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Some closed hyperbolic 5-manifolds

Jacopo G. Chen

Abstract

We give an explicit construction of a family of closed arithmetic hyperbolic 5-manifolds, tessellated by $117 964 800 = 512 \cdot 16 \cdot 14400$ copies of a Coxeter simplicial prism. We proceed to study various properties of these manifolds, such as the volume and the first Betti number. We also describe a related family of 5-manifolds with a larger volume, but a simpler construction.

Some closed hyperbolic 5-manifolds

Abstract

We give an explicit construction of a family of closed arithmetic hyperbolic 5-manifolds, tessellated by copies of a Coxeter simplicial prism. We proceed to study various properties of these manifolds, such as the volume and the first Betti number. We also describe a related family of 5-manifolds with a larger volume, but a simpler construction.

Paper Structure

This paper contains 19 sections, 19 theorems, 23 equations, 3 figures, 2 tables.

Table of Contents

  1. A hyperbolic Coxeter prism
  2. Some 4-manifolds
  3. Long's manifolds
  4. The adjacency graph
  5. Closed 5-manifolds
  6. The coloring method
  7. Counting good independent sets
  8. Linear binary codes
  9. Even smaller manifolds
  10. Classification of the 5-manifolds
  11. Properties
  12. Orientability
  13. Volume
  14. First homology
  15. Parallelizability
  16. ...and 4 more sections

Key Result

Theorem 1

There exist some closed orientable hyperbolic $5$-manifolds having $b_1 = 0$ and volume $<250000.0$.

Figures (3)

  • Figure 1.1: A schematic three-dimensional representation of the simplicial prism $P$ within the $120$-cell prism $Q$.
  • Figure 1.2: A three-dimensional representation of an orientable manifold with right-angled corners, obtained by arranging copies of $Q$ (here, a square prism) on a non-orientable hypersurface (here, a Möbius strip). Note that we will consider only $4$-manifolds without boundary, unlike the Möbius strip; hence, only the blue faces are relevant to our case.
  • Figure 2.1: Subgroup lattice of $G$. Solid lines denote normal subgroups, labeled by the quotient group or by an integer $n$, standing for $\mathbb Z_n$.

Theorems & Definitions (39)

  • Theorem
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • ...and 29 more