Geoffroy Horel
We compute the automorphisms of the Bousfield-Kan completion at a prime p of the little two-disks operads and show that they are given by the pro-p Grothendieck-Teichmüller group. We also show that the Grothendieck-Teichmüller group acts faithfully on the p-complete stable little disks operad.
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We define a simplicial enrichment on the category of differential graded Hopf cooperads (the category of dg Hopf cooperads for short). We prove that our simplicial enrichment satisfies, in part, the axioms of a simplicial model category structure on the category of dg Hopf cooperads. We use this simplicial model structure to define a model of mapping spaces in the category of dg Hopf cooperads and to upgrade results of the literature about the homotopy automorphism spaces of dg Hopf cooperads by dealing with simplicial monoid structures. The rational homotopy theory of operads implies that the…
Algebraic groups of homotopy classes of automorphisms and operadic Koszul duality
本文利用操作子Koszul对偶理论证明了单连通空间有理化同伦自同构类所对应的两个先验不同的代数群同构，并由此得到若干映射同伦类集合的Eckmann-Hilton对偶同构。
Automorphisms of framed operads
本文研究了框架operad的自同构，通过建立原operad与半直积operad的同伦自同构群之间的联系，成功计算了框架小圆盘operad的自同构。
Every Endomorphism of the Framed Little Disk Operad is an Automorphism
本文利用单李群自映射分类，将 Horel-Krannich-Kupers 关于小圆盘操作子自同态均为自同构的结果推广到框架化情形，并进一步考察了瑞士奶酪操作子及若干半直积情形。
Etale cohomology, purity and formality with torsion coefficients
本研究借助伽罗瓦群在平展上同调的作用，证明带挠系数的dg-Operad与dg-代数的有限次数形式化结论，可应用于小圆盘算子奇异链等构造，形式化次数由系数域基数决定。
Plethysm for characters of relative operads and props
本文将对称函数特征标理论推广至相对 operad 与 PROP，提出相对 plethysm 和连通盒积以刻画相应复合结构，并用于计算自由群及 IA-自同构群上同调的特征标。
Cyclic operads through modules
本文提出通过模计算循环operad映射空间的方法，并应用于计算循环Batalin-Vilkovisky（Hopf余）operad的同伦自同构空间。
The Torsion of Automorphisms of Nilpotent Spaces
研究Nilpotent空间中自同构torsion的消失性质，证明其与算术群的commensurable关系，并修正Sullivan的证明。
In this paper, we prove that the group of homotopy automorphisms of the profinite completion of the operad of little $2$-disks is isomorphic to the profinite Grothendieck-Teichmüller group. In particular, the absolute Galois group of $\mathbb{Q}$ acts faithfully on the profinite completion of $E_2$ in the homotopy category of profinite weak operads.
We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichmüller group. Using a result of Drummond-Cole, we deduce that the Grothendieck-Teichmüller group acts nontrivially on $\overline{\mathcal{M}}_{0,\bullet+1}$, the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little 2-disks operad is formal.
Two remarks on spaces of maps between operads of little cubes
本文研究小d立方体operad $E_d$的导出映射空间，证明其等价于非酉$E_d$的映射空间且$E_d$的自同态均为自同构，并探讨了相关变体。
Grothendieck-Teichmüller Symmetries of Cyclic Operads and Tangles
本文研究了循环操作符与结的Grothendieck-Teichmüller对称性，通过操作符模型揭示了 profinite 结的算术对称性，并给出了形式性的另一种证明。
We study a special type of $E_\infty$-operads that govern strictly unital $E_\infty$-coalgebras (and algebras) over the ring of integers. Morphisms of coalgebras over such an operad are defined by using universal $E_\infty$-bimodules. Thus we obtain a category of $E_\infty$-coalgebras. It turns out that if the homology of an $E_\infty$-coalgebra have no torsion, then there is a natural way to define an $E_\infty$-coalgebra structure on the homology so that the resulting coalgebra be isomorphic to the initial $E_\infty$-coalgebra in our category. We also discuss some invariants of the $E_\infty…
In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for 2-torsion part, using combinatorial group theory.
