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

Abstract

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

arXiv commentrevised version