Spaces of polynomials without 3-fold real roots.
type: Item
instance of: scholarly article
title: Spaces of polynomials without 3-fold real roots.
zbMATH Open document ID: 1039.55007
author: Koichi Hirata
author: Kohhei Yamaguchi
published in: Journal of Mathematics of Kyoto University
publication date: 2002-01-01T00:00:00Z
review text: Let \(P_n^d(\mathbb{R})\) denote the space of all monic real coefficient polynomials of degree \(d\) that have no real roots of multiplicity \(\geq n\). The authors study the homotopy type of this space for \(n=3\). The loop space \(\Omega S^{n-1}\) has, up to homotopy, a natural CW complex structure whose \(k(n-2)\)-skeleton is called the James \(k\)-th stage filtration and is denoted \(J_k(\Omega S^{n-1})\). It is known that there is a homotopy equivalence \(P_n^d(\mathbb{R})\simeq J_{[d/n]}(\Omega S^{n-1})\) for \(n\geq 4\) (induced by a map called the jet embedding); the authors ask if this holds for \(n=3\). They answer their question affirmatively when \(1\leq d \leq 17\). In addition, they show that the jet embedding is \([d/3]\)-connected and induces an integral homology equivalence (for \(n=3\) and all \(d\geq 1\)).
reviewed by: Bruce Hughes
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55P15
zbMATH DE Number: 1998378
zbMATH Keywords: spaces of polynomials
zbMATH Keywords: loop spaces
zbMATH Keywords: James filtration
zbMATH Keywords: jet embedding
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283847
OpenAlex ID: W1576084894
DOI: 10.1215/KJM/1250283847
description: scientific article; zbMATH DE number 1998378