Extend the Hurewicz theorem by showing that if \(X\) is an \((n-1)\)-connected CW complex, then the Hurewicz homomorphism \(h: \pi_{n+1}(X) \to H_{n+1}(X)\) is surjective when \(n > 1\), and when \(n = 1\) show there is an isomorphism \(H_2(X) / h\bigl(\pi_2(X)\bigr) \approx H_2\bigl(K(\pi_1(X), 1)\bigr)\).
Solution
Let \(X\) be \((n-1)\)-connected. Attach cells of dimensions at least \(n+2\) to \(X\) to kill \(\pi_{n+1},\pi_{n+2},\dots\), obtaining a \(K(\pi_nX,n)\), say \(K\). The inclusion \[ X\hookrightarrow K \] preserves \(\pi_n\).
For \(n>1\), Exercise 22 gives \[ H_{n+1}(K)=0. \] In the relative cellular chain complex \(C_*(K,X)\), the first possible generators occur in degree \(n+2\). Hence the exact sequence contains \[ H_{n+2}(K,X)\xrightarrow{\partial}H_{n+1}(X)\longrightarrow H_{n+1}(K)=0. \] Thus \(\partial\) is surjective. The generators of \(H_{n+2}(K,X)\) come from the attached \((n+2)\)-cells, and their boundaries are precisely the Hurewicz classes of their attaching maps in \(\pi_{n+1}(X)\). Consequently \[ h:\pi_{n+1}(X)\twoheadrightarrow H_{n+1}(X) \] is surjective.
When \(n=1\), perform the same construction to obtain \(K(\pi_1X,1)\) by attaching cells of dimension at least \(3\). The same relative exact sequence now gives \[ H_2(X)/\operatorname{im}\partial\cong H_2(K(\pi_1X,1)). \] Again \(\operatorname{im}\partial\) is exactly the subgroup generated by Hurewicz images of attaching maps in \(\pi_2(X)\), hence \[ \operatorname{im}\partial=h(\pi_2(X)). \] Therefore \[ \boxed{H_2(X)/h(\pi_2(X))\cong H_2(K(\pi_1(X),1)).} \] This proves both assertions.