Rational homotopy theory and geometric structures on smooth manifolds
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 geodesics, by D. Sullivan in the theory of Riemannian symmetric spaces and by P. Deligne, P. Griffiths, J. Morgan and D. Sullivan in Kählerian geometry. In the author's opinion, further investigations in these directions are promising. The paper is organized as follows: 1. Introduction; 2. Algebraic part: minimal models, formality, Massey products, calculational techniques; 3. Brief description of the correspondence \({\mathfrak I}\to k- MDGA_{(c)}\); 4. Geometric part: formality and geometric structures; 5. Minimal models of homogeneous spaces; 6. Minimal models of fat bundles over compact symplectic manifolds.
rational homotopy theory of homogeneous spaces
scientific article; zbMATH DE number 639201
scientific article