Mod p stable orders of finite CW-complexes

Let p be a prime and [X, X] the group consisting of classes of stable self-maps on a space X. The mod p stable order of X denoted by ¦X¦p is defined to be the order of the stable identity map in the group [X, X] ⊗ Z(p), where Z(p) is the ring of integers localized at p. Let Xnn + k be a finite CW-complex with nontrivial cells of dimensions between n and n + k. In this paper we prove in Theorem 1.1 that ¦Xnn + k¦p ⩽ p[k(2(p − 1))] + v + ε, where ε = 0 if p is odd, and is 2 if p = 2, while v = min{j ¦ pjH̃∗(Xnn + k; Z(p)) = 0}. As an application, we determine in Theorem 1.2 the mod p stable order of stunted lens spaces L2n − 12(n + m)mod pv, where p is an odd prime.