** Authors:**
ROBERT E. GOMPF

** Abstract: **
A topological structure is introduced that seems likely to provide a complete topological characterization of compact symplectic manifolds. The article
begins with a leisurely introduction to symplectic manifolds from a
topological viewpoint. It then focuses on Thurston's construction of a
symplectic structure on the total space of a fiber bundle. This is
generalized to a technique for putting a symplectic structure on the domain
of a J-holomorphic map. A topological structure called a hyperpencil on a
compact 2n-manifold is then defined; this is motivated by the special case of
a linear system of curves on an algebraic manifold, and it generalizes the
notion of a Lefschetz pencil on a 4-manifold (although the critical points of
a hyperpencil can be much more complicated). A deformation class of
hyperpencils determines an isotopy class of symplectic forms, via the above
generalization of Thurston's construction. This correspondence seems to be
essentially an inverse to the technique of Donaldson and Auroux for
constructing linear systems on symplectic manifolds. The likely end result
is that any symplectic form whose cohomology class is rational should be
realized up to scale by a hyperpencil. This would topologically characterize
symplectic manifolds as being those smooth manifolds admitting hyperpencils,
and put a dense subset of all symplectic forms on a manifold (up to scale and
isotopy) in bijective correspondence with the set of all hyperpencils on it
modulo a suitable equivalence relation.

** Keywords: **

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