Local Matrix Homotopies and Soft Tori

TL;DR

This paper introduces toroidal matrix links to solve local connectivity problems in matrix representations involving the Soft Torus, combining techniques from matrix geometry, optimization, and C*-algebra theory.

Contribution

It presents a novel approach using toroidal matrix links to address local connectivity in matrix representations of the Soft Torus, expanding the tools available in matrix algebra topology.

Findings

Solved local connectivity problems in matrix representations involving the Soft Torus.

Introduced toroidal matrix links as analogs of free homotopies in matrix form.

Combined techniques from matrix geometry, optimization, and C*-algebra classification.

Abstract

We present solutions to local connectivity problems in matrix representations of the form $C([-1,1]^{N}) \to A_{n,\varepsilon} \leftarrow C_{\varepsilon}(\mathbb{T}^{2})$ for any $\varepsilon\in[0,2]$ and any integer $n\geq 1$, where $A_{n,\varepsilon}\subseteq M_n$ and where $C_{\varepsilon}(\mathbb{T}^{2})$ denotes the {\bf Soft Torus}. We solve the connectivity problems by introducing the so called toroidal matrix links, which can be interpreted as normal contractive matrix analogies of free homotopies in differential algebraic topology. In order to deal with the locality constraints, we have combined some techniques introduced in this document with some techniques from matrix geometry, combinatorial optimization, classification and representation theory of C$^*$-algebras.