An investigation is carried out on inequivalent quantizations and statistics for a quantum mechanical system of identical particles occupying a topological space-time M undergoing spatial topology changes. In this paper a new concept has been introduced related to the construction of inequivalent quantizations for a system which is called history-like path for particles. This a collection of non-causal homotopical paths. Each homotopy class of paths will be labelled with a configuration space of the system in a space-like which is a deformation retract of Mi region that is a sub-space-time of M in which there is no singular slice (a slice M in M that contains a singular point). The labels determine the generators and their relations in constructions the fundamental group 1 (Qn) of the system. By using the new concept two methods are proposed to construct inequivalent quantizations of a system of identical particles undergoing a spatial topology change either via a local-local (L-L) or a global-local (G-L) quantization. The first method corresponds to constructions of inequivalent quantization of the system in each region Mi in M. Each of them is in a (1-1)-correspondence with irreducible unitary representations (IUR s) of a fundamental group (Qn(i)) Bn(i) constructedby generators in each of the region corresponding to the posible history-like homotopy paths. A possible statistics for N-identical particles in M is determined by a sub-representationof IUR of braid group Bn(i). The sub-representation is a representation of a subgroup n(i) which is a subgroup of Bn(i) only generated by permutation of particles. The second method is based on a conctruction of inequivalent quantization of the system M. They are in the (1-1)-correspondence with IUR s of the fundamental group 1(Qn()) Bn() raised by generators corresponding to the history-like homotopical paths of the system in M and these inequivalent quantizations are considered valid for all regions Mi in M. A possible statistic for N-identical particles in M is determined by a sub-representation of IUR of braid group Bn() i.e. a representation of n() C Bn().