Braids of algebraic functions and the cohomology of swallowtails

Abstract

There is an interesting connection between the theory of algebraic functions and Artin's braid theory: the space G n of nth-degree polynomials not having multiple roots is the space K(π, 1) for the group B(n) of braids on n strands: π 1 (G n) = B(n), π i (G n) = 0 for i > 1 . (1) This connection can be used in both directions: both for the study of braid groups and for the study of algebraic functions. Here are some examples. A) Along with the monodromy group, which describes the rearrangements of the leaves of a function when going round its ramification locus, there is a finer invariant of an algebraic function, namely, the braid group of the function. This group takes into account not only the rearrangement of the function values after going round the ramification locus, but also how they go round each other in the plane of function values. The monodromy group is a representation of the fundamental group of the complement of the ramification manifold in the permutation group. The braid group Artin braid group. B) The space G n can be regarded as the space of hyperelliptic curves of degree n. On the one hand, one can derive from this remark the representation of the braid group in the group of symplectic integer-valued matrices (namely, matrices of auto-morphisms of the homology of a curve induced by contours in the coefficient space). It can be shown that this representation is a representation on the entire symplectic group in the cases n = 3, 4, 6 and only in those cases. On the other hand, we obtain information on the branching of hyperelliptic inte-grals as functions of the paramters: relations between the Picard-Lefschetz matrices follow from the relations between the generators of the braid group. C) The space G n can be regarded as the set of regular values of the map Σ 1n . Thus, the relation (1) and the theorems stated below provide us with information on the topology of the simplest singularities of complex analytic maps.