Sean A Broughton

Milnor Numbers and the Topology of Polynomial Hypersurfaces

SummaryLetF: ℂn + 1→ℂ be a polynomial. The problem of determining the homology groupsHq(F−1(c)), c ∈ℂ, in terms of the critical points ofF is considered. In the “best case” it is shown, for a certain generic class of polynomials (tame polynomials), that for allc∈ℂ,F−1(c) has the homotopy type of a bouquet of μ-μcn-spheres. Here μ is the sum of all the Milnor numbers ofF at critical points ofF and μc is the corresponding sum for critical points lying onF−1(c). A “second best” case is also discussed and the homology groupsHq(F−1(c)) are calculated for genericc∈ℂ. This case gives an example in which the critical points “at infinity” ofF must be considered in order to determine the homology groupsHq(F−1(c)).

Symmetries of Accola-Maclachlan and Kulkarni surfaces

For all g 2 there is a Riemann surface of genus g whose automorphism group has order 8g+8, establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and Maclachlan established the existence of such surfaces; we shall call them Accola-Maclachlan surfaces. Later Kulkarni proved that for suciently large g the Accola-Maclachlan surface was unique for g = 0;1; 2 mod 4 and produced exactly one additional surface (the Kulkarni surface) for g = 3 mod 4. In this paper we determine the symmetries of these special surfaces, computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves. Explicit equations of these real forms of Accola-Maclachlan surfaces are given in all but one case.

Symmetries of Riemann surfaces on which PSL(2, q) acts as a Hurwitz automorphism group

Let X be a compact Riemann surface and Aut(X) be its automorphism group. An automorphism of order 2 reversing the orientation is called a symmetry. The authors together with D. Singerman have been working on symmetries of Riemann surfaces in the last decade. In this paper, the symmetry type St(X) of X is defined as an unordered list of species of conjugacy classes of symmetries of X, and for a class of particular surfaces, St(X) is found. This class consists of Riemann surfaces on which PSL(2, q) acts as a Hurwitz group. An algorithm to calculate the symmetry type of this class is provided.