A classification of special 2-fold coverings

TL;DR

This paper classifies special 2-fold coverings over surfaces using symplectic group actions, identifying two distinct orbits characterized by Arf-invariants and matrix congruences, with explicit subgroup generators.

Contribution

It introduces a novel classification scheme for special 2-fold coverings via symplectic group actions and characterizes the orbits using Arf-invariants and matrix congruences.

Findings

Two orbits of special 2-fold coverings are identified with specific sizes.

The action of Sp(Z_2,2g) on coverings is explicitly described.

Generators of subgroups fixing each orbit are explicitly constructed.

Abstract

Starting with an O(2)-principal fibration over a closed oriented surface F_g, g>=1, a 2-fold covering of the total space is said to be special when the monodromy sends the fiber SO(2) = S^1 to the nontrivial element of Z_2. Adapting D Jonhson's method [Spin structures and quadratic forms on surfaces, J London Math Soc, 22 (1980) 365-373] we define an action of Sp(Z_2,2g), the group of symplectic isomorphisms of (H_1(F_g;Z_2),.), on the set of special 2-fold coverings which has two orbits, one with 2^{g-1}(2^g+1) elements and one with 2^{g-1}(2^g-1) elements. These two orbits are obtained by considering Arf-invariants and some congruence of the derived matrices coming from Fox Calculus. Sp(Z_2,2g) is described as the union of conjugacy classes of two subgroups, each of them fixing a special 2-fold covering. Generators of these two subgroups are made explicit.