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Spin characteristic classes and reduced $ K{\rm Spin}$ group of a low-dimensional complex

Authors: Bang He Li and Hai Bao Duan
Journal: Proc. Amer. Math. Soc. 113 (1991), 479-491
MSC: Primary 55R50; Secondary 57R20
MathSciNet review: 1079895
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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

$\displaystyle ({q_1} \cup {q_2} + {q_2})\bmod 3 + U_3^1 \cup ({q_1}\bmod 3) \equiv 0$

, 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}$.

References [Enhancements On Off] (What's this?)

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Keywords: Spin bundle, Spin characteristic classes, Classifying space of a topological group, Steenrod reduced power operations, Wu classes of a manifold, Postnikov decomposition of a map
Article copyright: © Copyright 1991 American Mathematical Society

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