For a highly connected space, the first nonzero homotopy group agrees with the first nonzero homology group.Suggest a slogan
Let \(n\geq 2\) and let \(X\) be an \((n-1)\)-connected space. Then \(\tilde H_i(X) = 0\) for \(i < n\), and the Hurewicz map \(\pi_n(X)\to H_n(X)\) is an isomorphism [@Hat02]. For \(n = 1\) and \(X\) path-connected, the Hurewicz map \(\pi_1(X)\to H_1(X)\) induces an isomorphism from the abelianization of \(\pi_1(X)\) onto \(H_1(X)\) [@Hat02].