Show that $\pi_{n}(B)$ is isomorphic to $\pi_{n}(E)$.

Question

Let $(E, p, B)$ be a covering space with $B$ and $E$ both path-connected. Show that for all $n \geq 2,$ $\pi_{n}(B)$ is isomorphic to $\pi_{n}(E)$. Could anyone give me a hint for the solution please?

For a covering map $p:E\to B$ where $B$ and $E$ are path-connected, note that $p$ is a fibration with discrete fiber $F$. (Path-connected of $B$ and $E$ is important; otherwise, $p$ might not be a fibration.) For a discrete space $F$, $\pi_n(F)$ is trivial for every positive integer $n$. Then you can use the same argument as in my answer here: https://math.stackexchange.com/questions/3379347. — Batominovski, Oct 04 '19 at 07:59
@WETutorialSchool could you please look at this question also https://math.stackexchange.com/questions/3379461/calculating-pi-1x/3379839?noredirect=1#comment6954756_3379839 — Intuition, Oct 04 '19 at 08:30
In regard to your fist comment ..... will I use that theorem that "If $0 \rightarrow A \rightarrow B \rightarrow 0$ is exact with a map$ h: A \rightarrow B$, then $h$ is an isomorphim "@WETutorialSchool — Intuition, Oct 07 '19 at 09:20
Yes, that is exactly why $\pi_n(E)\cong \pi_n(B)$ for $n>1$. — Batominovski, Oct 07 '19 at 16:46

score 3 · Accepted Answer · answered Oct 03 '19 at 09:10

3

Hint: Lifting property of covering spaces.

answered Oct 03 '19 at 09:10

Ayman Hourieh

39,603

where can I find a good explanation for it? – Intuition Oct 03 '19 at 09:15
It's usually proved in any introduction to covering spaces. See proposition 1.30 in Hatcher's book. – Ayman Hourieh Oct 03 '19 at 09:20
And how can we relate it to long exact sequences? Do you have a reference for this? – Intuition Oct 03 '19 at 09:22
You don't need to worry about the long exact sequence. Your statement is literally a consequence of the lifting property. Look it up. – Ayman Hourieh Oct 03 '19 at 09:24
My professor said that the proof includes using long exact sequences ... but I can not see why? – Intuition Oct 03 '19 at 09:29
You need the homotopy lifting property to use the long exact sequence, so using the sequence feels backwards. Still, if you insist: The long exact sequence involves homotopy groups of three spaces, $B$, $E$ and $F$. Show that most of the groups of $F$ are trivial. – Ayman Hourieh Oct 03 '19 at 09:36
How do I need the homotopy lifting property to use the long exact sequence? – Intuition Oct 03 '19 at 09:42
Is the question I am asking is proposition 4.1 in Hatcher? – Intuition Oct 03 '19 at 09:45

Show that $\pi_{n}(B)$ is isomorphic to $\pi_{n}(E)$.

1 Answers1

Linked