The free loop space $LX$ of a space $X$ is the space of continuous maps from $S^1$ to $X$. The circle group $S^1$ acts on $LX$ by rotation, and we study the space of homotopy orbits, $LX_{hS^1}= ES^1\times_{S^1}LX$, sometimes called the Borel construction. The aim of the thesis is to compute the cohomology and complex $K$-theory of $LX_{hS^1}$ when $X$ is a projective space (complex or quaternion). The method is Morse theory on $LX$ as developed by Klingenberg.

We obtain generators for the cohomology $H^*(X_{hS^1};\mathbb{F}_p)$ as a module over $H^*(BS^1;\mathbb{F}_p)$, when $X$ is a quaternion projective space. Moreover, we find the $K$-theory of $LX_{hS^1}$ when $X$ is a complex projective space, and this is one of the first such calculations for a non-trivial $X$. Our result is that $K^0(X_{hS^1})\stackrel{\sim}{=} K^0(BS^1)$, and $K^1(X_{hS^1})$ is free abelian. We conjecture that $K^1(X_{hS^1})$ could be the completion of some ''representation theory'' type of group, in analogy with the classical result of Atiyah for the classifying space of a compact Lie group.