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Obstruction Class for the Existence of a Conformal Spin Structure in a Strict Sense

Abstract

Let V be a pseudo-Riemannian n-dimensional manifold or, more generally, let \((\xi ,Q)\) be a real fibre bundle whose base space is a paracompact space endowed with a non-degenerate quadratic form Q, (that is, with a structure group \({\mathrm {O}}(p,q),\) \(n=p+q\).) Let \(K_{p,q}\) denote the obstruction class for the existence of a \(\mathrm {Pin}(p,q)\)-spin structure on V or over \(\xi .\) Let \(K_{\mathrm {Conf}}(p,q)\) denote the obstruction class for the existence of a conformal spin structure in a strict sense on V or over \(\xi ,\) (simply: a \(C_{n}^{s}(p,q)\)-spin structure), if \(n=2r,\) or of a conformal special spin structure, if \(n=2r+1.\) This short self-contained paper will recall the determination of the obstruction class \(K_{p,q}\) on V,  or over \(\xi ,\) for n even or odd. Then, the obstruction class \(K_{p+1,q+1}\) for the existence of a \(\mathrm {Pin}(p+1,q+1)\)-spin structure over \(\xi _{j}\), (Greub’s j-extension of \(\xi ,\) where j denotes the identity mapping from \({\mathrm {O}}(p,q)\) into \({\mathrm {O}}(p+1,q+1)\)), will be determined in order to express \(K_{\mathrm {Conf}}(p,q),\) for \(n=2r\) or \(n=2r+1,\) in terms of the Stiefel–Whitney classes \(w_{i}(p,q),\) \(i=1,2,\) of \(\xi \), decomposed as the Whitney sum \(\xi =\xi ^{+}\oplus \xi ^{-},\) where the restriction of Q to \(\xi ^{+}\) is positive definite and the restriction of Q to \(\xi ^{-}\) is negative. If \(n=2r,\) we find again results obtained in previous publications [4, 5, 7], by different methods.

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Acknowledgements

The author wants to thank Max Karoubi, Université Paris VI, for his kind interest and support and José Bertin, Institut Fourier, Université de Grenoble, for a critical reading of the paper. He wants also to express all his grateful thanks to Rafał Abłamowicz for his judicious remarks on the preliminary draft of the paper and his generous help in the preparation of the files. Thanks also are due to an anonymous reviewer for his useful and constructive comments and to Jacques Helmstetter, Institut Fourier, Université de Grenoble, for his thorough inspection of the paper.

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Correspondence to Pierre Anglès.

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Communicated by Rafał Abłamowicz

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Anglès, P. Obstruction Class for the Existence of a Conformal Spin Structure in a Strict Sense. Adv. Appl. Clifford Algebras 30, 18 (2020). https://doi.org/10.1007/s00006-020-1044-2

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Keywords

  • Spin structure
  • Pin\((p, q)\)-spin structure
  • Conformal spin structure
  • \(C_{n}(p, q)\)-spin structure
  • Obstruction class

Mathematics Subject Classification

  • 11E88
  • 15A66
  • 17B37
  • 20C30
  • 16W55