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

Abstract

This note studies relations between Spin bundles, over a CW-complex of dimension ≤ 9 \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 M with dimension ≤ 8 \leq 8 are in one to one correspondence with the pairs of cohomology classes ( q 1 , q 2 ) ∈ H 4 ( M ; Z ) × H 8 ( M ; Z ) ({q_1},{q_2}) \in {H^4}(M;\mathbb {Z}) \times {H^8}(M;\mathbb {Z}) satisfying \[ ( q 1 ∪ q 2 + q 2 ) mod 3 + U 3 1 ∪ ( q 1 mod 3 ) ≡ 0 ({q_1} \cup {q_2} + {q_2})\bmod 3 + U_3^1 \cup ({q_1}\bmod 3) \equiv 0 \] , where U 3 1 ∈ H 4 ( M ; Z 3 ) U_3^1 \in {H^4}(M;{\mathbb {Z}_3}) is the indicated Wu-class of M M . As an application a computation is made for K Spin ~ ( M ) \widetilde {K\operatorname {Spin} }(M) , where M M is an eight-dimensional manifold with understood cohomology rings over Z , Z 2 , \mathbb {Z},{\mathbb {Z}_2}, , and Z 3 {\mathbb {Z}_3} .