When we study the Berry phase, we're dealing with the time dependent parameters with manifold $M$ and the adiabatic transport of the state, which for $N\times N$-dimensional hamiltonian defines the mapping $$ \tag 1 M \to CP^{N-1}, $$ where $CP^{N-1}$ is the space of N complex unit vectors defined up to the phase. Suppose mapping $$ \tag 2 M \to S^{2N-1} $$ where $S^{2N-1}$ is the space of complex unit vectors with definite phase. During adiabatic transport this mapping is possible if the phase can be globally defined. I want to clarify if there exists a lifting between these mappings $(1),(2)$: $$ f: \quad M \to CP^{N-1} \ \text{to} \ \tilde{f}:\quad M \to S^{2N-1} $$ Suppose $M = S^{n}, n > 1$. Then in order to check whether the lifting is possible, have I to compare the homotopy groups $$ \pi_{i}(CP^{N-1}) \ \ \text{with} \ \ \pi_{i}(S^{2N-1}), \quad i = 1, ...,n, $$ or only $$ \pi_{n}(CP^{N-1}) \ \ \text{with} \ \ \pi_{n}(S^{2N-1})? $$