Cyclic actions and divisible polynomials (Q860418)

The paper under review considers piecewise linear, closed, oriented, even-dimensional manifolds \(M^{2n}\) with an orientation preserving PL transformation of period \(m\) (i.e., a PL action of a cyclic group \(G_m, m>2\), preserving orientation). The author of the paper first studies the equivariant signature. The main result is stated as follows: if the dimensions of the components of the fixed point set \(F\) are either 0 or \(2n-2\), the action is regular and \(\text{Sign}(g, M)\in {\mathbb Z}\), then \(\text{Sign}(g,M)\equiv \operatorname{Sign} F \bmod 2^{\rho (m)}\); in particular, if \(p\) is an odd prime and \(m=2p^e\), then \(\text{Sign}(g,M)\equiv \text{Sign}(F \pitchfork F)\mod p\), where \(\rho(m)=\phi (m)-1\) if \(m=2^e\) and \(\rho(m)=\phi(m)\) if \(m\not=2^e\), and \(\phi\) is Euler's totient function. Then the main result is applied to study the divisibility of polynomials.