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

TL;DR

This paper investigates the cohomology of the free loop space of complex projective spaces, revealing that its Serre spectral sequence does not collapse despite the existence of a cross section in the natural fibration.

Contribution

It demonstrates that the Serre spectral sequence for the free loop space of complex projective space does not collapse, providing new insights into its topological structure.

Findings

Spectral sequence does not collapse despite a cross section

Cohomology of free loop space is more complex than expected

Provides new understanding of loop space topology

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