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 Lustig's are direct consequences of the work of Birman and Hilden (1973) on equivariant homotopies for surface homeomorphisms.It is well known that Γ₂ acts naturally on the Z₂ symplectic vector space of rank 4, H₁(M₂, Z₂). We identify this action with Lustig's 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.