An algebraic model for \(G\)-simple homotopy types
type: Item
instance of: scholarly article
title: An algebraic model for \(G\)-simple homotopy types
zbMATH Open document ID: 0564.55010
DOI: 10.1007/BF01450698
author: Mel Rothenberg
author: Georgia Visnjic Triantafillou
published in: Mathematische Annalen
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163937
review text: The authors' aim is to construct algebraic models for G-complexes X (G finite acting on X with nonvoid 1-connected fixed point sets \(X^ H\), for any subgroup H) which are adequate for the description of G-(simple) homotopy types and then to obtain classification results for G-(simple) homotopy types up to finite ambiguity, along the lines indicated by \textit{D. Sullivan} [Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 269-331 (1978; Zbl 0374.57002)] for ordinary homotopy types. The generalization is twofold: first, in the passage to the equivariant case, they obtain G- versions of results due to \textit{D. Quillen} [Ann. Math., II. Ser. 90, 205-295 (1969; Zbl 0191.537)] and \textit{H. J. Baues} and \textit{J. M. Lemaire} [Math. Ann. 225, 219-245 (1977; Zbl 0322.55019)] and establish the finiteness of the G-homotopy types with given G-DG Lie minimal model, integral structure and torsion bound. The same result was previously obtained by the second author [Astérisque 113/114, 312-337 (1984; see the previous review)] in terms of G-DG algebra models. The second point (motivating the Lie approach) is the construction of a G-DG Lie simple model for finite G-complexes, which is used in the place of the minimal Lie model in order to obtain the corresponding finiteness result for G- simple homotopy types. Finiteness results for G-(simple) homotopy types of \(Z_{p^ k}\) actions on spheres and of semilinear finite group actions are derived.
reviewed by: Ştefan Papadima
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 57Q10
zbMATH DE Number: 3899745
zbMATH Keywords: equivariant differential graded Lie algebras
zbMATH Keywords: G-homotopy types
zbMATH Keywords: algebraic models for G-complexes
zbMATH Keywords: Lie minimal model
zbMATH Keywords: integral structure
zbMATH Keywords: torsion bound
zbMATH Keywords: finite G-complexes
zbMATH Keywords: finiteness result for G-simple homotopy types
zbMATH Keywords: actions on spheres
zbMATH Keywords: semilinear finite group actions
MaRDI profile type: Publication
cites work: On the chain algebra of a loop space
cites work: Q5562621
cites work: Arithmetic subgroups of algebraic groups
cites work: Homotopy limits, completions and localizations
cites work: Equivariant cohomology theories
cites work: Q3205985
cites work: Q3871593
cites work: Systems of Fixed Point Sets
cites work: Q4057814
cites work: Q5550390
cites work: Whitehead torsion
cites work: Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces
cites work: Rational homotopy theory
cites work: Q3862116
cites work: Infinitesimal computations in topology
cites work: Equivariant Minimal Models
cites work: Rationalization of Hopf G-spaces
cites work: Norms of Units in Group Rings
Recommended article: Q3678311
Recommended article: Algebraic models for homotopy types
Recommended article: Group-theoretic algebraic models for homotopy types
Recommended article: On algebraic models for homotopy 3-types
Recommended article: Algebraic models in the homotopy theory of classifying spaces
Recommended article: An explicit model for the homotopy theory of finite-type Lie \(n\)-algebras
Recommended article: An algebraic model for homotopy fibers
Recommended article: Models for torsion homotopy types
Recommended article: Q3410369
Recommended article: Homotopy \(G\)-algebra structure on the cochain complex of Hom-type algebras
description: scientific article; zbMATH DE number 3899745