LECTURES ON ZARISKI VAN-KAMPEN THEOREM 1. Introduction ...

LECTURES ON ZARISKI VAN-KAMPEN THEOREM 13
2Ri
l 1
l n
l i
l i+1
... ...
σ i
2Ri
l 1
l n
˜li+1
... ...
˜l i
Figure 5.1. The braid monodromy
In particular, the fundamental group of a fiber is the free group generated by n
elements. We put
C ′ := p −1 (M ′ n),
and let p ′ : C ′ → M ′ n be the restriction of p to C ′ . We can construct a section of
p ′ : C ′ → M ′ n by
S ↦→ (S, 2Ri),
because, if S ∈ M ′ n, then 2Ri /∈ S. Then the monodromy action of the braid
group π 1 (M ′ n,S b )=B n on the free group π 1 (C \ S b , 2Ri) =F n is just the one
described in the previous section. Indeed, π 1 (C\S b , 2Ri) is the free group generated
by the homotopy classes of the loops l 1 ,...,l n indicated in the upper part of
Figure 5.1. By the movement of the points in S b that represents σ i ∈ B n , the i-th
and (i + 1)-st points interchange their positions by going around their mid-point
counter-clockwise, while the other points remain still. Hence the loops l i and l i+1
are dragged, and deform into the new loops ˜l i and ˜l i+1 indicated in the lower part
of Figure 5.1, while other loops does not change. The homotopy classes of loops ˜l i
and ˜l i+1 are written as a word of the homotopy classes of original loops:
Therefore we get the action (4.1).
[˜l i+1 ]=[l i ], [˜l i ]=[l i ][l i+1 ][l i ] −1 .