I made several attempts to articulate the connections between manifold theory and semiotics in the context of numerous list discussions that interwove each other through diverse nooks of the web in the early years of the present millennium.
Here is one place where I sifted out the more settled portions of several earlier discussions and reposted them in a more organized fashion to the Inquiry, Ontology, and Peirce lists, among a few others.
Excerpts from Serge Lang, Differential and Riemannian Manifolds,
Springer-Verlag, New York, NY, 1995.
Chapter 2. Manifolds
2.1. Atlases, Charts, Morphisms
2.2. Submanifolds, Immersions, Submersions
Using the concepts and terminology from Lang’s text, I explained the connection between manifold theory and semiotics in the following way.