Local matrix homotopies and soft tori

Abstract

We present solutions to local connectivity problems in matrix representations of the form $C\left(\right[-1,1{]}^N)\to C^{\ast }(u_{\epsilon },v_{\epsilon })$, with $C_{\epsilon }\left({\mathbb{T}}^2\right)\twoheadrightarrow C^{\ast }(u_{\epsilon },v_{\epsilon })$ for any $\epsilon \in [0,2]$ and any integer $n\ge 1$, where $C^{\ast }(u_{\epsilon },v_{\epsilon })\subseteq M_n$ is an arbitrary matrix representation of the universal $C^{\ast }$-algebra $C_{\epsilon }\left({\mathbb{T}}^2\right)$ that denotes the 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.

To deal with the locality constraints, we have combined some techniques introduced in this article with some techniques from matrix geometry, combinatorial optimization, and classification and representation theory of $C^{\ast }$-algebras.