One-parameter fixed point indices

Donco Dimovski

Vol. 164 (1994), No. 2, 263–297

DOI: 10.2140/pjm.1994.164.263

Abstract

Let F : X × I → X be a PL homotopy, where X is a compact connected PL n-dimensional manifold, in the euclidean space ℝⁿ, n ≥ 4, and let P : X ×I → X be the projection. A fixed point of F is a point (x,t) ∈ X ×I such that F(x,t) = x. The set of all the fixed points of F is denoted by Fix(F). For a family V of isolated circles of fixed points of F we define two indices: ind₁(F,V )—which is an element in the first homology group H₁(E), where E is the space of paths in X ×I ×X from the graph of F to the graph of P; and ind₂(F,V )—which is an element in the group ℤ₂ with two elements. We prove that there is a compact neighborhood N of V and a homotopy from F to H relX × I∖N such that Fix(H) = Fix(F)∖V if and only if ind₁(V,F) = 0 and ind₂(V,F) = 0. The indices ind₁(V,f) and ind₂(V,F) are defined via the degrees, deg ₁(g) and deg ₂(g), for maps g : S¹ × S^m → S^m. Moreover, we show how to modify F to create circles of fixed points with prescribed indices.

Mathematical Subject Classification 2000

Primary: 55M20

Secondary: 57Q35

Milestones

Received: 12 March 1992

Revised: 20 November 1992

Published: 1 June 1994