Assuming parallelism between CPn-1 model in two dimensions and SU(N) gauge model in four dimensions, we conclude: i) The vacuum of an SU(N) gauge model is two-fold degenerate and a topological symmetry breaking occurs. ii) In a lattice gauge model a new kind of phase transition occurs at the crossover point gT from strong to weak coupling. Although the system is disordered both for g≥ gT and g≤ gT, the mechanisms to disorder the system are different. For g≥ gT vortices with finite thickness disorder the system, while for g≤ gT half-instanton gas disorders the system (for SU(2)). For SU(3), one-third-instanton gas disorders the system. The relation between an instanton and a Z(2) monopole as well as the relation between SU(2) theory and SU(2)/ Z(2) theory is discussed. We also derive constraints on the number of flavor from requirement of stability of half (one-third)-instanton gas.