Cochain model for thickenings and its application to rational LS-category
type: Item
instance of: scholarly article
title: Cochain model for thickenings and its application to rational LS-category
zbMATH Open document ID: 0964.55013
author: Pascal Lambrechts
published in: Manuscripta Mathematica
publication date: 2001-07-13T00:00:00Z
review text: To any finite CW complex \(X\) there are associated compact manifolds (with boundary), known as thickenings. For example, if \(X\) is a smooth manifold of dimension \(k\), then one can smoothly embed \(X\) in Euclidean space, \({\mathbb R}^{n+1}\) for any sufficiently large \(n\), and then take a tubular neighbourhood to obtain a so-called \((n+1)\)-thickening of \(X\). This paper focusses on the boundary (as a manifold) of a thickening. This is again a compact manifold that is denoted \(\partial M\). Clearly \(\partial M\) reflects the topology of \(X\), although in general it is not even determined up to homotopy type by \(X\). The author shows that, subject to some mild technical hypotheses on the type of thickening, the cochain complex of \(\partial M\) with coefficients in a field is determined, as a module, by that of \(X\). We refer to the article for the precise statement. A stronger result is proven in the rational homotopy setting: For a stable thickening (\(n \geq 2k\)), a rational model (in the sense of rational homotopy theory) for \(\partial M\) can be constructed directly from a certain type of rational model for \(X\). It follows from this that, for a stable thickening, the rational homotopy type of \(\partial M\) is determined by that of \(X\). The methods used to prove this result are the `semi-free DG module' techniques, as summarized in [\textit{Y. Félix, S. Halperin} and \textit{J.-C. Thomas}, in A handbook of algebraic topology, 829-865 (1995; Zbl 0868.55016)]. An intriguing corollary is developed by combining the results of this paper with those of [\textit{Y. Félix, S. Halperin} and \textit{J.-M. Lemaire}, Topology 37, No. 4, 749-756 (1998; Zbl 0897.55001)]: Let \(X\) be a finite CW complex, and \(\partial M\) the boundary of a stable thickening of \(X\). Then \(\text{cat}_0(X) = \text{e}_0(\partial M) - 1\). Here, \(\text{cat}_0(X)\) denotes rational category and \(\text{e}_0(\partial M)\) denotes the so-called rational Toomer invariant. Fro…
reviewed by: Gregory Lupton
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 57P10
Mathematics Subject Classification ID: 55M30
zbMATH DE Number: 1556760
zbMATH Keywords: thickening
zbMATH Keywords: boundary of a thickening
zbMATH Keywords: cochain complex
zbMATH Keywords: rational homotopy type
zbMATH Keywords: Lyusternik-Shnirel'man category
zbMATH Keywords: rational Toomer invariant
MaRDI profile type: Publication
full work available at URL: s002290070017
OpenAlex ID: W2016574710
DOI: 10.1007/S002290070017
description: scientific article; zbMATH DE number 1556760