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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Spin characteristic classes and reduced $K\textrm {Spin}$ group of a low-dimensional complex
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by Bang He Li and Hai Bao Duan PDF
Proc. Amer. Math. Soc. 113 (1991), 479-491 Request permission

Abstract:

This note studies relations between Spin bundles, over a CW-complex of dimension $\leq 9$, and their first two Spin characteristic classes. In particular by taking Spin characteristic classes, it is proved that the stable classes of Spin bundles over a manifold $M$ with dimension $\leq 8$ are in one to one correspondence with the pairs of cohomology classes $({q_1},{q_2}) \in {H^4}(M;\mathbb {Z}) \times {H^8}(M;\mathbb {Z})$ satisfying \[ ({q_1} \cup {q_2} + {q_2})\bmod 3 + U_3^1 \cup ({q_1}\bmod 3) \equiv 0\], where $U_3^1 \in {H^4}(M;{\mathbb {Z}_3})$ is the indicated Wu-class of $M$. As an application a computation is made for $\widetilde {K\operatorname {Spin} }(M)$, where $M$ is an eight-dimensional manifold with understood cohomology rings over $\mathbb {Z},{\mathbb {Z}_2},$, and ${\mathbb {Z}_3}$.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 113 (1991), 479-491
  • MSC: Primary 55R50; Secondary 57R20
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1079895-1
  • MathSciNet review: 1079895