LECTURES ON ZARISKI VAN-KAMPEN THEOREM 1. Introduction ...
LECTURES ON ZARISKI VAN-KAMPEN THEOREM 1. Introduction ...
LECTURES ON ZARISKI VAN-KAMPEN THEOREM 1. Introduction ...
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❝<br />
<strong>LECTURES</strong> <strong>ON</strong> <strong>ZARISKI</strong> <strong>VAN</strong>-<strong>KAMPEN</strong> <strong>THEOREM</strong> 15<br />
l 2<br />
❝<br />
❝<br />
l 1<br />
<br />
l<br />
α<br />
0<br />
❝<br />
l 3<br />
l 4<br />
❝<br />
Figure 5.2. The generators of π 1 (F b , ˜b)<br />
<br />
<br />
Figure 5.3. A lasso around a deleted point<br />
Hence the monodromy action of π 1 (∆ × 2ε ,b)onπ 1(F b , ˜b) is given by<br />
l γ i = l i+p.<br />
We will return to this example when we calculate the local fundamental group of<br />
the curve singularity C.<br />
5.4. Semi-direct product. In order to use the monodromy action in the calculation<br />
of the fundamental group of the total space, we need the concept of semi-direct<br />
product of groups. Hence let us recall briefly the definition.<br />
Suppose that a group H acts on a group N from right. We denote this action<br />
by<br />
n ↦→ n h (n ∈ N,h ∈ H)<br />
We can define a product on the set N × H by<br />
(n 1 ,h 1 )(n 2 ,h 2 )=(n 1 n (h−1 1 )<br />
2 ,h 1 h 2 ).
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