# Symplectic domination

## TL;DR

This paper constructs a symplectic manifold of the same dimension as a given manifold, with a map of positive degree, using deep results from Riemannian geometry and symplectic topology.

## Contribution

It introduces a novel construction of symplectic manifolds related to negatively curved manifolds and symplectic divisors, expanding methods in symplectic topology.

## Key findings

- Constructs a symplectic manifold of the same dimension as the original manifold.
- Establishes a map of strictly positive degree from the symplectic manifold to the original.
- Utilizes deep theorems from Ontaneda and Donaldson to achieve the construction.

## Abstract

Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is a theorem of Ontaneda that gives a Riemannian manifold N of tightly pinched negative curvature which admits a map to M of degree equal to one; the second is a result of Donaldson on the existence of symplectic divisors. Given Ontaneda's negatively curved manifold N, the twistor space Z is symplectic. The manifold S is then a suitable multisection of the twistor space, found via Donaldson's theorem.