We prove that the existence of a Spin-structure on an oriented real vector bundle and the number of them can be obtained in terms of 2-fold coverings of the total space of the SO(n)-principal bundle associated to the vector bundle. Basically we use theory of covering spaces. We give a few elementary applications making clear that the Spin-bundle associated to a Spin-structure is not sufficient to classify such structure, as pointed out by [6].
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) ∼ S 1 to the nontrivial element of Z 2 . Adapting D Johnson's method [11], 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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