  We study some topological aspects of non-abelian gauge theories intimately
connected to the Lie algebras of the gauge groups and the homotopy theory in
the generalized gauge orbit space. The physics connection to the
non-perturbative solution to strong CP problem as originally proposed by the
author is also discussed. Some relevant topological formulas are also given and
discussed. A result from the physics application is that the usual gauge orbit
space on the compactified space can contain at most a $Z_2$ monopole structure
in the SP(2N) gauge theories. Some relevance to the open universe is also
discussed. We expect that our results may also be useful to the other studies
of non-abelian gauge theories in general.
