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Fundamental groups of 2-complexes with nonpositive planar sectional curvature

Lycka Drakengren

Abstract

We show that the finite simply connected 2-complexes of nonpositive planar sectional curvature are collapsible. Moreover, we show that each finite connected 2-complex with negative planar sectional curvature and fundamental group $\mathbb{Z}$ can be collapsed to a 1-dimensional cycle.

Fundamental groups of 2-complexes with nonpositive planar sectional curvature

Abstract

We show that the finite simply connected 2-complexes of nonpositive planar sectional curvature are collapsible. Moreover, we show that each finite connected 2-complex with negative planar sectional curvature and fundamental group can be collapsed to a 1-dimensional cycle.
Paper Structure (4 sections, 7 theorems, 4 equations, 1 figure)

This paper contains 4 sections, 7 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Collapsibility
  4. Conformally negatively curved complexes with fundamental group $\mathbb{Z}$

Key Result

Theorem 1.1

A finite, simply connected, conformally nonpositively curved 2-complex is collapsible.

Figures (1)

  • Figure 1: Forming a loop $\gamma$ by connecting a subpath of $q$ along an edge.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 16 more