The bar construction of an algebra as an E-infinite Hopf algebra

The universal Hopf operads of the bar construction

The goal of this memoir is to prove that the bar complex B(A) of an E-infinity algebra A is equipped with the structure of a Hopf E-infinity algebra, functorially in A. We observe in addition that such a structure is homotopically unique provided that we consider unital operads which come equipped with a distinguished 0-ary operation that represents the natural unit of the bar complex. Our constructions rely on a Reedy model category for unital Hopf operads. For our purpose we define a unital Hopf endomorphism operad which operates functorially on the bar complex and which is universal with th…

无限型曲面是非 Hopf 的

Surfaces of infinite-type are non-Hopfian

本文提出用度一真映射是否必同伦于同胚这一拓扑 Hopf 型性质，刻画有限型定向曲面，并据此表明无限型曲面非 Hopf。

Brace Bar-Cobar Duality

Using Kadeishvili's formulas with appropriate signs, we show that the classical cobar construction from coalgebras to algebras Ω: CoAlg -> Alg can be enhanced to a functor from Hopf algebras to E_2 algebras (for a certain choice of E_2 operad) Ω: HopfAlg -> E_2 Alg, which, unlike its classical counterpart, is not strictly adjoint, but homotopically equivalent to the left adjoint of the enhanced bar construction B: E_2 Alg -> HopfAlg studied by Gerstenhaber--Voronov and Fresse.

作为殆Hopf环的B型和D型Coxeter群无限族的模2上同调

The mod 2 cohomology of the infinite families of Coxeter groups of type B and D as almost Hopf rings

该研究刻画了二元域系数下B型、D型Coxeter群无限族上同调直和的Hopf环与殆Hopf环结构，给出生成元关系表示、加法基与Steenrod代数作用，并从几何构造及子群限制两方面描述生成元。

Infinitesimal and $B\_\infty$-algebras, finite spaces, and quasi-symmetric functions

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop …

Hopf 代数、Steinberg 模与 $SL_n(\mathbb Z)$ 和 $GL_n(\mathbb Z)$ 的不稳定上同调

Hopf algebras, Steinberg modules, and the unstable cohomology of $SL_n(\mathbb Z)$ and $GL_n(\mathbb Z)$

本文证明整数一般线性群以 Steinberg 模为系数的同调直和构成交换 Hopf 代数，并由此构造 $SL_n(\mathbb Z)$ 不稳定上同调类的无限族。