  The overlap Dirac operator, which satisfies the Ginsparg-Wilson relation,
realizes exact chiral symmetry on the lattice without any unphysical doubler
modes. To perform the path integrals, one should, however, note that the
overlap fermion determinant has discontinuities where the topology is
ill-defined, its locality is doubtful, and the numerical cost is suddenly
enhanced.
  An interesting solution would be to concentrate on a fixed topological sector
in the full configuration space. In this thesis, we test a gauge action which
automatically satisfies Luescher's admissibility condition, as well as an
additional (large) negative mass Wilson fermion action in the quenched study
(with no light quark). Both of them would keep the topology along the HMC
simulations. The quark potential and the topology stability are investigated
with different lattice sizes and different couplings.
  The results of quenched QCD in the $\epsilon$-regime are also presented as an
example of the lattice studies with fixed topology. Remarkable quark mass and
topology dependences of meson correlators allow us to determine the fundamental
parameters of the effective theory, in which the exact chiral symmetry with the
Ginsparg Wilson relation plays a crucial role.
