  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.
