Spin characteristic classes and reduced 𝐾𝑆𝑝𝑖𝑛 group of a low-dimensional complex

Li, Bang He; Duan, Hai Bao

doi:10.1090/S0002-9939-1991-1079895-1

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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