Title: 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 $S^n$,
$\mathbb{C}P^n$ 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 $H^*(\Lambda(SU(3)/T^2);\mathbb{Z})$ and
$H^*(\Lambda(Sp(2)/T^2);\mathbb{Z})$.