A fiberwise analogue of the Borsuk–Ulam theorem for sphere bundles over a 2-cell complex

For a vector bundle α, let indα denote the largest integer m for which there exists a Z / 2 -map from S m − 1 to S ( α ) . We prove that the equality ind α = dim α holds for every vector bundle α over the complex S n − 1 ∪ k e n , where n ⩾ 2 and k ≠ 0 , if and only if either k is even and n ≠ 2 , 3 , 4 , 8 or k is odd.