On the homology of spaces of sections of complex projective bundles (Q1060739)

Let X be a connected CW complex, \(\xi\) : \(V\to X\) an \((n+1)\)-dimensional vector bundle over X, and \(P\xi\) : P(V)\(\to X\) the associated projective bundle. Assume that \(P\xi\) has a section u and let \(\Gamma\) u be the space of sections vertically homotopic to u. The author gives some explicit (co-)homology groups of \(\Gamma\) u for various X such as closed orientable surfaces, complex projective spaces, or lens spaces obtained from groups of odd order. The method of proof is to identify the k- invariants in a Moore-Postnikov decomposition for the space of sections.