Cohomology of the free loop space of a complex projective space (Q2466888)

For a connected based space \((X,x_0)\), let \(\Lambda (X)\) denote the space consisting of all continuous maps \(\omega :S^1\to X\) with the compact open topology and let \(ev:\Lambda (X)\to X\) be the the evaluation map defined by \(ev(\omega )=\omega (x_0)\). It is known that this gives the evaluation fibration sequence \(\xi_X:\Omega X\to \Lambda (X) @>ev>> X\) with cross section. It follows from a result of Ziller that the Serre spectral sequence for the case \(X=\mathbb C \text{P}^n\) does not collapse at \(E_2\). In this paper, the author shows this fact by using only an elementary diagram chasing. The proof is very short and easy to understand.