The purpose of this paper is two-fold: On the one side we would like to fill a gap on the classification of vector bundles over $5$‑manifolds. Therefore it will be necessary to study quaternionic line bundles over $5$‑manifolds which are in $\textrm{1-1}$ correspondence to elements in the cohomotopy group $\pi^4(M) = [M,S^4]$ of $M$. From results in [22, 24] this group fits into a short exact sequence, which splits into $H^4(M ; \mathbb{Z}) \oplus \mathbb{Z}_2$ if $M$ is spin. The second intent is to provide a bordism theoretic splitting map for this short exact sequence, which will lead to a $\mathbb{Z}_2$‑invariant for quaternionic line bundles. This invariant is related to the generalized Kervaire semi-characteristic of [23].

中文翻译：

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$ \ operatorname {spin} 5 $-流形的向量捆绑和同代

本文的目的是双重的：一方面，我们要填补流形超过$ 5 $的向量束分类的空白。因此，有必要研究超过$ 5 $的四元线束，这些歧管在$ \ textrm {1-1} $对应于同型组$ \ pi ^ 4（M）= [M，S ^ 4] $ of $ M $。从[22，24]中的结果中，该组符合一个简短的精确序列，如果$ M $是自旋的，则将其拆分为$ H ^ 4（M; \ mathbb {Z}）\ oplus \ mathbb {Z} _2 $。第二个目的是为此简短的精确序列提供bordism理论分裂图，这将导致四元离子线束的$ \ mathbb {Z} _2 $-不变。该不变量与[23]的广义Kervaire半特征有关。