Estimates of the isovariant Borsuk-Ulam constants of connected compact Lie groups
type: Item
instance of: scholarly article
title: Estimates of the isovariant Borsuk-Ulam constants of connected compact Lie groups
zbMATH Open document ID: 1406.57030
author: Ikumitsu Nagasaki
publication date: 2018-12-21T00:00:00Z
review text: A \(G\)-equivariant map \(f : X \to Y\) is called \(G\)-isovariant if \(f\) preserves the isotropy subgroups, that is, \(G_{f(x)} = G_x\) for every \(x \in X\). Isovariant maps play crucial roles in equivariant topology and transformation groups. Determining existence or non-existence of isovariant maps and their classification up to isovariant homotopy are important problems in the subject. In the paper under review, the author investigates non-existence of \(G\)-isovariant maps between \(G\)-representation spheres. The isovariant version of the Borsuk-Ulam theorem was first studied by \textit{A. G. Wasserman} in [Topology Appl. 38, No. 2, 155--161 (1991; Zbl 0736.55001)]. A compact Lie group \(G\) is called a Borsuk-Ulam group if the isovariant Borsuk-Ulam theorem holds for \(G\)-representations; namely, if the inequality \[ \dim V- \dim V^G \leq \dim W - \dim W^G \] holds whenever there exists a \(G\)-isovariant map \(f : S(V ) \to S(W)\). It is known that finite solvable groups and compact tori are Borsuk-Ulam groups. However, not much is known about this property for other compact groups. As an approach to this problem, the author defines the isovariant Borsuk-Ulam constant \(c_G\) for a (nontrivial) compact Lie group \(G\) as the supremum of \(c \in \mathbb{R}\) such that \[ c(\dim V - \dim V^G) \leq \dim W- \dim W^G \] holds whenever there exists a \(G\)-isovariant map \(f : S(V ) \to S(W)\). The author discusses some properties of \(c_G\) and establishes lower bounds in case of connected compact Lie groups. This leads us to some Borsuk-Ulam type results for isovariant maps. One of the main results of the paper is: Theorem. Let \(G\) be a compact connected Lie group and \(\{G_i\}\) all simple factors of \(G\). Then \(c_G = \min_i\{c_{G_i} \}\).
reviewed by: Mahender Singh
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 22E99
zbMATH DE Number: 6996442
zbMATH Keywords: isovariant map
zbMATH Keywords: Borsuk-Ulam theorem
zbMATH Keywords: Borsuk-Ulam constant
zbMATH Keywords: transformation groups
zbMATH Keywords: representation theory
MaRDI profile type: Publication
full work available at URL: s10114-018-6437-y
OpenAlex ID: W2792909485
Wikidata QID: Q115385200
cites work: On the existence of Borsuk-Ulam theorems
cites work: Topological methods for variational problems with symmetries
cites work: A Borsuk-Ulam theorem for compact Lie group actions
cites work: Q3508457
cites work: Q3704110
cites work: Q5658847
cites work: Diagram cohomology and isovariant homotopy theory
cites work: Äquivariante Whiteheadtorsion
cites work: Introduction to Lie Algebras and Representation Theory
cites work: The weak isovariant Borsuk-Ulam theorem for compact Lie groups
cites work: Q5401186
cites work: A Hopf type classification theorem for isovariant maps from free \(G\)-manifolds to representation spheres
cites work: Q3563461
cites work: Fixed-point sets of group actions on finite acyclic complexes
cites work: Smooth compact Lie group actions on disks
cites work: The classification of 𝐺-spaces
cites work: Isovariant homotopy equivalences of manifolds with group actions
cites work: Isovariant maps and the Borsuk-Ulam theorem
DOI: 10.1007/S10114-018-6437-Y
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6996442