Klein Foams

TL;DR

This paper studies Klein foams, showing their dianalytic functions match those on associated Klein surfaces, and constructs their moduli space as an analytic space with a specific topological structure.

Contribution

It establishes the equivalence of dianalytic function fields and constructs the moduli space of Klein foams as an analytic space.

Findings

Dianalytic functions on Klein foams coincide with those on Klein surfaces.

Constructed the moduli space of Klein foams as an analytic space.

Proved the set of topologically equivalent Klein foams forms a specific analytic space.

Abstract

Klein foams are analogues of Riemann and Klein surfaces with one-dimensional singularities. We prove that the field of dianalytic functions on a Klein foam $\Omega$ coincides with the field of dianalytic functions on a Klein surface $K_{\Omega}$. We construct the moduli space of Klein foams and we prove that the set of classes of topologically equivalent Klein foams form an analytic space homeomorphic to $\mathbb{R}^{n}/\mathsf{Mod}$, where $\mathsf{Mod}$ is a discrete group.