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Mapping fiber, loop and suspension graphs in naive discrete homotopy theory

So Yamagata

TL;DR

This paper investigates how $A$-homotopy theory in graphs contrasts with and mirrors classical homotopy by constructing mapping fiber graphs and establishing a discrete analogue of the Puppe sequence. It defines $Mf_1$ (and $Mf_2$) for pointed graph maps, develops the loop and suspension machinery, and then proves a long exact sequence in the naive discrete homotopy category $\mathbf{hGraph}_*$ under a technical subloop-length bound and surjectivity assumptions. The work connects $A$-homotopy groups $A_n(G,u_0)$ to geometric models via $|X_G|$, leverages exponential laws and adjunctions, and provides a concrete combinatorial framework for exact sequences in a discrete setting. Overall, it advances explicit constructions that bridge $A$-homotopy and classical homotopy theories in graphs and yields a discrete Puppe sequence with potential applications to combinatorial topology.

Abstract

Discrete homotopy theory or A-homotopy theory is a combinatorial homotopy theory defined on graphs, simplicial complexes, and metric spaces, reflecting information about their connectivity. The present paper aims to further understand the (non-)similarities between the A-homotopy and ordinary homotopy theories through explicit constructions. More precisely, we define mapping fiber graphs and study their basic properties yielding, under a technical condition, a discrete analogous of Puppe sequence in a naive discrete homotopy theory.

Mapping fiber, loop and suspension graphs in naive discrete homotopy theory

TL;DR

This paper investigates how -homotopy theory in graphs contrasts with and mirrors classical homotopy by constructing mapping fiber graphs and establishing a discrete analogue of the Puppe sequence. It defines (and ) for pointed graph maps, develops the loop and suspension machinery, and then proves a long exact sequence in the naive discrete homotopy category under a technical subloop-length bound and surjectivity assumptions. The work connects -homotopy groups to geometric models via , leverages exponential laws and adjunctions, and provides a concrete combinatorial framework for exact sequences in a discrete setting. Overall, it advances explicit constructions that bridge -homotopy and classical homotopy theories in graphs and yields a discrete Puppe sequence with potential applications to combinatorial topology.

Abstract

Discrete homotopy theory or A-homotopy theory is a combinatorial homotopy theory defined on graphs, simplicial complexes, and metric spaces, reflecting information about their connectivity. The present paper aims to further understand the (non-)similarities between the A-homotopy and ordinary homotopy theories through explicit constructions. More precisely, we define mapping fiber graphs and study their basic properties yielding, under a technical condition, a discrete analogous of Puppe sequence in a naive discrete homotopy theory.
Paper Structure (3 sections, 11 theorems, 38 equations, 5 figures)

This paper contains 3 sections, 11 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Discrete homotopy theory of graphs
  3. Mapping fiber graphs

Key Result

Theorem 2.8

For a graph $G$, define $X_G$ as the topological space obtained from $G$ regarded as a one-dimensional cell complex by attaching 2-cells along the boundary of each 3- and 4-cycle. Then,

Figures (5)

  • Figure 1: The graph $I_{\infty}^{\otimes 2}$.
  • Figure 2: The graph $I_{\geq M}^{\otimes 2}$.
  • Figure 3: $A$-homotopy $C_3 \otimes I_1 \to C_3$ between $c_0$ and $\mathrm{id}_{C_3}$.
  • Figure 4: $A$-homotopy $C_4 \otimes I_1 \to C_4$ between $c_0$ and $\mathrm{id}_{C_4}$.
  • Figure 5: The path $\omega$ of length $l$.

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8: BKLW
  • Theorem 2.9: CKa
  • Definition 2.10
  • ...and 26 more