On genus and cancellation in homotopy (Q1182828)

The author proves the following Theorem. Suppose given \(\alpha_ i,\beta_ j\in\pi_{4n-1}(S^{2n})\), \(1\leq i\leq H\), \(1\leq j\leq H\), and assume that they are elements of infinite order for \(i\leq m\), \(j\leq m'\), and elements of finite order for \(i > m\), \(j > m'\). Denote by \(C_{\alpha_ i}\), \(C_{\beta_ j}\) the corresponding mapping cones. Then \(\bigvee^ H_{i=1}C_{\alpha_ i}\simeq \bigvee^ H_{j=1}C_{\beta_ j}\) if and only if: (i) \(m=m'\); (ii) \(C_{\alpha_ i}\simeq C_{\sigma(i)}\), \(i=1,\dots,m\), for some permutation \(\sigma\) of \(\{1,2,\dots,m\}\); (iii) \(\bigvee^ H_{i=m+1}C_{\alpha_ i}\simeq\bigvee^ H_{j=m+1}C_{\beta_ j}\).