arXiv subjects
Publications and source records attributed to Jonathan Treviño-Marroquín.
We establish a connection between the graphs and their geometric realizations of their clique complexes, proving that graphs with their canonical pseudotopological structure, are weakly homotopy equivalent to the geometric realization of their clique complex.
Pseudotopological spaces are the Cartesian closed hull of the category of Čech closure spaces. In this paper, we give a direct proof that the model category of the pseudotopological spaces constructed by Rieser is Quillen equivalent to the category of simplicial sets. In addition to noting that every pseudotopological space is weak homotopy equivalent to a topological CW complex, we prove that any weak equivalence of pseudotopological spaces can be converted to a weak equivalence of topological spaces. Finally, combining these ingredients, we construct rational homotopy for simply connected pseudotopological spaces. In this paper, we also prove that the cartesian product of two pseudotopological CW complexes is a CW complex.
In algebraic topology, the fundamental groupoid is a classical homotopy invariant which is defined using continuous maps from the closed interval to a topological space. In this paper, we construct a semi-coarse version of this invariant, using as paths a finite sequences of maps from $\mathbb{Z}_1$ to a semi-coarse space, connecting their tails through semi-coarse homotopy. In contrast to semi-coarse homotopy groups, this groupoid is not necessarily trivial for coarse spaces, and, unlike coarse homotopy, it is well-defined for general semi-coarse spaces. In addition, we show that the semi-coarse fundamental groupoid which we introduce admits a version of the Van Kampen Theorem.
We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like' structures to compact metric spaces, something which is impossible in coarse geometry. We first study homotopy in this context, and we then construct homology groups which are invariant under semi-coarse homotopy equivalence. We further show that any undirected graph $G=(V,E)$ induces a semi-coarse structure on its set of vertices $V_G$, and that the respective semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of undirected graphs.
Limit and Pseudotopological spaces are two generalizations of topological spaces which are defined by indicating what filters converge under some axioms. In this article, we introduce covering spaces and set forth some necessary conditions for a construction for a universal covering space.
We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second homotopy group from algebraic topology. We illustrate that our approach can be effective by computing this (digital) second homotopy group for a digital 2-sphere.