On the rational homotopical nilpotency index of principal bundles
type: Item
instance of: scholarly article
title: On the rational homotopical nilpotency index of principal bundles
zbMATH Open document ID: 1398.55006
author: Yanlong Hao
author: Liu Xiugui
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2018-02-06T00:00:00Z
review text: Let \(G\) be a compact connected topological group. For a principal \(G\)-bundle \(p : E \to X\) of simply connected CW complexes with \(E\) finite, let \(\mathrm{Aut}(X)\) denote the space of all self-fibre homotopy equivalences and let \(\mathrm{Aut}(p)_{0}\) be its rationalization. If the commutative differential graded \(\mathbb Q\)-algebra \((A, d)\) denotes a Sullivan model of \(X\), then it is well known that the minimal model of \(G\) is necessarily of the form \((\Lambda V, D)\) with \(V\) finite dimensional and concentrated in odd degrees. Then, the bundle \(p\) is modeled by an inclusion \((A, d) \to (A \otimes \Lambda V, D)\) for which \(DV \subset A\). A set \(\{v_{i} \}_{i\in I}\) generates the relative Sullivan algebra \((A \otimes \Lambda V, D)\), if \(v_{i} \notin A\), \(D(v_{i})\in A\) for each \(i\) and \(A\) and the linear space \(W\) spanned by \(\{v_{i} \}_{i\in I}\) generates multiplicatively \((A \otimes \Lambda V, D)\). Thus we can define a linear map of degree 1, \[ f:W\to H^{*}(A,d)\qquad f(v)=[Dv]. \] Let \(k\) be the number of dimensions \(m\) such that \(f:W^{m}\to H^{m+1}(A,d)\) is injective and let \(N(p)\) be the minimum of the different values \(k\) resulting from different choices of generators of \((A \otimes \Lambda V, D)\). The aim of the paper under review is to prove the following result \[ n-N(p)\leq \mathrm{Hnil}_{\mathbb Q}(\mathrm{Aut}(p)_{0})\leq n \] where \(n\) is the number of non-trivial rational homotopy groups of the space \(G\). Here Hnil\((\mathrm{Aut}(p)_{0})\) denotes the homotopical nilpotency of the space \(\mathrm{Aut}(p)_{0}\) defined as the least integer \(n\) such that the \((n+1)^{th}\)-commutator is nullhomotopic.
reviewed by: Mahmoud Benkhalifa
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55R80
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55P62
zbMATH DE Number: 6835197
zbMATH Keywords: fibre homotopy equivalence
zbMATH Keywords: Samelson-Lie algebra
zbMATH Keywords: Sullivan minimal model
zbMATH Keywords: derivation
MaRDI profile type: Publication
full work available at URL: s40590-016-0098-6
OpenAlex ID: W2326743691
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DOI: 10.1007/S40590-016-0098-6
description: scientific article; zbMATH DE number 6835197