We investigate the existence of symplectic non-Kählerian structures on compact solvmanifolds and prove some results which give strong necessary conditions for the existence of Kählerian structures in terms of rational homotopy theory. Our results explain known examples and generalize the Benson-Gordon theorem (Benson and Gordon (1990); our method allows us to drop the assumption of the complete solvability of G).
We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a simply connected closed manifold . We prove that the loop homology of is isomorphic to the Hochschild cohomology of the cochain algebra with coefficients in . Some explicit computations of the loop product and the string bracket are given.
Given a relative map f: (X,A) → (X,A) on a pair (X,A) of compact polyhedra and a closed subset Y of X, we shall give some criteria for Y to be the fixed point set of some map relatively homotopic to f.