On the computations of the Nielsen number

Boju Jiang

Vol. 106 (1983), No. 1, 105–113

DOI: 10.2140/pjm.1983.106.105

Abstract

The aim of this note is to show the following

Main Theorem. Let X be a compact connected ANR, f : X → X be a map. Suppose there is an integer n such that f_πⁿ(π₁(X)) ⊂ J(fⁿ). Then any two fixed point classes of f have the same index. Hence

	L(f) = 0 implies N(f) = 0, while
	L(f)≠0 implies N(f) = ♯Coker(H1(X)H1(X)).

Here L(f) and N(f) are the Lefschetz number and Nielsen number of f respectively, and J(F) ⊂ π₁(X) stands for the Jiang subgroup of f.

Mathematical Subject Classification 2000

Primary: 55M20

Milestones

Received: 29 September 1981

Published: 1 May 1983