"Robustness" of Nielsen-Thurston Classification

[WWGD]

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.

 

[WWGD]

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 )