Proposition 5.14.
If \(p \colon (\widetilde{X}, \tilde{x}) \to (X, x)\) is a based covering, then \(p_* \colon \pi_n(\widetilde{X}, \tilde{x}) \to \pi_n(X, x)\) is an isomorphism for \(n \geq 2\text{.}\)
Section 5.4 Coverings Revisited
Proof.
Surjectivity. Consider a map \(f \colon (S^n, s_0) \to (X, x_0)\text{.}\) The lift \(\tilde{f} \colon (S^n, s_0) \to (\widetilde{X}, \tilde{x})\) exists, since \(f_* \pi_1(S^n, s_0) = 0\) is contained in \(p_* \pi_1(\widetilde{X}, \tilde{x})\text{.}\)
Injectivity. A similar argument to the proof that \(p_* \colon \pi_1(\widetilde{X}, \tilde{x}) \to \pi_1(X, x)\) is injective.
Example 5.15.
Since \(p \colon \mathbb{R} \to S^1\) is a covering, we have \(p_* \colon \pi_n(\mathbb{R}, 0) \to \pi_n(S^1, 1)\) an isomorphism for \(n \geq 2\text{.}\) Since \(\pi_n(\mathbb{R}, 0) = 0\text{,}\) we conclude \(\pi_n(S^1, 1) = 0\) for \(n \geq 2\text{.}\)
Definition 5.16.
A path-connected space with \(\pi_n = 0\) for all \(n \geq 2\) is called aspherical.
