Show that the relative form of the Hurewicz theorem in dimension \(n\) implies the absolute form in dimension \(n-1\) by considering the pair \((CX, X)\) where \(CX\) is the cone on \(X\).
Solution
Assume the relative Hurewicz theorem in dimension \(n\). Let \(X\) be \((n-2)\)-connected, as in the absolute Hurewicz theorem in dimension \(n-1\), and consider the cone pair \((CX,X)\). Since \(CX\) is contractible, the long exact homotopy sequence gives \[ \pi_i(CX,X)\cong \pi_{i-1}(X)\qquad(i\ge2). \] Hence \((CX,X)\) is \((n-1)\)-connected. Relative Hurewicz therefore gives an isomorphism \[ h:\pi_n(CX,X)\xrightarrow{\cong}H_n(CX,X). \] The homology long exact sequence of the pair, again using contractibility of \(CX\), gives \[ H_n(CX,X)\cong \widetilde H_{n-1}(X). \] Under these two boundary isomorphisms, naturality of the Hurewicz maps makes the square \[ \begin{array}{ccc} \pi_n(CX,X)&\xrightarrow{h}&H_n(CX,X)\\ \downarrow\partial&&\downarrow\partial\\ \pi_{n-1}(X)&\xrightarrow{h}&H_{n-1}(X) \end{array} \] commute. The three other arrows are isomorphisms, so the bottom Hurewicz map is an isomorphism. This is exactly the absolute Hurewicz theorem in dimension \(n-1\).