Weierstrass points and Z2 homology

In a recent paper (1993), Lustig established a beautiful connection between the six Weierstrass points on a Riemann surface M2 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(π1M2) of M2 on the set of Weierstrass points of M2 and a virtual splitting of the natural homomorphism Aut(π1M2) → Out(π1M2). Our discussion in this paper begins with the observation that these two results of Lustigʹs are direct consequences of the work of Birman and Hilden (1973) on equivariant homotopies for surface homeomorphisms. well known that Γ2 acts naturally on the Z2 symplectic vector space of rank 4, H1(M2, Z2). We identify this action with Lustigʹs action by constructing a natural correspondence between pairs of distinct Weierstrass points on M2 and nonzero elements in H1(M2,Z2). In this manner, the well-known exceptional isomorphism of finite group theory, S6 ≅ Sp(4, Z2), arises from a natural isomorphism of Γ2 spaces.