For \(X\) a connected CW complex with \(\pi_i(X) = 0\) for \(1 < i < n\) for some \(n \geq 2\), show that \(H_n(X) / h\bigl(\pi_n(X)\bigr) \approx H_n\bigl(K(\pi_1(X), 1)\bigr)\), where \(h\) is the Hurewicz map.
Solution
Let \(G=\pi_1(X)\). Since \[ \pi_i(X)=0\qquad(1<i<n), \] we can construct a \(K(G,1)\) from \(X\) by first attaching \((n+1)\)-cells along representatives of elements of \(\pi_n(X)\), killing \(\pi_n\), and then attaching cells of dimensions at least \(n+2\) to kill all remaining higher homotopy groups. Denote the resulting space by \(K\).
The relative cellular chain complex \(C_*(K,X)\) has no generators below degree \(n+1\). Hence the long exact sequence of the pair gives \[ H_{n+1}(K,X)\xrightarrow{\partial}H_n(X) \longrightarrow H_n(K)\longrightarrow0. \] The relative \((n+1)\)-cells are attached precisely along representatives of \(\pi_n(X)\). Under the connecting map, the class of such a cell goes to the homology class of its attaching map, namely its Hurewicz image. Consequently \[ \operatorname{im}\partial=h(\pi_n(X)). \] Since \(K\simeq K(G,1)\), exactness yields \[ \boxed{H_n(X)/h(\pi_n(X))\cong H_n(K(\pi_1(X),1)).} \]