Cyclic actions and divisible polynomials

Let g:M2n⟶M2ng:M2n⟶M2n be an orientation preserving PLPL map of period m>2m>2. Suppose that the cyclic action defined by gg is locally linear PLPL, fixing a locally flat submanifold FF with components only of dimension 0 or 2n−22n−2, and regular. Let ϕ(m)ϕ(m) be Euler’s number and ρ(m)=ϕ(m)−1ρ(m)=ϕ(m)−1 if mm is a power of 2 and ρ(m)=ϕ(m)ρ(m)=ϕ(m) otherwise. If Sign(g,M) is a rational integer, then Sign(g,M)≡SignF(mod2ρ(m)). This congruence is used to show that a codimension-2 locally flat submanifold of cohomology complex projective nn-space fixed by gg must have degree one if m≠4m≠4 or 1010 and n<ϕ(m)+4n<ϕ(m)+4.