Lusternik-Schnirelmann category and the connectivity of \(X\)
type: Item
instance of: scholarly article
title: Lusternik-Schnirelmann category and the connectivity of \(X\)
zbMATH Open document ID: 1250.55001
author: Nicholas A. Scoville
published in: Algebraic \& Geometric Topology
publication date: 2012-05-16T00:00:00Z
Mathematics Subject Classification ID: 55M30
Mathematics Subject Classification ID: 55P05
zbMATH DE Number: 6035483
zbMATH Keywords: Lusternik-Schnirelmann category
zbMATH Keywords: categorical sequence
zbMATH Keywords: cone length
zbMATH Keywords: cofibration
reviewed by: Yuli Rudyak
MaRDI profile type: Publication
full work available at URL: agt.2012.12.435
OpenAlex ID: W2040737382
cites work: Q5558293
cites work: Introduction to homotopy theory
cites work: Q4421932
cites work: The Lusternik-Schnirelmann category of 𝑆𝑝(3)
cites work: Categorical sequences
cites work: Mixing categories
cites work: Higher order Whitehead products
cites work: The Approximation of Numbers as Sums of Reciprocals
cites work: Q3225696
cites work: Spaces with Lusternik-Schnirelmann category \(n\) and cone length \(n + 1\)
cites work: A minimum dimensional counterexample to Ganea's conjecture
cites work: Compositional Methods in Homotopy Groups of Spheres. (AM-49)
DOI: 10.2140/AGT.2012.12.435
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review text: The author continues to research a new numerical topological invariant of Lusternik-Schnirelmann type -- the \(\omega\)-weighted cone length \({\text{cl}}^{\omega}(X)\) -- introduced in [\textit{N. Scoville}, JP J. Geom. Topol. 11, No. 3, 209--233 (2011; Zbl 1237.55002)]NEWLINENEWLINETheorem. Let \(a\geq 1\) be an integer and \(q\) a rational number with \(q\geq 1/a\). Then for any \(\omega\) there exists an \(a\)-connected space \(X_q\) with \({\text{cl}}^{\omega}(X)=q\).NEWLINENEWLINEThis theorem is based on the following fact. Recall the definition of a categorical sequence for \(X\), see [\textit{R. Nendorf}, \textit{N. Scoville} and \textit{J. Strom}, Algebr. Geom. Topol. 6, 809--838 (2006; Zbl 1131.55001)].NEWLINENEWLINECorollary. Let \(X\) be a space with \({\text{cat}} X=n\) and let \(\sigma_X=\{m_1, \ldots, m_n\}\) be a categorical sequence. Then \(a+\displaystyle \sum_{k=1}^n 1/(m_k-1)\leq {\text{cl}}^{\omega}(X)\) where \(a=1\) for \(m_1=1\) and \(a=0\) otherwise. Applications to several spaces are presented.
description: scientific article; zbMATH DE number 6035483