14

MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN

base point b

= {

b1, . . . ,

br}

in

Ur

by

b1

=

1, ... ,

be

=

e, and

be+j

=

-3 +(2m+ 1- 2j)H,

for j

=

e, ... ,

2m.

Next, fix generators of

II1(lP'1"'-.b,

0). For each 1

:S

j

:S

r

let D1 be the disc of

diameter

!

around b1 (so that D 1, ... , Dr are disjoint). Define loops /'l, ... , /'r

in the complex plane with the initial and final point 0 in the following way:

(1) 1'1

=

{31,

')'2 = f3! 1f32, ... , "fe = f3;.!

1

f3e,

where {3j is the circle in the

counterclockwise direction with diameter [0, b1

+

!

J

on the real axis;

(2) for e j

:S

r the path

/'j

goes on a straight line from 0 towards b1

,

then travels on a circle of diameter

!

p

1 in the counterclockwise

direction around b1

,

and returns on a straight line to 0.

'Ye+m-1

l'e+m

'Ye+m+l

/'r

These loops are homotopic to the loops those from

(2.12).

Therefore they

represent generators of the fundamental group

II1(IP'1"nDj,

0), subject only

to the relation ')'1

· · ·

'Yr

=

1. If

a

is an r-tuple with

Ia

n

Dj

I

=

1 for j

=

1, ...

,r,

then /'1, ...

,f'r

also represent generators of

r =

Ill(IP'1"'-.a,O).

In-

deed,

II1(IP'1"nDj,

0)

~

r

via the inclusion

IP'1"nDj

____.

IP'1"'-.a.

Furthermore,

for such a, we may use

/'l, . . . ,

'Yr also to represent free generators of

f'

=

III(IP'1"'-(a

U {oo}),O). The canonical epimorphism

-X.:f' ____. r

induced by the

inclusion

-X:

IP'1 "(aU { oo}) ____. IP'1 "'-.a maps the class of

/'j

in

f'

onto the class of

/'j

in

r.

Using b and

/'I, ...

''Yr. define the Hurwitz spaces

1i

and

it.

and the maps

A, W,

and~

as in

(2.13).