The cohomology of free loop spaces of homogeneous spaces

Abstract: The free loops space $\Lambda X$ 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 $S<sup>n$, $\mathbb{C}P<sup>n$ 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 $SU(n+1)/T<sup>n$ and $Sp(n)/T<sup>n$. 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 $H<sup>*(\Lambda(SU(3)/T<sup>2);\mathbb{Z})$ and $H<sup>*(\Lambda(Sp(2)/T<sup>2);\mathbb{Z})$.