Some notes on spaces realized as classifying spaces
type: Item
instance of: scholarly article
title: Some notes on spaces realized as classifying spaces
author: Yang Bai
author: Liu Xiugui
author: Sang Xie
published in: Topology and its Applications
publication date: 2024-09-27T00:00:00Z
review text: Let \( B\mathrm{aut}_1 X\) denote the classifying space for the monoid of self homotopy equivalences of \(X\) which are homotopic to the identity on \(X\). The authors show that a certain class of rational spaces \(Y\), for which \(\pi_*(Y) \otimes \mathbb{Q}\) has dimension \(2\), cannot be realized up to rational homotopy as the classifying space \( B\mathrm{aut}_1 X\) of a space \(X\) for which \(\pi_*(X)\otimes \mathbb{Q}\) is finite dimensional. It is also shown that the Eilenberg-MacLane space \( K(\mathbb{Q}^r, n) \) (\( r\geq 2, n \geq 2\)) can be realized up to rational homotopy as the classifying space of an elliptic space, that is, a space \(X\) for which both \( \pi_*(X) \otimes \mathbb{Q}\) and \(H^*(X, \mathbb{Q})\) are finite dimensional, if and only if \( X = \prod_r S^{n-1}\), with \(n\) even.
reviewed by: Jean Baptiste Gatsinzi
Mathematics Subject Classification ID: 55P62
zbMATH DE Number: 7919656
zbMATH Keywords: rational homotopy theory
zbMATH Keywords: classifying spaces
zbMATH Keywords: Sullivan minimal models
zbMATH Keywords: elliptic spaces
zbMATH Keywords: higher order Whitehead products
zbMATH Keywords: \( L_\infty \)-algebras
MaRDI profile type: Publication
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DOI: 10.1016/J.TOPOL.2024.109030
description: scientific article; zbMATH DE number 7919656