Moduli spaces for generic low-dimensional complexes
type: Item
instance of: scholarly article
title: Moduli spaces for generic low-dimensional complexes
zbMATH Open document ID: 0833.55010
DOI: 10.1016/0022-4049(94)90115-5
author: Barbu Berceanu
author: Ştefan Papadima
published in: Journal of Pure and Applied Algebra
publication date: 1994-09-22T00:00:00Z
review text: A space (DGA) \(S\) \(((A,d))\) is marked if there is a specified isomorphism \(H^* (S; \mathbb{Q}) \cong H\) \((H^* (A, d)\) \(\cong H)\) for a given algebra \(H\). In this paper, the authors describe a moduli space classifying marked homotopy types of spaces (DGA's). The moduli space is defined in terms of differentials on the rational homotopy theoretic bigraded model associated to \(H\). Because the structure of the bigraded model is rather complicated in high degrees, the authors focus on situations where only low dimensional information is required. In particular, they introduce notions of aspheric and \({\mathbf L}\)-fibred algebras in order to make low dimensional information (e.g. the Maltsev completion of \(\pi_1\)) deterministic. Indeed, then a `finite part' of the moduli space may be used in place of the full moduli space for certain purposes of classification. A main result proved here is the fact that, given an aspheric and \({\mathbf L}\)-fibred cohomology algebra for marked spaces \(S\) and \(S'\), the `finite' moduli space description of equivalence of \(S\) and \(S'\) corresponds to isomorphism of appropriate lower central series quotients of the Mal'tsev completions of \(\pi_1 (S)\) and \(\pi_1 (S')\). The authors apply these ideas to construct invariants for certain classes of links whose link algebras are \({\mathbf L}\)-fibred and aspheric.
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 643789
zbMATH Keywords: differential graded algebras
zbMATH Keywords: fundamental group
zbMATH Keywords: marked homotopy types
zbMATH Keywords: differentials on the rational homotopy theoretic bigraded model
zbMATH Keywords: aspheric
zbMATH Keywords: \({\mathbf L}\)-fibred algebras
zbMATH Keywords: lower central series quotients of the Mal'tsev completions
zbMATH Keywords: links
reviewed by: John Oprea
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90115-5
OpenAlex ID: W2051737640
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description: scientific article; zbMATH DE number 643789