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

LECTURES ON ZARISKI VAN-KAMPEN THEOREM 11
σ i
σ i+1
σ i
≀
homotopic
σ i+1
σ i
σ i+1
Figure 4.5. σ i σ i+1 σ i = σ i+1 σ i σ i+1
The fact that B n is generated by σ 1 , ... , σ n−1 is easy to see. The relations
actually hold can be checked easily by drawing braids. The difficult part is that
any other relations among the generators can be derived from these relations. See
[1] for the proof.
We can define an action from right of the braid group B n on the free group F n
by the following
⎧
⎪⎨ a j if j ≠ i, i +1
a σi
j := a i a i+1 a −1
i if j = i
(4.1)
⎪⎩
a i if j = i +1.
Check that this definition is compatible with the defining relation of the braid
group. In the next section, we will explain the geometric meaning of this action.
5. Monodromy on fundamental groups
We denote the conjunction of paths α : I → X and β : I → X on the topological
space X in such a way that αβ is defined if and only if α(1) = β(0).
5.1. Fundamental groups and locally trivial fiber spaces. Let p : E → B be
a locally trivial fiber space. Suppose that p : E → B has a section
s : B → E;
that is, s is a continuous map satisfying p ◦ s =id B . We choose a base point ˜b of
E and b of B in such a way that ˜b = s(b) holds. We then put
F b := p −1 (b).