Symmetric ideals in group rings and simplicial homotopy

A braided simplicial group

By studying braid group actions on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described group. We also give certain representation of higher differentials in the Adams spectral sequence for the homotopy groups of the 2-sphere.

Homotopy of profinite groups

We study simplicial profinite groups with a view towards applications in profinite combinatorial group theory. This approach provides a natural framework to the concept of pro-$\mathfrak{C}$-presentation of a pro-$\mathfrak{C}$-group $G$ as a 1-truncation of its free simplicial pro-$\mathfrak{C}$-resolution. The category of simplicial pro-$\mathfrak{C}$-groups has a closed simplicial model category structure. This yields a possibility to define some old and new derived functors as left Quillen derived functors from this simplicial model category. When $\mathfrak{C}$ is L-groups, than may const…

刚性化单纯复形与实现群作用

Rigidifying simplicial complexes and realizing group actions

本文证明有限群在有限展示群及有限生成阿贝尔群上的作用均可实现为某空间自同伦等价群在同伦群上的作用，并证明任意有限连通单纯复形可在保持同伦型的前提下刚性化，使其自同构群约化为任意给定子群。

有限约化群的Brauer对的单纯复形

The simplicial complex of Brauer pairs of a finite reductive group

本文研究有限约化群的Brauer对偏序集诱导的单纯复形，明确其在定义特征下与Tits building同伦型一致，在非定义特征下与广义Harish-Chandra理论确定的复形同伦等价，拓展了Brown复形相关研究。

On minimal models in integral homotopy theory

This paper takes its starting point in an idea of Grothendieck on the representation of homotopy types. We show that any locally finite nilpotent homotopy can be represented by a simplicial set which is a finitely generated free group in all degrees and whose maps are given by polynomials with rational coefficients. Such a simplicial set is in some sense a universal localisation/completion as all localisations and completions of the homotopy is easily contructed from it. In particular relations with the Quillen and Sullivan approaches are presented. When the theory is applied to the Eilenberg-…

立方体复形的同伦等价边界

Homotopy equivalent boundaries of cube complexes

本文研究有限维 CAT(0) 立方体复形的多种边界，通过 Roller 边界商上的偏序构造单纯 Roller 边界，并证明其与 Tits、单纯边界在同伦意义下 $Aut(X)$-等变等价，从而说明度量扰动不改变等变同伦类型。