  Let $\Lambda (\mathbb{C}P^n)$ denote the free loop space of the complex
projective space $\mathbb{C}P^n$, i. e. $\mathbb{C}P^n$ is the projective space
of the vector space $\mathbb{C}^{n+1}$ of dimension $n+1$ over the complex
numbers $\mathbb{C}$ and $\Lambda(\mathbb{C}P^n)$ is the function space
$\mathrm{map}(S^1,\mathbb{C}P^n)$ of unbased maps from a circle $S^1$ into
$\mathbb{C}P^n$ topologized with the compact open topology. In this note we
show that despite the fact that the natural fibration
$\Omega(\mathbb{C}P^n)\hookrightarrow
\Lambda(\mathbb{C}P^n)\stackrel{eval}{\longrightarrow}\mathbb{C}P^n$ has a
cross section its Serre spectral sequence does not collapse: Here $eval$ is the
evaluation map at a base point * $\in \mathbb{C}P^n$.
