A classification of special 2-fold coverings

Details A classification of special 2-fold coverings A classification of special 2-fold coverings

2009-04-07

arxiv.org/abs/0904.1194v1

Geom. Topol. Monogr. 14 (2008) 27-47

Mathematics

Algebraic Topology

This is the version published by Geometry & Topology Monographs on 29 April 2008

Scientific paper

10.2140/gtm.2008.14.27

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.

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