REAL HILBERTIANITY-TOTALLY REAL NUMBERS 7
{2.2)
Punctured spheres. Let a 1, ... ,
ar E
IP'1
be the branch points of (the
Galois cover)
x:
X
---
IP'1,
and set a = { a 1, ... ,
ar}.
Then
x
restricts to an
( unramified) topological covering of the punctured sphere
IP'1'-...a.
Choose a base
point ao on this punctured sphere, and consider the (topological) fundamental
group r = Ill
(IP'1'-...a,
ao), based at ao (with the composition law: 'Yl/'2 is the
path 'Yl followed by 'Y2)·
Depending on the choice of a base point
p0
E
x-
1
(a0
),
we get an epimorphism
t:
r ---
Aut(X/IP'1)
as follows. For each path 'Y representing an element
['Y]
of r,
let p1 be the endpoint of the unique lift of 'Y to X "-X-
1(a)
with initial point
p0
.
Then,
t
sends
['Y]
to the unique element o: of
Aut(X/IP'1)
with o:(Po) = p1.
{2.3)
Related equivalence classes of covers. Let 71.ab = Hr(G)ab be the
set of equivalence classes
[x]
of all Galois covers
x:
X ---
IP'1
with
r
branch points
and with
Aut(X/IP'1)
~G.
Let 71.in =
Hr(G)in
be the set of equivalence classes
[x,
h] of pairs
(x,
h) where
x:
X ---
IP'1
is a Galois cover with
r
branch points, and
h:
Aut(X/IP'1)
---
G
is an isomorphism. Here
(x, h)
and
(x': X'
--- IP'\
h')
are
equivalent if there is an isomorphism
8: X ---X'
with
x'
o
8
=
x
and
h'
o
8.
=
h,
where
8.:
Aut(X/IP'1)
--- Aut(
X'
/1P'1)
is the isomorphism o:
~
8
o o: o
8-1.
Let
A: 11.in--- 71.ab be the map sending
[x,
h] to
[x].
{2.4)
G-covers. Think of points of 11.in as equivalence classes [a,
a0
,
f]
of
triples (a, a0
,
f).
Here a= { a 1, ... , ar} is a set of
r
points of
IP'1,
and a0 E
IP'1'-...a,
and
f:
r = II
1
(IP'1'-...a,
a0
)
--- G is an epimorphism that does not factor through
the canonical map r---
II1((JP1,a)
U
{ai},a0
),
for any i. (The latter condition
means that the corresponding cover
x
has exactly r branch points). Call two
such triples (a,
a0
,
f)
and
(a,
ii0
, /)
equivalent if a=
a
and there is a path
w
from
ao to ii0 in
IP'1'-...a
such that /ow*=
f.
Here
w*:Ill(IP'1'-...a,ao)--- Ill(IP'1'-...a,iio)
is the isomorphism 'Y
~ w- 1')'w.
(2.5)
Covers versus cycle descriptions. Here is the correspondence be-
tween the above pairs and triples
[FVl, §1.2].
Given
[x,
h] E 11.in, with
x:
X ---
IP'1
as above, let a be the set of branch points of
x,
and choose
a0
E
IP'1'-...a
andpo E
x-
1
(ao). Set r =
II1(1P'1'-...a,ao)
as above, and define J:r--- Gas
f
=
hot,
where
t:
r---
Aut(X/IP'1)
is the map from
(2.2).
Recall that
t
is canon-
ical up to composition with inner automorphisms of
Aut(X/IP'1).
Thus
h
and
f
determine each other up to inner automorphisms of
G.
This is compatible with
the equivalence of pairs (resp., triples).
(2.6)
The topology on H.in. To specify a neighborhood
N
of the point
[a,a0 ,J] ofH.in, where a= {a1, ...
,ar},
choose pairwise disjoint open discs
D1, ... , Dr around a1, ... , an with ao
tJ.
D1 U · · · U Dr. Then
N
consists of
all points
[a,
ao, /] such that
a
has exactly one point in each Di, and
j
is the
composition of the canonical isomorphisms
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