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A fake Klein bottle with bubble

Wajid Mannan

TL;DR

The paper investigates whether a finite 2-complex can share the fundamental group and Euler characteristic of the Klein bottle with a bubble, $K^8$, yet be homotopy-distinct. It constructs a finite 2-complex $X$ with $(X)ong(K^8)$ and $(X)=(K^8)$ but with $H_2(( ilde{X}))$ not a free $[(X)]$-module, thereby providing a negative answer in this setting and connecting to Wall's $D(2)$ problem. The approach compares two presentations of the Klein bottle group, $=$ and $$, shows they define the same group and yield a complex with matching Euler characteristic, and then analyzes the second homology of the universal cover via a kernel map in a noncommutative setting to prove non-freeness. This work ties into the existence of finite $D(2)$ complexes and highlights subtle obstructions to achieving freeness of the relation module even for the Klein bottle group, informing broader questions in low-dimensional topology.

Abstract

We resolve the question of the existence of a finite 2-complex with the same fundamental group and Euler characteristic as a Klein bottle with a bubble, but homotopically distinct to it.

A fake Klein bottle with bubble

TL;DR

The paper investigates whether a finite 2-complex can share the fundamental group and Euler characteristic of the Klein bottle with a bubble, , yet be homotopy-distinct. It constructs a finite 2-complex with and but with not a free -module, thereby providing a negative answer in this setting and connecting to Wall's problem. The approach compares two presentations of the Klein bottle group, and , shows they define the same group and yield a complex with matching Euler characteristic, and then analyzes the second homology of the universal cover via a kernel map in a noncommutative setting to prove non-freeness. This work ties into the existence of finite complexes and highlights subtle obstructions to achieving freeness of the relation module even for the Klein bottle group, informing broader questions in low-dimensional topology.

Abstract

We resolve the question of the existence of a finite 2-complex with the same fundamental group and Euler characteristic as a Klein bottle with a bubble, but homotopically distinct to it.
Paper Structure (3 sections, 6 theorems, 22 equations, 4 figures)

This paper contains 3 sections, 6 theorems, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Presentations of the Klein bottle group
  3. Non-freeness of $H_2(\widetilde{X_\mathcal{Q}})$

Key Result

Lemma 2.1

Mann3 For a group $G$, let $a,b \in G$ satisfy (abrel1), (abrel2). Then $ba=1$.

Figures (4)

  • Figure 1: The universal cover of $K^\circ$
  • Figure 2: The Klein bottle as a Cayley complex
  • Figure 3: Cellular decomposition lifted to universal cover of Klein bottle
  • Figure 4: Conjugation action on relator / boundary of associated 2-cell

Theorems & Definitions (10)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof