On the Topology of T-manifolds of Higher Codimension

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 Brugallé, López 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 Brugallé, López de Medrano and Rau (2024). In addition, we present a new description of patchworking à la Viro for T-manifold of codimension 2. We use this method to construct a family of maximal real algebraic curves in $\mathbb RP^3$.