Vector bundles and cohomotopies of $\operatorname{spin} 5$-manifolds

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].

Keywords

framed bordism, classification of vector bundles, $5$-manifolds

2010 Mathematics Subject Classification

55N22, 55R15, 55S37, 57R25

Full Text (PDF format)

Received 25 July 2019

Received revised 19 January 2020

Accepted 24 February 2020

Published 2 September 2020