Weierstrass points and Z 2 homology

Abstract

In a recent paper (1993), Lustig established a beautiful connection between the six Weierstrass points on a Riemann surface M 2 of genus 2 and intersection points of closed geodesics for the associated hyperbolic metric. As a consequence, he was able to construct an action of the mapping class group Out( π 1 M 2) of M 2 on the set of Weierstrass points of M 2 and a virtual splitting of the natural homomorphism Aut( π 1 M 2) → Out( π 1 M 2). Our discussion in this paper begins with the observation that these two results of Lustig's are direct consequences of the work of Birman and Hilden (1973) on equivariant homotopies for surface homeomorphisms. It is well known that Γ 2 acts naturally on the Z 2 symplectic vector space of rank 4, H 1( M 2, Z 2). We identify this action with Lustig's action by constructing a natural correspondence between pairs of distinct Weierstrass points on M 2 and nonzero elements in H 1( M 2, Z 2). In this manner, the well-known exceptional isomorphism of finite group theory, S 6 ≅ Sp(4, Z 2), arises from a natural isomorphism of Γ 2 spaces.