再论pinor，spinor和可定向性
Revisiting pinors, spinors and orientability
摘要：
本文研究了不可定向的偶维流形上的pin结构与其可定向的双覆盖上的pin结构之间的关系，其中可定向的双覆盖要求后者在片交换下是不变的.我们表明，有没有一个简单的双射，但自然映射引起的拉回既不是内射，也不是满射：因此，我们找到的条件，以恢复一个完整的对应。我们还展示了如何描述这样一个对应使用旋量，而不是pinor的双重覆盖：这是在一定意义上是可能的，但在某种程度上，包含一个明确的参考pinor。然后，我们考虑表面的例子，详细计算的真正的射影平面，克莱因瓶和莫比乌斯带。
英文摘要：
We study the relations between pin structures on a non-orientable even-dimensional manifold, with or without boundary, and pin structures on its orientable double cover, requiring the latter to be invariant under sheet-exchange. We show that there is not a simple bijection, but that the natural map induced by pull-back is neither injective nor surjective: we thus find the conditions to recover a full correspondence. We also show how to describe such a correspondence using spinors instead of pinors on the double cover: this is in a certain sense possible, but in a way that contains anyhow an explicit reference to pinors. We then consider the example of surfaces, with detailed computations for the real projective plane, the Klein bottle and the Moebius strip.
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