An infinite-dimensional phenomenon in finite-dimensional metric topology

The infinite topology of the hyperelliptic locus in Torelli space

Genus $g$ Torelli space is the moduli space of genus $g$ curves of compact type equipped with a homology framing. The hyperelliptic locus is a closed analytic subvariety consisting of finitely many mutually isomorphic components. We use properties of the hyperelliptic Torelli group to show that when $g\geq 3$ these components do not have the homotopy type of a finite CW complex. Specifically, we show that the second rational homology of each component is infinite-dimensional. We give a more detailed description of the topological features of these components when $g=3$ using properties of genu…

关于纤维化空间的无穷维空间拓扑

On Topology of the Infinite-Dimensional Space of Fibrations

本文研究流形上光滑纤维化模空间的代数与微分拓扑，通过无穷维主丛揭示其 Fréchet 流形结构，并借助“同伦核心”方法给出低维情形的显式同伦计算，完成三维及以下情形的分类。

The topology of the zero locus of a genus 2 theta function

Mess showed that the genus 2 Torelli group $T_2$ is isomorphic to a free group of countably infinite rank by showing that genus 2 Torelli space is homotopy equivalent to an infinite wedge of circles. As an application of his computation, we compute the homotopy type of the zero locus of any classical genus 2 theta function in $\mathfrak{h}_2 \times \mathbb{C}^2$, where $\mathfrak{h}_2$ denotes rank 2 Siegel space. Specifically, we show that the zero locus of any such function is homotopy equivalent to an infinite wedge of 2-spheres.

Topology of open nonpositively curved manifolds

This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples.

On constructions preserving the asymptotic topology of metric spaces

We prove that graph products constructed over infinite graphs with bounded clique number preserve finite asymptotic dimension. We also study the extent to which Dranishnikov's property C, and Dranishnikov and Zarichnyi's straight finite decomposition complexity are preserved by constructions such as unions, free products, and group extensions.

单模随机双曲流形的拓扑与动力学

Topology and dynamics of unimodular random hyperbolic manifolds

研究单模随机双曲流形端点空间与测地线流动力学关系，证明有两个无限体积端点意味着递归，无限多个则意味着正漂移和熵，还给出适用于确定性双曲流形的暂态准则，方法基于德劳内图和容量概念。

Spaces of Pants Decompositions for Surfaces of Infinite Type

We study the pants complex of surfaces of infinite type. When $S$ is a surface of infinite type, the usual definition of the pants graph $\mathcal{P}(S)$ yields a graph with infinitely many connected-components. In the first part of our paper, we study this disconnected graph. In particular, we show that the extended mapping class group $\mathrm{Mod}(S)$ is isomorphic to a proper subgroup of $\mathrm{Aut}(\mathcal{P})$, in contrast to the finite-type case where $\mathrm{Mod}(S)\cong \mathrm{Aut}(\mathcal{P}(S))$. In the second part of the paper, motivated by the Metaconjecture of Ivanov, we se…

A Fréchet topology on measured laminations and Earthquakes in the hyperbolic plane

We prove that the bijective correspondence between the space of bounded measured laminations $ML_b(\mathbb{H})$ and the universal Teichmüller space $T(\mathbb{H})$ given by $λ\mapsto E^λ|_{S^1}$ is a homeomorphism for the Fréchet topology on $ML_b(\mathbb{H})$ and the Teichmüller topology on $T(\mathbb{H})$, where $E^λ$ is an earthquake with earthquake measure $λ$. A corollary is that earthquakes with discrete earthquake measures are dense in $T(\mathbb{H})$. We also establish infinitesimal versions of the above results.