Vector bundles and cohomotopies of spin 5manifolds
Abstract
The purpose of this paper is twofold: 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 $11$ 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 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 semicharacteristic.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 DOI:
 10.48550/arXiv.1812.06547
 arXiv:
 arXiv:1812.06547
 Bibcode:
 2018arXiv181206547K
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Algebraic Topology;
 Mathematics  Differential Geometry
 EPrint:
 17 pages, comments are welcome, changed misleading title