#### Volume 7, issue 4 (2007)

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String cohomology groups of complex projective spaces

### Iver Ottosen and Marcel Bökstedt

Algebraic & Geometric Topology 7 (2007) 2165–2238
##### Abstract

Let $X$ be a space and write $LX$ for its free loop space equipped with the action of the circle group $\mathbb{T}$ given by dilation. The equivariant cohomology ${H}^{\ast }\left(L{X}_{h\mathbb{T}};ℤ∕p\right)$ is a module over ${H}^{\ast }\left(B\mathbb{T};ℤ∕p\right)$. We give a computation of this module when $X=ℂ{P}^{r}$ for any positive integer $r$ and any prime number $p$. The computation does not use the fact that $ℂ{P}^{r}$ is formal, nor does it use the Jones isomorphism and negative cyclic homology.

##### Keywords
free loop spaces, string topology, Morse theory
##### Mathematical Subject Classification 2000
Primary: 58E05, 55P35, 55N91, 18G50