Cyclic actions and divisible polynomials

Let g:M²ⁿ?M²ⁿ be an orientation preserving PL map of period m>2. Suppose that the cyclic action defined by g is locally linear PL, fixing a locally flat submanifold F with components only of dimension 0 or 2n−2, and regular. Let ?(m) be Euler’s number and ρ(m)=?(m)−1 if m is a power of 2 and ρ(m)=?(m) otherwise. If is a rational integer, then . This congruence is used to show that a codimension-2 locally flat submanifold of cohomology complex projective n-space fixed by g must have degree one if m≠4 or 10 and n<?(m)+4.