#### Publication Date

8-25-2016

#### Abstract

In this document we solve some local connectivity problems in matrix representations of the form C(T^N) -> M_n and C(T^N) -> M_n <- C([-1, 1]^N) using the so called toroidal matrix links, which can be interpreted as normal contractive matrix analogies of free homotopies in algebraic topology. In order to deal with the locality constraints, we have combined some techniques introduced in this document with several versions of the Basic Homotopy Lemma L.2.3.2, T.2.3.1 and C.2.3.1 obtained initially by Bratteli, Elliot, Evans and Kishimoto in [4] and generalized by Lin in [19] and [22]. We have also implemented some techniques from matrix geometry, combinatorial optimization and noncommutative topology developed by Loring [24, 27], Shulman [27], Bhatia [2], Chu [8], Brockett [5], Choi [7, 6], Effros [6], Exel [11], Eilers [11], Elsner [12], Pryde [31, 30], McIntosh [30] and Ricker [30].

#### Degree Name

Mathematics

#### Level of Degree

Doctoral

#### Department Name

Mathematics & Statistics

#### First Advisor

Loring, Terry A.

#### First Committee Member (Chair)

Buium, Alexandr

#### Second Committee Member

Boyer, Charles

#### Third Committee Member

Packer, Judith

#### Language

English

#### Keywords

Matrix homotopy, relative lifting problems, matrix representation, noncommutative semialgebraic sets, K-theory, amenable C*-algebra, joint spectrum.

#### Document Type

Dissertation

#### Recommended Citation

Vides Romero, Fredy Antonio. "Toroidal Matrix Links: Local Matrix Homotopies and Soft Tori." (2016). https://digitalrepository.unm.edu/math_etds/51