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Computations regarding the torsion homology of Oeljeklaus-Toma manifolds

Dung Phuong Phan, Tuan Anh Bui, Alexander D. Rahm

TL;DR

By extending the Silver–Williams paradigm to Oeljeklaus–Toma manifolds, the paper studies the growth of torsion in towers of OT manifolds and shows that, in the Inoue-type and certain OT cases, the torsion in degrees $1$ and $2$ grows exponentially with the covering index, governed by the Mahler measure of a defining polynomial. The authors develop and implement a computational framework based on free resolutions over $\mathbb{Z}^2$ and $\mathbb{Z}^4$ and Wall's twisted tensor product to compute the homology of the semidirect product $\mathbb{Z}^4 \rtimes_{\overline{\varphi}} \mathbb{Z}^2$, thereby enabling $H_*(X_{m,n})$ computations for OT manifolds. In the explicit $p=2$ case with $K=\mathbb{Q}(\sqrt[4]{2})$, they obtain concrete descriptions of $H_1(X_{m,n})$, including periodicity in $n$ and $m$ tied to the factorization of $-a_m$, where $a_m=\frac{1}{2}(-1+\sqrt{2})^m+\frac{1}{2}(-1-\sqrt{2})^m$. These results provide a practical computational tool for higher-dimensional OT manifolds and reveal a deep link between number field arithmetic (Mahler measures, units) and topological torsion phenomena.

Abstract

This article investigates the torsion homology behaviour in towers of Oeljeklaus-Toma (OT) manifolds. This adapts an idea of Silver and Williams from knot theory to OT-manifolds and extends it to higher degree homology groups. In the case of surfaces, i.e. Inoue surfaces of type $S^0$, the torsion grows exponentially in both $H_1$ (as was established by Braunling) and $H_2$ (our result) according to a parameter which already plays a role in Inoue's classical paper, and we obtain that the torsion vanishes in all higher degrees. This motivates our presented machine calculations for OT-manifolds of higher dimension.

Computations regarding the torsion homology of Oeljeklaus-Toma manifolds

TL;DR

By extending the Silver–Williams paradigm to Oeljeklaus–Toma manifolds, the paper studies the growth of torsion in towers of OT manifolds and shows that, in the Inoue-type and certain OT cases, the torsion in degrees and grows exponentially with the covering index, governed by the Mahler measure of a defining polynomial. The authors develop and implement a computational framework based on free resolutions over and and Wall's twisted tensor product to compute the homology of the semidirect product , thereby enabling computations for OT manifolds. In the explicit case with , they obtain concrete descriptions of , including periodicity in and tied to the factorization of , where . These results provide a practical computational tool for higher-dimensional OT manifolds and reveal a deep link between number field arithmetic (Mahler measures, units) and topological torsion phenomena.

Abstract

This article investigates the torsion homology behaviour in towers of Oeljeklaus-Toma (OT) manifolds. This adapts an idea of Silver and Williams from knot theory to OT-manifolds and extends it to higher degree homology groups. In the case of surfaces, i.e. Inoue surfaces of type , the torsion grows exponentially in both (as was established by Braunling) and (our result) according to a parameter which already plays a role in Inoue's classical paper, and we obtain that the torsion vanishes in all higher degrees. This motivates our presented machine calculations for OT-manifolds of higher dimension.
Paper Structure (10 sections, 7 theorems, 81 equations, 1 table)

This paper contains 10 sections, 7 theorems, 81 equations, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Proof of Theorem \ref{['thm_1']}
  4. An algorithm computing the homology of $X_{m,n}$
  5. A resolution of $\mathbb{Z}$ over $\mathbb{Z}^2$
  6. A resolution of $\mathbb{Z}$ over $\mathbb{Z}^4$
  7. The isomorphism from $O_K \rtimes_\varphi \mathbb{Z}<u^{im},v^{jn}>$ to $\mathbb{Z}^4 \rtimes_{\overline{\varphi}} \mathbb{Z}^2$
  8. An algorithm for computing the homology of $\mathbb{Z}^4 \rtimes_{\overline{\varphi}} \mathbb{Z}^2$
  9. An explicit example when $p=2$
  10. Proof of Theorem 4 and Proposition 3

Key Result

Theorem 1

For Inoue surfaces of type $S^{0}$ (resp. Oeljeklaus--Toma manifolds with $r_{1},r_{2}=1$), the torsion homology growth satisfies as $n\longrightarrow+\infty$, where $M(f)$ is the Mahler measure of the minimal polynomial as described above in the text. This is $>1$, so the orders of the torsion part of both $H_{1}$ and $H_{2}$ grow exponentially with $n$.

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 2
  • Remark
  • Proposition 3
  • Theorem 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 3 more