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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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# "Robustness" of Nielsen-Thurston Classification

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