Homotopy types and Nielsen numbers of periodic homotopy idempotents on tori
type: Item
instance of: scholarly article
title: Homotopy types and Nielsen numbers of periodic homotopy idempotents on tori
zbMATH Open document ID: 1327.55003
author: Yasuzo Nishimura
author: N. Nishimura
published in: Journal of Fixed Point Theory and Applications
publication date: 2015-10-07T00:00:00Z
review text: Let \(f:X\to X\) be a continuous self-map of a compact polyhedron, and let \(k\) be a positve integer. \(f\) is a periodic homotopy idempotent of period \(k\), if \(f^r\) is homotopic to \(f^{r+k}\) for some integer \(r\geq 0\). It is a periodic homotopy equivalence if \(f^k\) is homotopic to the identity. The authors determine the Nielsen numbers of periodic homotopy idempotents and equivalences on the \(n\)-dimensional torus \(T^n\). For example, if \(f: T^4\to T^4\) is a periodic equivalence, then \(N(f)\) is \(0,1, 2,3,4,5,6,8,9,12\) or \(16\). If \(f:T^6\to T^6\) is a periodic homotopy idempotent, then \(N(f)\) is \(0,1,3,5,7,9,15\) or \(27\). In the case \(n=2\) the authors classify the homotopy types of periodic idempotents using a number-theoretic discussion.
reviewed by: Zdzisław Dzedzej
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 11D09
zbMATH DE Number: 6490682
zbMATH Keywords: periodic homotopy idempotent
zbMATH Keywords: Nielsen number
zbMATH Keywords: torus
Wikidata QID: Q114221822
MaRDI profile type: Publication
full work available at URL: s11784-013-0124-5
OpenAlex ID: W2098493848
cites work: Nielsen Numbers of Maps of Tori
cites work: Q5619504
cites work: Q3935145
cites work: Splitting homotopy idempotents which have essential fixed points
cites work: Homotopy periodicity and coherence
cites work: Lectures on Nielsen fixed point theory
cites work: Q4820492
cites work: Q5668937
cites work: On the Diophantine Equation mX 2 - nY 2 = ± 1
DOI: 10.1007/S11784-013-0124-5
description: scientific article; zbMATH DE number 6490682