Maps with symplectic graphs

Abstract

We consider the homotopy type of maps between symplectic surface whose graphs form symplectic submanifolds of the product. We give a purely topological model for this space in terms of maps with constrained numbers of pre-images. We use this to show that the dependence of the homotopy type on the area forms of each surface is quantized- it changes only when the parameters pass certain discrete levels. When the domain is a sphere or torus, and its total area is smaller than the range, we compute the full homotopy type of the low degree components. We also give an example, showing that the homotopy type of the space of sections of a symplectic fibration F must sometimes change as we deform F. Much of this work generalizes to n-dimensional manifolds equipped with volume forms.