"Robustness" of Nielsen-Thurston Classification

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WWGD
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Hi all,

Let X be a nice-enough topological space so that it admits a universal cover ## \tilde X ##

When does a homeomorphism ## h: X \rightarrow X ## give rise to a homeomorphism

of the universal cover to itself, i.e., we have ## p: \tilde X \rightarrow X ## , then, by

lifting properties this gives rise to (after choosing a specific sheet in the cover) to an

automorphism ## \tilde h : \tilde X \rightarrow \tilde X ## satisfying ## p \tilde h =hp ## ( I wish

I knew how to draw the diagram in here). Question: is ## \tilde h ## always a homeomorphism ?

Thanks.
 

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WWGD
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Sorry, I got it, the answer is yes. And the title is for another question I wanted to make but never did. I was trying to ask initially (when I wrote the title) under what condition claases in Thurston-Nielsen were preserved: if we have h with ## h^n \simeq Id ## and g is isotopic with h, does it follow that ## g^n \simeq Id ## (meaning g^n is isotopic to the identity )
 

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