  The search for a Unified description of all interactions has created many
developments of mathematics and physics. The role of geometric effects in the
Quantum Theory of particles and fields and spacetime has been an active topic
of research. This paper attempts to obtain the conditions for a Unified Gauge
Field Theory, including gravity. In the Yang Mills type of theories with
compactifications from a 10 or 11 dimensional space to a spacetime of 4
dimensions, the Kaluza Klein and the Holonomy approach has been used. In the
compactifications of Calabi Yau spaces and sub manifolds, the Euler number
Topological Index is used to label the allowed states and the transitions. With
a SU(2) or SL(2,C) connection for gravity and the U(1)*SU(2)*SU(3) or SU(5)
gauge connection for the other interactions, a Unified gauge field theory is
expressed in the 10 or 11 dimension space. Partition functions for the sum over
all possible configurations of sub spaces labeled by the Euler number index and
the Action for gauge and matter fields are constructed. Topological Euler
number changing transitions that can occur in the gauge fields and the
compactified spaces, and their significance are discussed. The possible limits
and effects of the physical validity of such a theory are discussed.
