Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2406.13241 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914841283788800 |
|---|---|
| author | Tian, Ye Wang, Shicheng Wang, Zhongzi |
| author_facet | Tian, Ye Wang, Shicheng Wang, Zhongzi |
| contents | 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 $1-ρ$,
where $$ρ:=\prod_{j=1}^\infty \left(1+2^{-j}\right)^{-1} = 0.41942\cdots.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Achirality of Sol 3-Manifolds, Stevenhagen Conjecture and Shimizu's L-series Tian, Ye Wang, Shicheng Wang, Zhongzi Geometric Topology Number Theory 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 $1-ρ$, where $$ρ:=\prod_{j=1}^\infty \left(1+2^{-j}\right)^{-1} = 0.41942\cdots.$$ |
| title | Achirality of Sol 3-Manifolds, Stevenhagen Conjecture and Shimizu's L-series |
| topic | Geometric Topology Number Theory |
| url | https://arxiv.org/abs/2406.13241 |