Title: Achirality of Sol 3-Manifolds, Stevenhagen Conjecture and Shimizu's L-series Show Chinese title

Title: 三维流形Sol的非手性，Stevenhagen猜想和Shimizu的L级数

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 $$\rho:=\prod_{j=1}^\infty \left(1+2^{-j}\right)^{-1} = 0.41942\cdots.$$ Show Chinese abstract

Abstract: 一个闭的可定向流形是{\em 非手性的}，如果它允许一个方向反转的同胚。 一个闭流形的共轭类是无向的，如果它包含一个无向元素，或者等价地，每个流形在$\CM$中都有一个无向的有限覆盖。 包含不可定向元素的每个共轭类必须是无向的。 自然地会想知道有多少个共轭类是无向的，以及有多少个无向类包含不可定向元素。 我们研究了Sol 3-流形的这个问题。 每个Sol 3-流形的共轭类$\CM$都有一个完整的拓扑不变量$D_{\CM}$，即$\CM$的判别式。 我们的主要结果是： (1) 在所有Sol 3-流形的共轭类中，有无限多个无向类；然而按判别式排序时，无向共轭类的密度为0。 (2) 在所有Sol 3-流形的无向共轭类中，按判别式排序时，包含不可定向元素的类的密度是$1-\rho$，其中$$\rho:=\prod_{j=1}^\infty \left(1+2^{-j}\right)^{-1} = 0.41942\cdots.$$