Symplectic Origami

TL;DR

This paper introduces origami manifolds, a class of manifolds with a folded symplectic structure, and explores their properties, classifications, and relationships with symplectic manifolds through cutting and folding techniques.

Contribution

It establishes an origami convexity theorem for torus actions, classifies toric origami manifolds via paper-like polyhedral objects, and analyzes their cobordism and cohomology.

Findings

Proved an origami convexity theorem for Hamiltonian torus actions

Classified toric origami manifolds using paper origami-like polyhedra

Computed cohomology for a special class of origami manifolds

Abstract

An origami manifold is a manifold equipped with a closed 2-form which is symplectic except on a hypersurface where it is like the pullback of a symplectic form by a folding map and its kernel fibrates with oriented circle fibers over a compact base. We can move back and forth between origami and symplectic manifolds using cutting (unfolding) and radial blow-up (folding), modulo compatibility conditions. We prove an origami convexity theorem for hamiltonian torus actions, classify toric origami manifolds by polyhedral objects resembling paper origami and discuss examples. We also prove a cobordism result and compute the cohomology of a special class of origami manifolds.