Concomitantly, some new and stimulating conjectures and problems have been
formulated in the last several years due to this influx of homotopical ideas.
Examples include the Lupton-Oprea conjecture [LOl] and the Benson-Gordon
conjecture [BG2], both of which are in the spirit of some older and still unsolved
problems (e.g. Thurston's conjecture and Sullivan's problern). These results,
problems and conjectures are scattered in various research articles. In this work,
we intend to present them in a unified way, stressing geometric techniques
flavored with the spice of homotopy theory.
Before starting our presentation, we emphasize some particular features of
this work. Here, we collect a majority of known results on the problem of
constructing symplectic manifolds with uo Kahlerian structure. With this in mind,
uilinanifolds, solvmanifolds, fiber bundles and surgery techniques are discussed.
We also present some relevant homotopy theory, e.g. the Dolbeault rational
homotopy theory. We give many examples with the aim of claryfying methods of
rational homotopy theory to geometers and attracting the attention of
"rationalists" to some interesting geometric problems. As an example of the latter,
we mention the existence theorems for symplectic fat bundles [ANT, TrK]. This
book is meant to be a kind of "bridge" for mathematicians working in two
different research areas, so we give proofs (especially geometric ones) of background
material where we can while simply providing motivation (and references) where
detailed proofs would bring the narrative to a halt. Our explicit aim is to clarify
the interrelations between certain aspects of symplectic geometry and homotopy
theory, so we try to present as much of the geometry "hidden" behind algebraic
calculations as possible.