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).
Previous Page Next Page