Connected one-dimensional CW complex and homotopy groups

Question

Prove that connected one-dimensional CW complex X has $\pi_n(X)=0, n \ge 2$. This is a problem from my exam, I tried to use the cellular approximation theorem but didn't solve it. Professor said it is very easy. Any help?

Answer 1

Here is a proof. I will be using Hatcher's "Algebraic Topology" as a reference.

1. The universal covering space of every connected graph is a connected graph (Lemma 1A.3). Every simply-connected graph is a tree (p. 86). Every tree is contractible (pretty much by the definition used by Hatcher). See Section 1.A for basic definitions and properties of trees. Hence, the universal covering space of every connected graph is contractible.

2. Every contractible space has trivial homotopy groups (since the latter are homotopy-invariant).

3. If $X, Y$ are path-connected, $p: X\to Y$ is a covering space, then $p$ induces isomorphisms of homotopy groups $\pi_i(X)\to \pi_i(Y)$ for all $i\ge 2$, see Proposition 4.1.

Now, we can finish: Let $Y$ be a connected graph, $X\to Y$ its universal covering. Then $\pi_i(X)=0$ for all $i>0$ and, therefore, $\pi_i(Y)=0$ for all $i>1$. qed