Calculation of Nielsen periodic numbers on infra-nilmanifolds

摘要

It is known that for weakly Jiang maps f on infra-nilmanifolds, the equality NFn(f) = N(f(n)) holds. For non-weakly Jiang maps this is no longer true. In this paper, we show formula expressing the Nielsen type numbers for periodic points on infra-nilmanifols as a linear combination of the ordinary Nielsen numbers. Let M be an infra-nilmanifold and f : M - M be a map. Suppose M-K is a regular covering of M which is a compact nilmanifold with pi(1)(M-K) = K. Assume that f*(K) subset of K. Then for any m, f(m) has lifts (f) over bar ((m)) : M-K - M-K on M-K. We prove that the prime Nielsen-Jiang periodic number NPn(f) can be expressed by Nielsen numbers N((f) over bar ((m))) of lifts (f) over bar ((m)) of f(m) with m vertical bar n. Our formula will be particularly useful for maps of infra-nilmanifolds which are not weakly Jiang. (C) 2021 Elsevier B.V. All rights reserved.