Geodesic
Revisiting Pinors and Orientability
Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 2, pp. 405-422 Cet article a éte moissonné depuis la source Biblioteca Digitale Italiana di Matematica

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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 then consider the example of surfaces, with detailed computations for the real projective plane, the Klein bottle and the Moebius strip. 
@article{BUMI_2012_9_5_2_a12,
     author = {Bonora, Loriano and Ferrari Ruffino, Fabio and Savelli, Raffaele},
     title = {Revisiting {Pinors} and {Orientability}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {405--422},
     year = {2012},
     volume = {Ser. 9, 5},
     number = {2},
     zbl = {1261.57022},
     mrnumber = {2977256},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2012_9_5_2_a12/}
}
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Bonora, Loriano; Ferrari Ruffino, Fabio; Savelli, Raffaele. Revisiting Pinors and Orientability. Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 2, pp. 405-422. http://geodesic.mathdoc.fr/item/BUMI_2012_9_5_2_a12/