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

+ 2 like - 0 dislike
762 views

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

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverf$\varnothing$owThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.