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

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 and the cohomology ring of the manifolds, where $r = 3$ or $4$. As an application, we got that two rank $3$ complex vector bundles over $4$-dimensional complex projective spaces $\C P^{4}$ are isomorphic if and only if they have the same Chern classes. Moreover, the Chern classes of rank $3$ complex vector bundles over $\C P^{4}$ are determined. Combing Thomas's and Switzer's results with our work, we can assert that complex vector bundles over $\C P^{4}$ are all classified.