Periodic mappings of complex projective space with an isolated fixed point (Q1307991)

A PL cohomology complex projective \(n\)-space is a piecewise linear closed orientable \(2n\)-manifold \(M^{2n}\), such that there is a class \(x\in H^2(M;\mathbb{Z})\) with the property that \(H^*(M;\mathbb{Z})= \mathbb{Z}[x]/(x^{n+ 1})\). The Pontryagin class of \(M^{2n}\), \(p_*(M^{2n})\in H^*(M;\mathbb{Q})\), is standard if \(p_*(M^{2n})= (1+ x^2)^{n+ 1}\). The complex projective \(n\)-space \(\mathbb{C} P^n\) has a standard Pontryagin class. Let \(p\) be an odd prime and let \(G_p\) denote the cyclic group of order \(p\). If \(M^{2n}\) admits a locally linear PL \(G_p\) action, then the number of components of the fixed-point set \(M^{G_p}\) is at most \(p\). If \(M^{G_p}\) consists of two components then the action is said to be of type II. If \(M^{G_p}= F^{2k_1}_1\cup F^{2k_2}_2\), then \(k_1+ k_2= n-1\), and we say that the action is of type \(\text{II}_k\) where \(k= \min(k_1,k_2)\). Actions of type \(\text{II}_0\) fix an isolated point and a codimension-2 locally flat submanifold \(F^{2n-2}\). An action of type \(\text{II}_0\) is regular if its restriction to the normal block bundle of \(M^{G_p}\) is a multiple of one irreducible complex representation of \(M^{G_p}\). The degree of \(F^{2n-2}\) in \(M^{2n}\) is the integer \(d\) if \(i_*[F]\) is dual to \(dx\) where \(i: F^{2n-2}\subset M^{2n}\) is the inclusion mapping. An action of type \(\text{II}_0\) is standard if it is regular and the degree of \(F^{2n-2}\) in \(M^{2n}\) is one. The main result of this paper is Theorem A. Suppose that \(M^{2n}\) is a PL cohomology projective \(n\)-space which admits a locally linear PL \(G_p\)-action of type \(\text{II}_0\) for \(p= 3\) or \(5\). If \(n< p+5\), then the degree of the fixed codimention-2 submanifold is one. If \(n< p+3\), then the action is standard and the Pontryagin class of \(M^{2n}\) is standard. Theorem B. If \(n< p+3\) and the action is standard, then the Pontryagin class of \(M^{2n}\) is standard. If \(n< p+1\) and the action is regular, then the action is standard and the Pontryagin class of \(M^{2n}\) is standard. Theorem C. If \(p=3\) or \(5\) and \(3\leq n< p+3\), then there are at most \(2^{[n/2]-1}\) PL homotopy complex projective \(n\)-spaces which admit a locally linear PL \(G_p\) action of type \(\text{II}_0\). Analogous theorems D--F are also proven in the paper.