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$7$-located locally $5$-large complexes are aspherical

Katherine Goldman, Piotr Przytycki

TL;DR

This work proves that any $7$-located locally $5$-large simplicial complex is aspherical by developing a robust combinatorial framework that partners $m$-location and flagness with disc and lunar diagram techniques. The authors define dwheels, planar wheels, and the $m$-located condition, then leverage disc diagrams and the $oldsymbol{ba}$-method to control interior valences and curvature. A Whitehead-type deformation strategy shows that finite subcomplexes sit inside contractible, flag supercomplexes, and that downward links are contractible, enabling an inductive contraction of balls $B_n(x)$ and finalizing the $K(\pi,1)$-type conclusion. The results apply to prominent examples, including triangulations of hyperbolic space and certain Artin-group contexts, marking significant progress in understanding asphericity under weaker local curvature hypotheses.

Abstract

We prove that $7$-located locally $5$-large simplicial complexes are aspherical.

$7$-located locally $5$-large complexes are aspherical

TL;DR

This work proves that any -located locally -large simplicial complex is aspherical by developing a robust combinatorial framework that partners -location and flagness with disc and lunar diagram techniques. The authors define dwheels, planar wheels, and the -located condition, then leverage disc diagrams and the -method to control interior valences and curvature. A Whitehead-type deformation strategy shows that finite subcomplexes sit inside contractible, flag supercomplexes, and that downward links are contractible, enabling an inductive contraction of balls and finalizing the -type conclusion. The results apply to prominent examples, including triangulations of hyperbolic space and certain Artin-group contexts, marking significant progress in understanding asphericity under weaker local curvature hypotheses.

Abstract

We prove that -located locally -large simplicial complexes are aspherical.

Paper Structure

This paper contains 6 sections, 13 theorems.

Table of Contents

  1. Introduction
  2. Location
  3. Disc diagrams
  4. Lunar diagrams
  5. Contractibility
  6. Downward links

Key Result

Lemma 2.4

Any homotopically trivial cycle embedded in $X^1$ is the boundary cycle of a disc diagram in $X$. Any minimal disc diagram is reduced.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Lemma 2.4: MW
  • Lemma 2.5: HL
  • Corollary 2.6
  • Remark 2.7
  • Definition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • ...and 17 more