Fold-Forms for Four-Folds

Details Fold-Forms for Four-Folds Fold-Forms for Four-Folds

2009-09-22

arxiv.org/abs/0909.4067v1

Mathematics

Symplectic Geometry

16 pages, no figures, to appear in the Journal of Symplectic Geometry

Scientific paper

This paper explains an application of Gromov's h-principle to prove the existence, on any orientable 4-manifold, of a folded symplectic form. That is a closed 2-form which is symplectic except on a separating hypersurface where the form singularities are like the pullback of a symplectic form by a folding map. We use the h-principle for folding maps (a theorem of Eliashberg) and the h-principle for symplectic forms on open manifolds (a theorem of Gromov) to show that, for orientable even-dimensional manifolds, the existence of a stable almost complex structure is necessary and sufficient to warrant the existence of a folded symplectic form.

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