# Vector bundles and cohomotopies of spin 5-manifolds

@article{Konstantis2018VectorBA, title={Vector bundles and cohomotopies of spin 5-manifolds}, author={Panagiotis Konstantis}, journal={arXiv: Geometric Topology}, year={2018} }

The purpose of this paper is two-fold: On the one side we would like to close 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 $1-1$ correspondence to elements in the first cohomotopy group $\pi^4(M)=[M,S^4]$ of $M$. From previous results 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…

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