特殊二重覆盖的分类
A classification of special 2-fold coverings
摘要：
从闭定向曲面F_g（g>=1）上的O（2）-主纤维化开始，当单值性将纤维SO（2）= S^1发送到Z_2的非平凡元素时，全空间的2重覆盖被称为特殊的。采用D Jonhson的方法[Spin structures and二次型on surfaces，J London Math Soc，2,2（1980）365-373]，我们定义了Sp（Z_2，2g）的作用，Sp（Z_2，2g）是（H_1（F_g;Z_2），.）的辛同构群，在特殊的2-重覆盖集上，它有两个轨道，一个轨道有2^{g-1}（2^g+1）个元素，另一个轨道有2^{g-1}（2^g-1）个元素。这两个轨道是通过考虑Arf-不变量和由Fox演算导出的矩阵的某些同余关系得到的。Sp（Z_2，2g）是两个子群的共轭类的并，每个子群固定一个特殊的2-重覆盖.这两个子群的生成元是显式的。
英文摘要：
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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