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Here is something I read:

"Let $p : \tilde{S} \rightarrow S$ be any covering space. By a lift of a closed curve $\alpha$ to $\tilde{S}$ we will always mean the image of a lift $\mathbb{R} \rightarrow \tilde{S}$ of the map $\alpha \circ \pi$ where $\pi : \mathbb{R} \rightarrow S^1$ is the usual covering map."

I was wandering if I could get a few examples of lifts in a visual sense as that is usually the way I can learn things at first. I would be very appreciative. It would help me understand the idea that I read better. Thanks!

Note I am not talking about a path lift.

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  • $\begingroup$ But path lifting is the way to understand general lifting. If certain obvious requirements about fundamental groups are satisfied, then a map from a locally path-connected & path-connected space can be lifted to a covering space. The lifting is constructed by explicitly lifting a single point (the basepoint) then using the path (and homotopy) lifting property. See proposition 1.33 in Hatcher. $\endgroup$ – user40167 Feb 18 '14 at 4:09
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It is useful to consider some graph examples. Consider the following pictures (Fig. 10.3 from Topology and Groupoids).

graphs

These give $3$-fold covers of $X$, which is two circles with a point in common: label those $a,b$. Make up a word in $a,b$, say $w=ababa^{-1}b^{-1}$. Now start at say the left hand vertex of one of the pictured graphs and follow that word through the graph. You are then "lifting" a path in $X$ to a path in the covering graph, and should give you a good feel for the process, and how the result differs between the two graphs, or, in the right hand graph, on where you start.

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