Show that a continuous map from covering space is a homeomorphism if fundamental group is finite.

Asked Sep 17 '17 at 14:45

Active Sep 24 '17 at 18:34

Viewed 287 times

I have this question:

Let $X$ be a topological space, $X$ path connected, locally path connected. Let $Y$ be the covering space with $p:Y \rightarrow X $ be the covering map.

I want to show that if $f$ is a continuous map from Y to Y such that $pf=f$ then $f$ is a homeomorphism in following cases:

a) fundamental group of $Y$ is finite

b) $p$ is normal covering

c)Image of $p$ has finite index.

How should I solve this problem. Any hints.

Edits

I found this question ( I did a Google search).

edited Sep 24 '17 at 18:34

asked Sep 17 '17 at 14:45

Tensor_Product

1,433

https://math.stackexchange.com/questions/256951/if-a-covering-map-has-a-section-is-it-a-1-fold-cover – Tsemo Aristide Sep 17 '17 at 14:56
How this question is related. – Tensor_Product Sep 17 '17 at 15:27
One problem that i see with what you ask is that $pf$ goes from Y to X while $f$ goes from Y to Y. Is there any relation between X and Y? – user1868607 Dec 19 '17 at 12:05
Oh i see it should be $pf = p$ – user1868607 Dec 19 '17 at 12:08

Show that a continuous map from covering space is a homeomorphism if fundamental group is finite.

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