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Semi-coarse Spaces: Fundamental Groupoid and the van Kampen Theorem

Jonathan Treviño-Marroquín

TL;DR

The paper introduces a semi-coarse fundamental groupoid for semi-coarse spaces by leveraging tails of symmetric maps from $\\mathbb{Z}_1$ and a refined string calculus. It defines a robust invariant $\\pi_{\\leq 1}(X)$ via delete-one-point and opposite-map relations, and shows functoriality with respect to bornologous maps. Because coarse spaces collapse many invariants, the authors develop a relative, tail-controlled framework $\\pi_{\\leq 1}(X,\\mathcal{U})$ and prove a semi-coarse van Kampen theorem under mild hypotheses (well-splitting covers and Lebesgue-type decompositions). The results yield a practical, scale-aware algebraic-topological tool that remains informative on coarse spaces and provides a pathway to analyze graphs and semi-pseudometric spaces through groupoid presentations. This work thus bridges coarse geometry and classical topological invariants, with potential implications for topological data analysis at prescribed scales.

Abstract

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.

Semi-coarse Spaces: Fundamental Groupoid and the van Kampen Theorem

TL;DR

The paper introduces a semi-coarse fundamental groupoid for semi-coarse spaces by leveraging tails of symmetric maps from and a refined string calculus. It defines a robust invariant via delete-one-point and opposite-map relations, and shows functoriality with respect to bornologous maps. Because coarse spaces collapse many invariants, the authors develop a relative, tail-controlled framework and prove a semi-coarse van Kampen theorem under mild hypotheses (well-splitting covers and Lebesgue-type decompositions). The results yield a practical, scale-aware algebraic-topological tool that remains informative on coarse spaces and provides a pathway to analyze graphs and semi-pseudometric spaces through groupoid presentations. This work thus bridges coarse geometry and classical topological invariants, with potential implications for topological data analysis at prescribed scales.

Abstract

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 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.
Paper Structure (6 sections, 29 theorems, 87 equations, 16 figures, 2 tables)

This paper contains 6 sections, 29 theorems, 87 equations, 16 figures, 2 tables.

Table of Contents

  1. Semi-coarse Spaces and its Homotopy
  2. Connectedness and Well-Splitting
  3. Semi-coarse Strings
  4. Semi-coarse Fundamental Groupoid
  5. Relative Fundamental Groupoid and Van Kampen Theorem
  6. Coarse Homotopy Fails in Semi-Coarse

Key Result

Proposition 1.3

Let $(X,\mathcal{V})$ and $(Y,\mathcal{W})$ be semi-coarse spaces, and suppose that $(X_i,\mathcal{V}_i)\subset (X,\mathcal{V})$, $i\in\{1,\ldots,n\}$, are subspaces of $(X,\mathcal{V})$ such that $\cup_{i=1}^n X_i = X$ and every set $V\in\mathcal{V}$ may be written in the form where each $V_i\in\mathcal{V}_i$. Now suppose that $f:X\rightarrow Y$ is a map such that the restrictions $f\mid_{X_i}:(

Figures (16)

  • Figure 1: Well-splitting example: Red points are $A-B$, blue points are $B-A$ and violet points are $A\cap B$. $X$ with the semi-coarse space induce by the graph.
  • Figure 2: No well-splitting example 1: Red points are $A-B$, blue points are $B-A$ and there are no points in $A\cap B$.
  • Figure 3: No well-splitting example 2: Red points are $A-B$, blue points are $B-A$ and violet points are $A\cap B$. $X$ with the semi-coarse space induce by the graph.
  • Figure 4: Every controlled set in $A\sqcup_{A\cap B}B$ and $X$, respectively, is contained in some red area. Observe that there are not controlled sets in the second and fourth quadrants in $A\sqcup_{A\cap B}B$, in contrast to $X$.
  • Figure 5: Representation of $(e_1,e'_1,e"_1)$
  • ...and 11 more figures

Theorems & Definitions (88)

  • Definition 1.1: Semi-coarse space; rieser2023semicoarse
  • Definition 1.2: Semi-coarse Subspace; 2.2.3 rieser2023semicoarse
  • Proposition 1.3: rieser2023semicoarse 2.2.4
  • Definition 1.4: Semi-coarse Product; 2.3.7 rieser2023semicoarse
  • Definition 1.5: Quotient Space; 2.4.1 and 2.4.2 rieser2023semicoarse
  • Theorem 1.6: rieser2023semicoarse 2.4.3
  • Definition 1.7: Disjoint Union; 2.5.1 and 2.5.2 rieser2023semicoarse
  • Proposition 1.8
  • proof
  • Definition 1.9: Product Extension; 2.5.9 and 2.5.10 rieser2023semicoarse
  • ...and 78 more