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

LECTURES ON ZARISKI VAN-KAMPEN THEOREM 9
Figure 4.1. A braid
We put
4. Braid groups
M n := C n \ (the big diagonal) = { (z 1 ,...,z n ) ∈ C n | z i ≠ z j (i ≠ j) }.
The symmetric group S n acts on M n by interchanging the coordinates. We then
put M n := M n /S n . This space M n is the space parameterizing non-ordered sets of
distinct n points on the complex plane C (sometimes called the configuration space
of non-ordered sets of distinct n points on C). By associating to a non-ordered set
of distinct n points {α 1 ,...,α n } the coefficients a 1 ,...,a n of
z n + a 1 z n−1 + ···+ a n−1 z + a n =(z − α 1 ) ···(z − α n ),
we obtain an isomorphism from M n to the complement to the discriminant hypersurface
of monic polynomials of degree n in C n . For example, M 4 is the complement
in an affine space C 4 with the affine parameter a 1 ,a 2 ,a 3 ,a 4 to the hypersurface defined
by
−27 a 4 2 a 1 4 +18a 4 a 3 a 2 a 1 3 − 4 a 4 a 2 3 a 1 2 − 4 a 3 3 a 1 3 + a 3 2 a 2 2 a 1
2
+144 a 4 2 a 2 a 1 2 − 6 a 4 a 3 2 a 1 2 − 80 a 4 a 3 a 2 2 a 1 +16a 4 a 2 4 +18a 3 3 a 2 a 1 − 4 a 3 2 a 2
3
We put
−192 a 4 2 a 3 a 1 − 128 a 4 2 a 2 2 + 144 a 4 a 3 2 a 2 − 27 a 3 4 + 256 a 4 3 =0.
P n := π 1 (M n ), B n := π 1 (M n ),
where the base points are chosen in a suitable way. The group P n is called the pure
braid group on n strings, and the group B n is called the braid group on n strings.
By definition, we have a short exact sequence
1 −→ P n −→ B n −→ S n −→ 1,
corresponding to the Galois covering M n → M n with Galois group S n . The point
of the configuration space M n is a set of distinct n points on the complex plane
C. Hence a loop in M n is a movement of these distinct points on C, which can be
expresses by a braid as in Figure 4.1, whence the name the braid group.
The product in B n is defined by the conjunction of the braids. In particular, the
inverse is represented by the braid upside-down. For i =1,...,n− 1, let σ i be the
element of B n represented by the braid given in Figure 4.4.