# Local homeomorphism and lifting

Assume $$X,Y,X',Y'$$ are nice spaces (not necessarily manifolds). If $$\pi_{X}:X\rightarrow X'$$ and $$\pi_{Y}:Y\rightarrow Y'$$ are local homeomorphisms, if $$f':X'\rightarrow Y'$$ is a continuous map, then does there exists a continuous map $$f:X\rightarrow Y$$ such that $$f'\circ \pi_{X}=\pi_{Y}\circ f$$?

Questions:

1. What about if we start of with $$f$$ and want an $$f'$$?

2. When do we have such a property?

3. What if $$\pi_{X}$$ and $$\pi_{Y}$$ are covering maps?

Thoughts:

1. If $$\pi_{X}$$ and $$\pi_{Y}$$ are bijective then yes.

2. Somehow if the answer is yes, then $$f$$ must locally be $$\pi_{Y}^{-1}\circ f'\circ \pi_{X}$$.

3. How to go from local property in 2. to global?

Motivation: Every continuous map $$X\rightarrow Y$$ induces a continuous map on its universal covers $$\tilde{X}\rightarrow \tilde{Y}$$

• Do you mean $f : X \to Y$? Apr 2 at 20:08
• @diracdeltafunk yup! Apr 2 at 20:33
• You should correct your mistake instead of commenting "yup". Apr 3 at 9:20
• @PaulFrost OK. It was a typo. The reader could have figured it out without much effort given some thought. Apr 3 at 18:48
• @monoidaltransform Of course the reader can do it. But I do no think it is better that many readers stumble over a typo than that the OP makes a corection. Apr 3 at 22:45

There is a difference with your question, however, since one typically works in the category of pointed spaces and asks for the existence of a lift $$f$$ with the prescribed values at a point. This makes answers much cleaner. For instance, if you take $$X=X'=[0,1]$$ and the base-points $$0\in X', y'_0=f'(0)\in Y'$$, then one usually asks for the existence of a lift $$f$$ for any choice of $$y_0:=f(0)\in \pi_Y^{-1}(y'_0)$$. The existence of $$f$$ is then called the path-lifting property. It is known that (assuming that $$Y, Y'$$ are nice as defined above), a local homeomorphism satisfies the path-lifting property if and only if it is a covering map. (See for instance section 5-6 "Covering spaces" in do Carmo's book "Differential geometry of curves and surfaces.")
Furthermore, for general maps of pointed "nice" spaces a lift $$f$$ exists (provided that $$\pi_X, \pi_Y$$ are covering maps) if and only if
$$f_*(\pi_1(X,x_0))\le \pi_{Y*}(\pi_1(Y, y_0)),$$ where, $$f(x_0)=y_0$$.
Since you are not working in the category of pointed topological spaces, things are less predictable and I do not think there is a general theory answering your questions. One can still take $$X=X'=[0,1], \pi_X=id$$ and ask for a "weak" path lifting property for local homeomorphisms $$Y\to Y'$$ (i.e. an existence of a lift $$f$$ for every map $$f': [0,1]\to Y'$$, just as in your question). One can prove, for instance, that a finite-to-one local homeomorphism $$\pi_Y: (0,1)\to S^1=Y'$$ satisfies the weak path-lifting property if and only if it is a covering map. (Moreover, even locally injective maps $$f': [0,1]\to S^1$$ suffice to disprove the weak path-lifting property.) On the other hand, the map $$(0,\infty)\to S^1, t\mapsto e^{it}$$ satisfies the weak path-lifting property but is not a covering map.