The homomorphism on fundamental group induced by a homotopy idempotent having essential fixed points

Ross Geoghegan

Vol. 95 (1981), No. 1, 85–93

DOI: 10.2140/pjm.1981.95.85

Abstract

f : X → X is a periodic homotopy idempotent if f is homotopic to f^k+1 for some positive integer k. Special cases are homotopy idempotents (k = 1) and period k homeomorphisms. Let X be a compact polyhedron and let f have an essential fixed point x; there is such, for example, when the Lefschetz number is nonzero. During a homotopy H : f≅f^k+1, x traces out a loop ω. Generalizing a theorem of Gottlieb, we show (Theorem 1.2) that the possible values of [ω] in π₁(X,x) are severely restricted. In particular, some power of f_♯([ω]) is a commutator. The theorem is applied in a sequel paper.

Mathematical Subject Classification 2000

Primary: 55P99

Secondary: 55M20

Milestones

Received: 13 November 1979

Published: 1 July 1981