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 * $\in \mathbb{ C}P^n$.
Categories: math.AT
