Fixed point classes of a fiber map

Cheng Ye You

Vol. 100 (1982), No. 1, 217–241

DOI: 10.2140/pjm.1982.100.217

Abstract

Let (E,p,B) be a fiber space with E, B and all fibers compact connected ANR’s. Let f : E → E be a fiber map, then f induces f : B → B. For each fixed point b of f, we define f_b = f|p⁻¹(b) : p⁻¹(b) → p⁻¹(b). Then p ∘ f = f ∘ p and i_b ∘ f_b = f ∘ i_b, where i_b is the inclusion map. We have Nielsen numbers N(f), N(f) and N(f_b). A product formula relating these Nielsen numbers was published by Brown in 1967. There have been several improvements of the formula since that time.

In this paper, we study the structure of the fixed point classes of f, and prove some theorems about the product formula of the Nielsen number of a fiber map, which imply results of Fadell and of Pak.

Mathematical Subject Classification 2000

Primary: 55R05

Secondary: 55M20

Milestones

Received: 22 October 1980

Published: 1 May 1982