On homotopical and homological $Z_n$-sets

Homological and homotopical Dehn functions are different

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical Dehn function measures fillings of curves by disks. Since the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions, but prior to this work there were no known examples of finitely-presented groups for which the two functions differ. This paper gives the first such examples, constructed …

A note on a topological approach to the $μ$-constant problem in dimension 2

We provide an example, which shows that studying homological and homotopical properties of cobordisms between arbitrary, that is not necessarily negative, graph manifolds is not enough to prove the $μ$-constant conjecture of Le Dung Trang in complex dimension 2.

局部紧群的几何不变量：同伦论视角

Geometric invariants of locally compact groups: the homotopical perspective

该研究将经典同伦Σ-集理论推广到局部紧Hausdorff群，定义Σ_topⁿ并证明其相关性质，推广了经典理论的多个核心结果。

Homological stability for automorphism groups

Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial and abelian twisted coefficients, with no further assumptions. This new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class …

同调不变量的可定义内容 II：Čech 上同调与同伦分类

The definable content of homological invariants II: Čech cohomology and homotopy classification

本文将描述集合论应用于同调代数与代数拓扑，提出可定义的Čech上同调函子并证明其能精化经典函子，解决了映射望远镜同伦分类与Borsuk-Eilenberg问题的非光滑性。

$\mathrm{CAT}(0)$ 立方复形中子水平集的多项式填充函数

Polynomial filling functions of sublevel sets in $\mathrm{CAT}(0)$ cube complexes

本文给出判据确保 CAT(0) 立方复形高子水平集有多项式积分同调填充函数，并验证 Higman-Thompson 群满足，从而这些群在每度 k≥2 有多项式同调填充函数。