On mapping cones of suspension elements of finite order in the homotopy groups of a wedge of spheres

Abstract:

The genus of the mapping cone ${C_f}$ of a map $f:{S^{m - 1}} \to \bigvee {S^n}(m > n > 1)$ representing a suspension element of finite order in ${\pi _{m - 1}}(\bigvee {S^n})$ is classified by a subgroup ${G_f}$ of ${\pi _{m - 1}}({S^n})$ depending only on the homotopy type of ${C_f}$. The group ${G_f}$ finds application in proving that the genus of ${C_f}$ is trivial whenever ${C_f}$ has sufficiently many $n$-cells, the number being limited by the torsion subgroup of ${\pi _{m - 1}}({S^n})$.