Applications of rational homotopy to geometry (results, problems, conjectures) (Q2365179)

During the last years, rational homotopy theory (especially Sullivan's minimal model) was applied successfully to the Thurston-Weinstein problem of constructing compact symplectic manifolds not admitting (positive definite) Kähler metrics [see, for example, the reviewer with \textit{Luis A. Cordero} and \textit{A. Gray}, Topology 25, 375-380 (1986; Zbl 0596.53030); the reviewer with \textit{M. J. Gotay} and \textit{A. Gray}, Proc. Am. Math. Soc. 103, 1209-1212 (1988; Zbl 0656.53034); \textit{K. Hasegawa}, ibid. 106, 65-71 (1989; Zbl 0691.53040); the reviewer with \textit{A. Gray} and \textit{J. W. Morgan}, Mich. Math. J. 38, 271-283 (1991; Zbl 0726.53028)], to Arnold's conjecture [\textit{C. McCord} and \textit{J. Oprea}, Topology 32, 701-717 (1993; Zbl 0798.58017)] as well as to some other geometric problems. In this paper, the author gives a clear presentation of the results proved in the above-mentioned papers and others, in a unified way, stressing geometric techniques. Moreover, some results of the author concerning Kähler solvmanifolds, published in Coll. Math. 73 (1997), are discussed in some detail. Also, relevant problems and conjectures concerning symplectic and Kähler geometry are considered.