Vol. 95, No. 1, 1981

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

Ross Geoghegan

Vol. 95 (1981), No. 1, 85–93

f : X X is a periodic homotopy idempotent if f is homotopic to fk+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 : ffk+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
Received: 13 November 1979
Published: 1 July 1981
Ross Geoghegan