Rational homotopy theory and geometric structures on smooth manifolds
type: Item
instance of: scholarly article
title: Rational homotopy theory and geometric structures on smooth manifolds
zbMATH Open document ID: 0830.55006
DOI: 10.1016/0034-4877(94)90013-2
published in: Reports on Mathematical Physics
publication date: 1996-02-05T00:00:00Z
review text: This work is devoted to geometric applications of rational homotopy theory and consists of two parts. The first part (sections 1-4) describes briefly the rational homotopy correspondence \[ M: {\mathfrak I}\to k- DGA_{(c)} \tag{1} \] and contains a survey of geometric results obtained by (1). We stress the technical aspects of the theory, with the aim of clarifying its methods and applications to physicists. The second part (sections 5 and 6) describes some results of the author on geometric applications of minimal models and contains an explicit exposition of rational homotopy theory of homogeneous spaces. In the author's opinion, there is no explicit exposition of calculational methods in the case of homogeneous spaces in the literature, and this attempt to systematize the theory may be useful for the broader audience. Roughly speaking, rational homotopy theory can be described as the correspondence (1), containing all rational algebraic-topology invariants of a simplicial complex \(X\in {\mathfrak I}\) (here and everywhere below \({\mathfrak I}\) denotes the category of simplicial complexes). Namely, \(M(X)\) determines the Postnikov tower of \(X\) tensored by \(Q\). In spite of the `simplicial' formulation, the theory of rational homotopy type is of great value for differential geometry (and therefore, for mathematical physics) because the minimal model \({\mathcal M}_\Omega\) of the de Rham algebra \(\Omega_{DR} (M)\) of a smooth manifold \(M\) normally contains more information than the cohomology algebra \(H^* (M)\), but in many cases is calculable. The original ideals of rational homotopy theory are due to \textit{D. Sullivan} [Inst. Haut. Étud. Sci., Publ. Math. 47(1977), 269-331 (1978; Zbl 0374.57002)], \textit{K.-T. Chen} [Algebra, Topol., Category Theory; Collect. Pap. Honor S. Eilenberg, 19-32 (1976; Zbl 0341.57034)], and to \textit{D. Quillen}. The first geometric applications were noticed by D. Sullivan and M. Vigué-Porrier in the theory of closed geo…
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 57T20
zbMATH DE Number: 639201
zbMATH Keywords: rational homotopy theory
zbMATH Keywords: minimal models
zbMATH Keywords: rational homotopy theory of homogeneous spaces
zbMATH Keywords: Postnikov tower
zbMATH Keywords: formality
zbMATH Keywords: Massey products
zbMATH Keywords: formality and geometric structures
author: Aleksy Tralle
MaRDI profile type: Publication
full work available at URL: 0034-4877(94)90013-2
OpenAlex ID: W2059485114
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description: scientific article; zbMATH DE number 639201