Lusternik-Schnirelmann category of products with half-smashes
type: Item
instance of: scholarly article
title: Lusternik-Schnirelmann category of products with half-smashes
zbMATH Open document ID: 1439.55002
author: Donald Stanley
author: Jeffrey Strom
published in: Algebraic \& Geometric Topology
publication date: 2020-04-07T00:00:00Z
review text: The Lusternik-Schnirelmann category (or just category) of a space \(X\), denoted as \(\mbox{cat}(X),\) is the least nonnegative integer \(n\) such that \(X\) admits a cover constituted by \(n + 1\) open subsets each of which is contractible to a point in \(X\). The original motivation of this numerical homotopy invariant is that, for any closed smooth manifold \(M\), \(\mbox{cat}(M)\) provides a lower bound of the number of critical points of any smooth map \(f : M\rightarrow \mathbb{R}.\) In the paper under review the authors prove that for any connected CW-complex \(X\) and any well-pointed path-connected space \(A\) satisfying a certain inequality which involves the dimension of \(X\), the connectivity of both spaces \(X\) and \(A\), and the category \(\mbox{cat}(X\times A)\), the following equality holds \[\mbox{cat}(X\times A)=\mbox{cat}(X\times (A\rtimes B))\] for every unpointed space \(B\). Here \(A\rtimes B\) denotes the half-smash product \(A\wedge B_+\). As a corollary, if \(n\) is less than or equal to three times plus one the connectivity of a well-pointed space \(A\), then \[\mbox{cat}(S^n\times (A\rtimes B))=\mbox{cat}(S^n\times A)\] for every unpointed space \(B\). This result enables the authors to provide a new list of examples that do not satisfy the Ganea condition.
Mathematics Subject Classification ID: 55M30
zbMATH DE Number: 7188029
zbMATH Keywords: Lusternik-Schnirelmann category
zbMATH Keywords: half-smash
zbMATH Keywords: Ganea conjecture
zbMATH Keywords: Ganea condition
zbMATH Keywords: cartesian product
zbMATH Keywords: homotopy theory
reviewed by: José Calcines
MaRDI profile type: Publication
full work available at URL: agt.2020.20.439
OpenAlex ID: W3009076980
cites work: The cone length and category of maps: pushouts, products and fibrations
cites work: A generalization of the homology and homotopy suspension
cites work: A proof of Ganea's conjecture for rational spaces
cites work: Ganea's Conjecture on Lusternik-Schnirelmann Category
cites work: Implications of the Ganea Condition
cites work: Rational L-S category and a conjecture of Ganea
cites work: Fibrewise construction applied to Ljusternik-Schnirelman category
cites work: Fibrewise suspension and Lusternik-Schnirelmann category
DOI: 10.2140/AGT.2020.20.439
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description: scientific article; zbMATH DE number 7188029