Starting with an SO(2)–principal fibration over a closed
oriented surface Fg, g≥ 1, a 2–fold covering of
the total space is said to be special when the monodromy
sends the fiber SO(2)∼ S1 to the nontrivial element of
Z2. Adapting D Johnson's method [Spin
structures and quadratic forms on surfaces, J London Math Soc, 22
(1980) 365-373] we define an action of Sp(Z2,2g),
the group of symplectic isomorphisms of
(H1(Fg;Z2),.), on the set of
special 2–fold coverings which has two orbits, one with
2g-1(2g+1) elements and one with
2g-1(2g-1) elements. These two orbits are
obtained by considering Arf-invariants and some congruence of the
derived matrices coming from Fox Calculus.
Sp(Z2,2g) 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.
Keywords
coverings, spin structures, quadratic
forms, Fox calculus