  This thesis contains work which appeared in several papers. Additionally to
the results in the papers it contains a detailed introduction and some further
proofs and remarks. The dissertation gives a description of the topology and
symplectic and algebraic geometry of Hitchin's hyperkaehler moduli space M of
rank 2 Higgs bundles with fixed determinant of odd degree over a fixed Riemann
surface. After the long introduction it describes a compactification of M in
great detail, using symplectic cutting (math.AG/9804083). Examining the
downward Morse flow of a natural circle action on M it shows the vanishing of
intersection numbers (math.AG/9805071). Examining the upward Morse flow it
explains a set of generators of the cohomology ring and a conjectured explicit
description of the cohomology ring (which was proven in math.AG/0003094). Then
finally it introduces the resolution tower for M, and shows that its direct
limit is homotopically equivalent with the classifying space of the gauge
group. In turn it yields another proof of the generation theorem (as in
math.AG/0003093) and also yields a purely algebraic geometric proof of the
Mumford conjecture about the cohomology ring of the moduli space of rank 2
stable bundles on curves. It finishes by proving homotopy stabilizations in the
resolution tower analogously to the Atiyah-Jones conjecture.
