Geodesic
Vector bundles and cohomotopies of $\operatorname{spin} 5$-manifolds
Homology, homotopy, and applications, Tome 23 (2021) no. 1, pp. 143-158.

Voir la notice de l'article provenant de la source International Press of Boston

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].
DOI : 10.4310/HHA.2021.v23.n1.a9
Classification : 55N22, 55R15, 55S37, 57R25
Keywords: framed bordism, classification of vector bundles, $5$-manifolds 
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     title = {Vector bundles and cohomotopies of $\operatorname{spin} 5$-manifolds},
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Panagiotis Konstantis. Vector bundles and cohomotopies of $\operatorname{spin} 5$-manifolds. Homology, homotopy, and applications, Tome 23 (2021) no. 1, pp. 143-158. doi : 10.4310/HHA.2021.v23.n1.a9. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2021.v23.n1.a9/

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