Classifications of special double-coverings associated to a non-orientable surface

This paper investigates some possible actions ”à la Johnson” on the set with 2 elements, denoted by E , of Spinstructures which are interpreted as special Z2-coverings of a trivial S−fibration over a non-orientable surface Ng+1. The group acting is first a group of orthogonal isomorphisms of (H1(Ng+1;Z2), ·). A second approach is to consider the subspace of E (with 2 elements) coming from special Z2-coverings of S 1 × Fg, where Fg −→ Ng+1 is the orientation covering. The group acting now is a subgroup of the symplectic group of isomorphisms of the symplectic space (H1(Fg+;Z2), ·). In both situations, we obtain results on the number of orbits and the number of elements in each orbit. Except in one case, these results do not depend on any necessary choices. We also compare both previous classifications to a third one : weak-equivalence of coverings When a surface Ng+1 is not orientable, the only S − principal fibration over Ng+1 admitting a Spin-structure is the trivial one. We denote it by Pn. Like in [BGHM], we interpret a Spin-structure as a special Z2-covering of the fibration Pn. “Special” means that the fibre of Pn is Z2-covered. The set of these Spin-structures, denoted by E , is an affine space. It has 2 elements. The purpose of this paper is to investigate the action on E of a group of homeomorphisms of the surface Ng+1. We know, for instance from [L], [G], [ZVC], that homeomorphisms of Ng+1 map onto automorphisms of (H1(Ng+1;Z2), ·), the Z2-vector space H1(Ng+1;Z2) endowed with the (non-degenerate) intersection form ·. The affine structure of E compels to fix basis in Z2-vector spaces. We obtain results on the number of orbits and the number of elements in each orbit. Except in one case, see theorem (24), these results neither depend on this choice of basis nor on the additional choices in the definition of the action. The group acting on E may be identified with the orthogonal group O(Z2, g). Let us remark that this identification depends on the choice of an orthogonal basis of (H1(Ng+1;Z2), ·). The action is an action ”à la Johnson”, i.e. obtained after fixing a section H1(Ng+1;Z2) −→ H1(Pn;Z2), so as to choose a particular way to lift any orthogonal automorphism of (H1(Ng+1;Z2), ·) to an automorphism of H1(Pn;Z2). A different approach to the classification of special Z2-coverings of Pn = S × Ng+1 is to consider the orientation covering Fg −→ Ng+1 and the classification, in [BGHM], of special Z2-coverings of Po = S 1 × Fg (Fg is a closed orientable surface). Instead of classifying the 2 elements of E , we are thus led to classify only 2 of them, namely those coming from special Z2-coverings of Po. This also constrains to restrict the symplectic group which acts. Surprisingly, although the action we will consider is a restriction (both of the group and of the space on which it acts), it is classified by the same invariant deduced from the classical