A Classification of Toric, Folded-Symplectic Manifolds

TL;DR

This paper classifies G-toric folded-symplectic manifolds with co-orientable fold hypersurfaces by associating them with unimodular maps with folds and characterizing their isomorphism classes via second cohomology groups.

Contribution

It introduces a classification framework for G-toric folded-symplectic manifolds using orbit space data and characteristic classes, extending classical results to the folded-symplectic setting.

Findings

Orbit space is a manifold with corners with a fold map to dual Lie algebra.

Isomorphism classes correspond to second cohomology groups with specific coefficients.

Generalizes Delzant's theorem and classifications of toric and origami manifolds.

Abstract

Given a $G$-toric, folded-symplectic manifold with co-orientable folding hypersurface, we show that its orbit space is naturally a manifold with corners $W$ equipped with a smooth map $\psi: W \to \frak{g}^*$, where $\frak{g}^*$ is the dual of the Lie algebra of the torus, $G$. The map $\psi$ has fold singularities at points in the image of the folding hypersurface under the quotient map to $W$ and it is a unimodular local embedding away from these points. Thus, to every $G$-toric, folded-symplectic manifold we can associate its orbit space data $\psi:W \to \frak{g}^*$, a unimodular map with folds. We fix a unimodular map with folds $\psi:W \to \frak{g}^*$ and show that isomorphism classes of $G$-toric, folded-symplectic manifolds whose orbit space data is $\psi:W \to \frak{g}^*$ are in bijection with $H^2(W; \mathbb{Z}_G\times \mathbb{R})$, where $\mathbb{Z}_G= \ker(\exp :\mathfrak{g} \to G)$ is the integral lattice of $G$. Thus, there is a pair of characteristic classes associated to every $G$-toric, folded-symplectic manifold. This result generalizes a classical theorem of Delzant, a classification of non-compact toric, symplectic manifolds due to Lerman and Karshon, and the classification of toric, origami manifolds, due to Cannas da Silva, Guillemin, and Pires, in the case where the folding hypersurface is co-orientable.