Classification of the bundle decompositions arising from equivariant maps

Determine, for the complex rank-(n-k) vector bundles E_i over the lens space L^{2r-1}(m) constructed from the pairwise distinct-degree \mathbb{Z}/m-equivariant maps in Proposition \ref{noncan-prop}, which pairs (i,j) satisfy E_i\cong E_j, and establish whether there exists a rank-(n-k) complex vector bundle F over L^{2r-1}(m) that is not isomorphic to any E_i.

Background

Assume that M_{r-(n-k)}\mid r and that the complex vector bundle n\lambda over the lens space L^{2r-1}(m) decomposes as n\lambda\cong E\oplus\epsilon^k, where \lambda is the line bundle associated with the canonical principal \mathbb{Z}/m-bundle S^{2r-1}\to L^{2r-1}(m) and \epsilon^k is the trivial rank-k bundle. Proposition \ref{noncan-prop} shows that this assumption produces infinitely many decompositions n\lambda\cong E_i\oplus\epsilon^k with rank-(n-k) bundles E_i.

Theorem \ref{mainthm} supplies infinitely many pairwise distinct equivariant degrees, but the associated bundles need not be pairwise nonisomorphic. Indeed, the paper observes that the set of homotopy classes [L^{2r-1}(m),B\mathbb{U}(n-k)] is finite because the lens space is rationally homotopy equivalent to a sphere and the relevant homotopy group of B\mathbb{U}(n-k) is torsion. The stated problem asks how the resulting bundles are distributed among isomorphism classes and whether additional rank-(n-k) bundles occur beyond the family {E_i}.