The thesis is organized as follows. At first, in the second chapter we will give an elementary
introduction to the QHE. We review the theories for the IQHE and FQHE. In particular, We
emphasis the role of topology in the QHE and show that the Hall conductance is related to
the topological invariant. It is also shown that the quasiparticles in the FQHE are anyons,
which satisfy fractional statistics.
In chapter 3, we first review the standard hierarchical theory of the FQHE. We consider
the FQHE on the sphere and use the projective coordinates on the sphere. The guiding
principle for constructing the state of the FQHE on the sphere is the rotational invariance of
the wave function.
Spin-nonpolarized FQHE will be also discussed in this chapter. When the magnetic field
is not strong enough, the electron spin may not be polarized. Halperin had proposed a class of
state with half spins reversed which are spin-singlet states. The hierarchical wave function for
the spin-non-polarized case, for example, the spin-singlet FQHE, is not still fully understood.
So it remains an interesting problem. We will construct the hierarchical wave function based on the Halperin spin-singlet state.
We will also calculate the spin of the quasiparticle in the FQHE on the sphere. However,
some questions remain unanswered. Is the spin of the quasiparticle topological-independent
or not? This problem will be answered in chapter 6.
In chapter 4, the hierarchical wave function on the torus will be analyzed in details.
We construct the wave function of the quasiparticles and discuss the fractional statistics on
the torus. Modular invariance is also discussed. The degeneracy of the hierarchical state is
obtained explicitly by constructing the wave function.
In chapter 5, we discuss the quantum mechanics and quantum Hall effect on Riemann
surfaces. In particular, we construct the Laughlin wave function on Riemann surfaces of high
genus. The degeneracy of the Laughlin states is obtained explicitly and our result about the
degeneracy is different from the one in the literature.
In chapter 6, we will calculate the spin of the quasiparticle on Riemann surfaces by
using the braid group on Riemann surfaces. We find that the spin of the quasiparticle is
topological-independent.