The geometric dimension for proper actions $\underline{\mathrm{gd}}(G)$ of a group $G$ is the minimal dimension of a classifying space for proper actions $\underline{E}G$. We construct for every integer $r\geq 1$, an example of a virtually torsion-free Gromov-hyperbolic group $G$ such that for every group $Γ$ which contains $G$ as a finite index normal subgroup, the virtual cohomological dimension $\mathrm{vcd}(Γ)$ of $Γ$ equals $\underline{\mathrm{gd}}(Γ)$ but such that the outer automorphism group $\mathrm{Out}(G)$ is virtually torsion-free, admits a cocompact model for $\underline E\mathrm{…
We compute the Poincare polynomial and the cohomology algebra with rational coefficeints of the manifold M_n of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the 2-local torsion in the cohomology of M_n. As was shown by E. Rains in arXiv:math/0610743 the cohomology of M_n does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of M_n is the operad o…
By analyzing how the Borel regulator classes vanish on various groups related to $\mathrm{GL}(n,\mathrm{Z})$, we define three series of secondary characteristic classes for subgroups of automorphism groups of free groups. The first case is the $\mathrm{IA}$-automorphism groups and we show that our classes coincide with higher $\mathrm{FR}$ torsions due to Igusa. The second case is the mapping class groups and our classes also turn out to be his higher torsions which are non-zero multiples of the Mumford-Morita-Miller classes of even indices. Our construction gives new group cocycles for these …
Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group
本文通过构造群胚中的模操作子$\mathbf{S}$，建立了高亏格泰希米勒塔的操作子模型，并证明了$\widehat\Gamma$子群在$\widehat{\mathbf{S}}$上的忠实作用，从而给出了$\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$的作用。
Stable homology of Higman--Thompson groups via scanning methods
本文通过扫描方法，利用操作群框架构建拓扑模型，证明了Higman-Thompson群的同调稳定性，并计算其稳定同调为Moore谱的无限循环空间的同调。
We study a family of Lie algebras {hO} which are defined for cyclic operads O. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper we introduce a hairy graph homology theory for O. We show that the homology of hO embeds in hairy graph homology via a trace map which generalizes the trace map defined by S. Morita. For the Lie operad we use the trace map to find la…
We prove a dichotomy between rationality and a natural boundary for the analytic behavior of the Reidemeister zeta function for automorphisms of non-finitely generated torsion abelian groups and for endomorphisms of groups $\mathbb Z_p^d,$ where $\mathbb Z_p$ the group of p-adic integers. As a consequence, we obtain a dichotomy for the Reidemeister zeta function of a continuous map of a topological space with fundamental group that is non-finitely generated torsion abelian group. We also prove the rationality of the coincidence Reidemeister zeta function for tame endomorphisms pairs of finitel…
We study torsion in homology of the random $d$-complex $Y \sim Y_d(n,p)$ experimentally. Our experiments suggest that there is almost always a moment in the process where there is an enormous burst of torsion in homology $H_{d-1}(Y)$. This moment seems to coincide with the phase transition studied in \cite{AL,LP,LP3} , where cycles in $H_d(Y)$ first appear with high probability. Our main study is the limiting distribution on the $q$-part of the torsion subgroup of $H_{d-1}(Y)$ for small primes $q$. We find strong evidence for a limiting Cohen--Lenstra distribution, where the probability that t…
Categorical symmetries of T-duality
本文研究了拓扑T对称的高阶范畴化结构，通过计算严格李2群的自同构群，揭示了拓扑T对称的高阶对称性，并证明该自同构群是整数分裂伪正交群的非中心范畴扩展。
Given an automorphism $ϕ:Γ\to Γ$, one has an action of $Γ$ on itself by $ϕ$-twisted conjugacy, namely, $g.x=gxϕ(g^{-1})$. The orbits of this action are called $ϕ$-twisted conjugacy classes. One says that $Γ$ has the $R_\infty$-property if there are infinitely many $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $Γ$. In this paper we show that SL$(n,\mathbb{Z})$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $Γ$ has the $R_\infty$-property where $Γ$ is a torsion free non-elementary hyperbolic group, or SL$(n,\math…
Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group
该研究通过分析标架Drinfeld-Kohno李代数operad的循环结构，揭示翻转的反射性质，推导Drinfeld结合子的两循环关系，进而给出分次Grothendieck-Teichmüller群$\textsf{GRT}_1$作用的新证明。
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