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On the cohomology of the free loop space of a complex projective space

Let $Λ(\mathbb{C}P^n)$ 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 $Λ(\mathbb{C}P^n)$ 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 $Ω(\mathbb{C}P^n)\hookrightarrow Λ(\mathbb{C}P^n)\stackrel{eval}{\longrightarrow}\mathbb{C}P^n$ has a cross section its Serre spectral sequence does not collapse: Here $eval$ is the evaluation map at a base point * $\in \mathbb{C}P^n$.