'Elementary' proof of $\tilde{X}$ is contractible iff $\pi_n(X) =0 \forall n \ge 2$.

Question

I have studied algebraic topology, but have not studied $\pi_n(X)$ any further than $n=1$. Is there a proof of $\tilde{X}\cong 1 \Leftrightarrow \pi_n(X) =0 \forall n \ge 2$ that does not require much heavy machinery ($\tilde{X}$ here is the universal cover of $X$, and $X$ is a CW complex) ? Even if not, could someone please link to (or ideally concisely document) a proof of the theorem, using as few lemmas or theorems about $\pi_n$ for $n \ne 1$ as possible?

Answer 1

Whitehead's theorem states that if a map of CW complexes induces isomorphisms on all homotopy groups, it's a homotopy equivalence. For $X$ connected, $\tilde X$ is simply connected with $\pi_n(\tilde X)=\pi_n(X)$ for $n\geq 2$. So if $\pi_n(X)=0$ for $n\geq 2,$ the inclusion of any point into $\tilde X$ induces isomorphisms on homotopy groups and is a homotopy equivalence, that is, $\tilde X$ is contractible. A proof of Whitehead's theorem can be found in Hatcher, 4.5.