Combinatorial homotopy of simplicial complexes and complex information systems

Kramer, Xenia; Laubenbacher, Reinhard

doi:10.1090/psapm/053/1602359

Cited by 11 publications

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“…Discrete homotopy theory is a discrete analogue of homotopy theory, associating a bigraded sequence of groups to a simplicial complex, capturing its combinatorial structure, rather than its topological structure. Originally called A-theory, it was developed in [13], [4], [5], and [3], which built on the work of Atkin [1], [2]. Discrete homotopy theory can be equivalently defined for finite connected graphs, resulting in an algebraic invariant of finite connected graphs and graph homomorphisms, in the same way that classical homotopy theory gives invariants of topological spaces and continuous maps: it associates a sequence A i (G) of groups to a finite connected graph.…”

Section: Introductionmentioning

confidence: 99%

Discrete homology theory for metric spaces

Barcelo

Capraro

White

2014

Bulletin of the London Mathematical Society

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Abstract. We define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an n-dimensional cube to a fixed metric space. We prove that the resulting homology theory satisfies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a 'fine enough' rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space.

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Section: Introductionmentioning

confidence: 99%

Discrete homology theory for metric spaces

Barcelo

Capraro

White

2014

Bulletin of the London Mathematical Society

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“…Subsequently, Q-analysis has been used in a variety of social and biological settings. As part of a research project on decision networks, Kramer and Laubenbacher, following suggestions in Atkin's papers, began developing a general connectivity theory of simplicial complexes, modeled on classical higher homotopy theory of spaces [6]. In this theory, the invariants of Q-analysis play the role of the set of connected components of the space.…”

Section: Introductionmentioning

confidence: 99%

Foundations of a Connectivity Theory for Simplicial Complexes

Barcelo

Kramer

Laubenbacher

et al. 2001

Advances in Applied Mathematics

119

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This paper lays the foundations of a combinatorial homotopy theory, called A-theory, for simplicial complexes, which reflects their connectivity properties. A collection of bigraded groups is constructed, and methods for computation are given. A Seifert-Van Kampen type theorem and a long exact sequence of relative A-groups are derived. A related theory for graphs is constructed as well. This theory provides a general framework encompassing homotopy methods used to prove connectivity results about buildings, graphs, and matroids.

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“…(3) (J 1 , ⊡)-homotopy restricts to Gph where it is called A-homotopy or discrete homotopy ( [20,5,4,2]). (4) (J + , ×)-homotopy restricts to DiGph.…”

Section: 5mentioning

confidence: 99%

“…(Symmetric) reflexive relations are equivalent to simple (undirected) graphs. Discrete homotopy theory, originally called A-theory, was developed for undirected graphs [20,5,4,2]. Barcelo, Capraro and White [3] developed a discrete homology theory for metric spaces that is compatible with the discrete homotopy theory.…”

Section: Introductionmentioning

confidence: 99%

Homological Algebra for Persistence Modules

Bubenik

Milićević

2021

Found Comput Math

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We useČech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain six homology and homotopy theories of closure spaces. 1 Closure spacesIn this section we provide background on EduardČech's closure spaces [9,25].2.1. Elementary definitions. We start with the elementary definitions we need to work with closure spaces. Definition 2.1. Let X be a set and let P(X) denote the collection of subsets of X. A function c : P(X) → P(X) is called a closure operation (or just closure) for X if the following axioms are satisfied:for all A, B ⊆ X. An ordered pair (X, c) where X is a set and c a closure for X is called a (Čech) closure space. Elements of X are called points.Lemma 2.2. Let (X, c) be a closure space. If A ⊆ B are subsets of X then c(A) ⊆ c(B). Proof. Note thatExample 2.3. Let X be a set. Then the identity map 1 P(X) : P(X) → P(X) is a closure operation for X. It is called the discrete closure for X. The closure operation defined by the map A → X for A = ∅ and ∅ → ∅ is called the indiscrete closure for X.Definition 2.5. Given a set X and a closure c for X we can associate to it an interior operation for X, int c : P(X) → P(X) given by int c (A) := X − c(X − A).The interior operation satisfies the following:1) int c (X) = X, 2) For all A ⊆ X, int c (A) ⊆ A, 3) For all A, B ⊆ X, int c (A ∩ B) = int c (A) ∩ int c (B). If int : P(X) → P(X) is a function satisfying the 3 conditions in Definition 2.5 and one defines c int : P(X) → P(X) by c int (A) := X − int(X − A), then c int is a closure operation for X.Proposition 2.6. [9, 14.A.12] A subset A of a closure space (X, c) is open if and only if int c (A) = A.Definition 2.7. [9, Definition 14.B.1] Let (X, c) be a closure space. Let A ⊆ X. We say B ⊆ X is a neighborhood of A if A ⊆ X − c(X − B) = int c (B). In the case that A = {x}, we say B is a neighborhood of x. The neighborhood system of A is the collection of all neighborhoods of A. Definition 2.8. [9, Definition 16.A.1] Let (X, c X ) and (Y, c Y ) be closure spaces. A map). If f is continuous at every point of X we say f is continuous. Equivalently, f is continuous if for every A ⊆ X, f (c X (A)) ⊆ c Y (f (A)). A continuous map f is called a homeomorphism if f is a bijection with a continuous inverse.

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Combinatorial homotopy of simplicial complexes and complex information systems

Cited by 11 publications

References 0 publications

Discrete homology theory for metric spaces

Discrete homology theory for metric spaces

Foundations of a Connectivity Theory for Simplicial Complexes

Homological Algebra for Persistence Modules

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