On the Topology of T-manifolds of Higher Codimension

TL;DR

This paper explores the topology of T-manifolds of various codimensions constructed via combinatorial patchworking, providing new bounds on their connected components and introducing a novel patchworking method for codimension 2.

Contribution

It offers improved bounds on the number of connected components of T-manifolds and presents a new Viro-style patchworking approach for codimension 2.

Findings

New bounds on connected components of T-curves and T-surfaces.

A novel patchworking method for T-manifolds of codimension 2.

Construction of maximal real algebraic curves in RP^3.

Abstract

This paper undertakes the study of the topology of T-manifolds of arbitrary codimension obtained by combinatorial patchworking with real phase structure as described by Brugall\'e, L\'opez de Medrano and Rau (2024). We prove new bounds on the number of connected components of T-curves and T-surfaces. For sufficiently high codimension, this improves the results of Brugall\'e, L\'opez de Medrano and Rau (2024). In addition, we present a new description of patchworking \`a la Viro for T-manifold of codimension 2. We use this method to construct a family of maximal real algebraic curves in $\mathbb RP^3$.