The cohomology of free loop spaces of homogeneous spaces

Abstract

The free loops space $ΛX$ of a space $X$ has become an important object of study particularly in the case when $X$ is a manifold.The study of free loop spaces is motivated in particular by two main examples. The first is their relation to geometrically distinct periodic geodesics on a manifold, originally studied by Gromoll and Meyer in $1969$. More recently the study of string topology and in particular the Chas-Sullivan loop product has been an active area of research. A complete flag manifold is the quotient of a Lie group by its maximal torus and is one of the nicer examples of a homogeneous space. Both the cohomology and Chas-Sullivan product structure are understood for spaces $S^n$, $\mathbb{C}P^n$ and most simple Lie groups. Hence studying the topology of the free loops space on homogeneous space is a natural next step. In the thesis we compute the differentials in the integral Leray-Serre spectral sequence associated to the free loops space fibrations in the cases of $SU(n+1)/T^n$ and $Sp(n)/T^n$. Study in detail the structure of the third page of the spectral sequence in the case of $SU(n)$ and give the module structure of $H^*(Λ(SU(3)/T^2);\mathbb{Z})$ and $H^*(Λ(Sp(2)/T^2);\mathbb{Z})$.