Topological classification of complex vector bundles over 8-dimensional spin$$^{c}$$ manifolds

Abstract

In this paper, complex vector bundles of rank r over 8-dimensional spin\(^{c}\) manifolds are classified in terms of the Chern classes of the complex vector bundles, where \(r = 3\) or 4. As an application, we see that two rank 3 complex vector bundles over 4-dimensional complex projective space \(\mathbb {C}P^{4}\) are isomorphic if and only if they have the same Chern classes. Moreover, the Chern classes of rank 3 complex vector bundles over \(\mathbb {C}P^{4}\) are determined. Together with results of Thomas and Switzer, this completes the classification of complex vector bundles of any rank over \(\mathbb {C}P^4\).