# How to check whether a lifting between given mappings is possible?

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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 $$\pi_{n}(CP^{N-1}) \ \ \text{with} \ \ \pi_{n}(S^{2N-1})?$$

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