Achirality of Sol 3-Manifolds, Stevenhagen Conjecture and Shimizu's L-series

TL;DR

This paper investigates the prevalence of achiral Sol 3-manifold classes, showing infinitely many exist but with zero density among all classes, and quantifying the proportion with non-orientable elements using a specific infinite product.

Contribution

It provides a complete classification of achiral Sol 3-manifold classes and quantifies their distribution based on discriminants, connecting topology with number theory.

Findings

Infinitely many achiral classes exist among Sol 3-manifolds.

Density of achiral classes among all classes is zero.

Proportion of achiral classes with non-orientable elements is approximately 58.04%.

Abstract

A closed orientable manifold is {\em achiral} if it admits an orientation reversing homeomorphism. A commensurable class of closed manifolds is achiral if it contains an achiral element, or equivalently, each manifold in $\CM$ has an achiral finite cover. Each commensurable class containing non-orientable elements must be achiral. It is natural to wonder how many commensurable classes are achiral and how many achiral classes have non-orientable elements. We study this problem for Sol 3-manifolds. Each commensurable class $\CM$ of Sol 3-manifold has a complete topological invariant $D_{\CM}$, the discriminant of $\CM$. Our main result is: (1) Among all commensurable classes of Sol 3-manifolds, there are infinitely many achiral classes; however ordered by discriminants, the density of achiral commensurable classes is 0. (2) Among all achiral commensurable classes of Sol 3-manifolds, ordered by discriminants, the density of classes containing non-orientable elements is $1-\rho$, where $$\rho:=\prod_{j=1}^\infty \left(1+2^{-j}\right)^{-1} = 0.41942\cdots.$$