The realization space of a \(\Pi\)-algebra: a moduli problem in algebraic topology
type: Item
instance of: scholarly article
title: The realization space of a \(\Pi\)-algebra: a moduli problem in algebraic topology
zbMATH Open document ID: 1054.55007
author: David Blanc
author: William Dwyer
author: Paul Goerss
published in: Topology
publication date: 2004-08-16T00:00:00Z
review text: Let \(\Pi\) denote the full subcategory of the pointed homotopy category on the finite wedges of spheres \(S^{n_{1}} \vee \dots \vee S^{n_{k}}\). A \(\Pi\)-algebra \(A\) is a contravariant set-valued functor defined on \(\Pi\) which takes wedges to products. Examples include the functors \(\pi_{\ast}X\) represented by pointed spaces \(X\) on the pointed homotopy category. The authors develop an obstruction theory for determining when a given \(\Pi\)-algebra \(A\) is realized by a space \(X\) in the sense that \(A \cong \pi_{\ast}X\). The main tool is a tower of partial moduli spaces of realizations which is effectively a Postnikov tower for the full moduli space of realizations of \(A\) in the category of simplicial spaces. The partial moduli are moduli of particular types of simplicial spaces. Simplicial spaces represent simplicial \(\Pi\)-algebras, and the Dwyer-Kan-Stover \(E_{2}\)-model structure for simplicial spaces is the natural context for manipulating their homotopy types, see [\textit{W.G. Dwyer, D.M. Kan} and \textit{C. R. Stover}, J. Pure Appl. Algebra 90, No. 2, 137--152 (1993; Zbl 0814.55008)].
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55S45
Mathematics Subject Classification ID: 55P65
zbMATH DE Number: 2091696
zbMATH Keywords: \(\Pi\)-algebras
zbMATH Keywords: simplicial spaces
MaRDI profile type: Publication
full work available at URL: j.top.2003.10.002
OpenAlex ID: W1990722760
DOI: 10.1016/J.TOP.2003.10.002
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description: scientific article; zbMATH DE number 2091696