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