Relative fixed point theory
type: Item
instance of: scholarly article
title: Relative fixed point theory
zbMATH Open document ID: 1218.55001
published in: Algebraic \& Geometric Topology
publication date: 2011-04-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0906.0762
review text: Relative fixed point theory deals with the fixed point sets of relative maps, i.e. maps of the form \(f:(X, A)\to (X,A)\). The crucial invariant is the relative Nielsen number \(N(f; X, A)\) defined by \textit{H. Schirmer} in [Pac. J. Math. 122, 459--473 (1986; Zbl 0553.55001)], which is a lower bound for the number of fixed points for all maps in the relative homotopy class of \(f\). The author of this paper defines some relative Lefschetz numbers and relative Reidemeister traces by using traces in bicategories with shadows. The author also illustrates how to compare her new invariants with existing relative Nielsen numbers. Moreover, a relative Lefschetz fixed point theorem and its inverse are given: If \((X,A)\) is a pair of compact manifolds such that \(\dim X\geq 3\) and \(\dim X - \dim A \geq 2\), then a relative map \(f: (X,A)\to (X,A)\) is relatively homotopic to a fixed point free map if and only if the relative Reidemeister trace of \(f\) is zero. The proof of this main theorem follows that in [\textit{J. R. Klein} and \textit{E. B. Williams}, Geom. Topol. 11, 939--977 (2007; Zbl 1132.57024)]. More detailed material about bicategories with shadows can be found in the author's book [Fixed point theory and trace for bicategories. Astérisque 333. Paris: Société Mathématique de France (SMF) (2010; Zbl 1207.18001)].
reviewed by: Xuezhi Zhao
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 55P25
zbMATH DE Number: 5877253
zbMATH Keywords: Reidemeister trace
zbMATH Keywords: Nielsen number
zbMATH Keywords: fixed point
zbMATH Keywords: Lefschetz number
zbMATH Keywords: bicategory
author: Kate Ponto
MaRDI profile type: Publication
OpenAlex ID: W3106122217
arXiv ID: 0906.0762
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DOI: 10.2140/AGT.2011.11.839
description: scientific article