Homotopy theory of dihedral and quaternionic sets
type: Item
instance of: scholarly article
title: Homotopy theory of dihedral and quaternionic sets
zbMATH Open document ID: 0965.55008
DOI: 10.1016/S0040-9383(99)00018-X
author: Jan Spaliński
published in: Topology
publication date: 2001-08-05T00:00:00Z
review text: A map \(f:A\to B\) of \(O(2)\)-spaces is called a weak equivalence if the induced map of fixed points \(f^H:A^H\to B^H\) is a weak equivalence for all finite cyclic and dihedral subgroups \(H\subset O(2)\). It is shown that the resulting homotopy category is equivalent to the homotopy category of dihedral sets as introduced by \textit{Z. Fiedorowicz} and \textit{J.-L. Loday} in [Trans. Am. Math. Soc. 326, No. 1, 57-87 (1991; Zbl 0755.18005)]. There is a similar relation between Pin(2)-spaces (where Pin(2) is the normalizer of \(S^1\) in \(S^3)\) and quaternionic sets. The central part of the paper is the construction of suitable subdivisions of simplicial sets. The results are proved via \textit{D. Quillen}'s theorem in [Ann. Math. (2) 90, 205-295 (1969; Zbl 0191.53702)] on adjoint functors between model categories inducing an equivalence between their homotopy categories.
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P10
zbMATH DE Number: 1436052
zbMATH Keywords: dihedral set
zbMATH Keywords: simplicial set
zbMATH Keywords: equivariant homotopy theory
zbMATH Keywords: model category
reviewed by: Klaus Heiner Kamps
MaRDI profile type: Publication
Wikidata QID: Q126571712
description: scientific article; zbMATH DE number 1436052