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