Geometry & Topology Monographs 14 (2008) 27–47

DOI: 10.2140/gtm.2008.14.27

arXiv: 0904.1194

Abstract

Starting with an SO(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)∼ 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(F_g;Z₂),.), 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(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

Mathematical Subject Classification

Primary: 57R15

Secondary: 53C27

References

Publication

Received: 7 June 2006
Revised: 10 2007
Accepted: 16 January 2007
Published: 29 April 2008

Authors