An analogue of the stabilization map for regular Z_p actions

Several authors have used the stabilization technique introduced by Boardman to analyze the fixed point structure of a smooth action of a compact Lie group on smooth manifolds. Using regular Z_p actions, defined herein, and the bordism groups of regular Z_p actions, we show that if there is a regular action of Z_p on a smooth manifold and the manifold is not a boundary in the Thom oriented cobordism group mod the ideal of those elements all of whose Pontrjagin numbers are 0 mod p, then the manifold must have some component of its fixed point set of dimension at least half that of the ambient manifold. As a coda to this, we study the relationship between this stabilization map and the factorization of the cyclotomie polynomials over Zp.