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3-manifold spine cyclic presentations with seldom seen Whitehead graphs

Gerald Williams

TL;DR

This paper addresses when cyclically presented groups yield spines of closed 3-manifolds by focusing on planar Whitehead graphs and their classifications. It introduces two new families, $\mathcal{H}(r,n)$ and $\mathcal{G}^{k/l}(n,f)$, and proves spine properties under planarity and parity constraints, linking them to lens spaces and Fractional Fibonacci groups. The results show $\mathcal{H}(r,n)$ yields an $n$-fold cyclic branched cover of the lens space $L(r,1)$ when $\gcd(n,r)=1$, while $\mathcal{G}^{k/l}(n,f)$ provides spines of type $(II.7)$ whose manifolds relate to $F^{k/l}(n)$, with hyperbolicity preserved from the base family in many cases. The work broadens the catalog of cyclic presentations that are 3-manifold spines, connects to McDermott's type Z, and uses face-pairing polyhedra and Seifert–Threlfall constructions to illuminate the geometry of these combinatorial objects.

Abstract

We consider a family of cyclic presentations and show that, subject to certain conditions on the defining parameters, they are spines of closed 3-manifolds. These are new examples where the reduced Whitehead graphs are of the same type as those of the Fractional Fibonacci presentations; here the corresponding manifolds are often (but not always) hyperbolic. We also express a lens space construction in terms of a class of positive cyclic presentations that are spines of closed 3-manifolds. These presentations then furnish examples where the Whitehead graphs are of the same type as those of the positive cyclic presentations of type $\mathfrak{Z}$, as considered by McDermott.

3-manifold spine cyclic presentations with seldom seen Whitehead graphs

TL;DR

This paper addresses when cyclically presented groups yield spines of closed 3-manifolds by focusing on planar Whitehead graphs and their classifications. It introduces two new families, and , and proves spine properties under planarity and parity constraints, linking them to lens spaces and Fractional Fibonacci groups. The results show yields an -fold cyclic branched cover of the lens space when , while provides spines of type whose manifolds relate to , with hyperbolicity preserved from the base family in many cases. The work broadens the catalog of cyclic presentations that are 3-manifold spines, connects to McDermott's type Z, and uses face-pairing polyhedra and Seifert–Threlfall constructions to illuminate the geometry of these combinatorial objects.

Abstract

We consider a family of cyclic presentations and show that, subject to certain conditions on the defining parameters, they are spines of closed 3-manifolds. These are new examples where the reduced Whitehead graphs are of the same type as those of the Fractional Fibonacci presentations; here the corresponding manifolds are often (but not always) hyperbolic. We also express a lens space construction in terms of a class of positive cyclic presentations that are spines of closed 3-manifolds. These presentations then furnish examples where the Whitehead graphs are of the same type as those of the positive cyclic presentations of type , as considered by McDermott.
Paper Structure (4 sections, 5 theorems, 23 equations, 9 figures)

This paper contains 4 sections, 5 theorems, 23 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Presentation complexes as spines of closed 3-manifolds
  3. Cyclic presentations $\mathcal{H}(r,n)$ as spines of type (I.5)
  4. Cyclic presentations $\mathcal{G}^{k/l}(n,f)$ as spines of type (II.7)

Key Result

Theorem A

Let $n\geq 4$, $fk\not \equiv 2 \bmod n$, and $(k,l)\in \{(k,1),(1,l),(5,2),(2,5)\}$. Then $\mathcal{G}^{k/l}(n,f)$ is a spine of a closed, oriented 3-manifold $M^{k/l}(n,f)$ if and only if $n$ and $f$ are even and $fk\equiv 0 \bmod n$.

Figures (9)

  • Figure 1: Whitehead graph for $\mathcal{H}(r,n)$ (where $(r,n)=1$)
  • Figure 2: Face pairing polyhedron for $\mathcal{H}(r,n)$
  • Figure 3: Heegaard diagram for the manifold $M(r,n)$
  • Figure 4: Heegaard diagram for the manifold $M(r,n)/\rho$
  • Figure 5: Whitehead graph for $\mathcal{G}^{k/l}(n,f)$ (where $\lambda = 2l+k-3$, $n$ even and $fk\equiv 0$ or $2\bmod n$)
  • ...and 4 more figures

Theorems & Definitions (12)

  • Theorem A
  • Remark 4.1
  • Example 4.2: Non-cyclic finite groups.
  • Lemma 4.3
  • Proof 1
  • Lemma 4.4
  • Proof 2
  • Theorem 4.5
  • Proof 3
  • Remark 4.6
  • ...and 2 more