homotopic to a point in C
∗ if and only if
the variation of Log z or, equivalently, of Arg z over γ is zero. If the variation
of the argument is 2πn, then γ is homotopic to a circle centered at 0 traveled
n times.
In the previous statement, C
∗ could be replace by a pointed open disc
centered at 0, for example 0 < |z| < 1 ; it suffices to argue in the Poincare
half-plane rather than in C.
Exercise 2. Extract from the preceding arguments a direct elementary
proof of corollary 2.
(iv) Coverings of a simply connected space. We are now ready to prove
one the main results of the theory:
Theorem 3. Every covering (X, B, p) of a simply connected and locally
connected space is trivial.
It all amounts to showing the existence of a global section with given value
α ∈ X at a given point a ∈ B. To define it at an arbitrary z ∈ B, connect a
to z by a path γ and consider the lifting μ of γ with initial point α. If γ is
replaced by another path γ
connecting a to z, the terminal point μ(1) does
not change because, B being simply connected, γ and γ
are fixed-endpoint
homotopic, and hence so are their liftings. A map f : B −→ X can, therefore,
be defined without any ambiguity by setting f (z) = μ(1). So p[f (z)] = z for
all z ∈ B. For good reasons, this is very much like the construction of a
primitive of a holomorphic function on a simply connected open set.
To show that f is a section, it suffices to prove that it is continuous at
every z ∈ B. But let D be a connected open neighbourhood of z over which X
is trivial. To calculate f (z
) for z
∈ D, choose once for all a path γ connecting
a to z. Let it be followed by a path γ
connecting z to z
in D. To lift the new
path to X, lift γ onto a path μ with initial point α and terminal point f (z)
by definition, then lift γ
onto a path with initial point f (z) ; its terminal
point is f (z
). But f (z) belongs to one of the connected components D i of
p
−1 (D), hence so does the lifting of γ
since its image is a connected subset
of p
−1 (D). As a result, f (z
) is the point of D i projecting onto z
, qed.