A classification of complex rank 3 vector bundles on CP5

Given integers $a_1,a_2,a_3$, there is a complex rank 3 topological bundle on $C{P}^5$ with i-th Chern class equal to $a_i$ if and only if $a_1,a_2,a_3$ satisfy the Schwarzenberger condition. Provided that the Schwarzenberger condition is satisfied, we prove that the number of isomorphism classes of rank 3 bundles V on $C{P}^5$ with $c_i\left(V\right)=a_i$ is equal to 3 if $a_1$ and $a_2$ are both divisible by 3 and equal to 1 otherwise.

This shows that Chern classes are incomplete invariants of topological rank 3 bundles on $C{P}^5$. To address this problem, we produce a universal class in the $\mathrm{tm}{f}_{(3)}$-cohomology of a Thom spectrum related to $BU\left(3\right)$, where $\mathrm{tm}{f}_{(3)}$ denotes topological modular forms localized at 3. From this class and orientation data, we construct a $Z/3$-valued invariant of the bundles of interest and prove that our invariant separates distinct bundles with the same Chern classes.