Domenico Marasco
We show that in the bounded cohomology of non-abelian free groups the Massey triple product is always trivial when the second factor is represented by the coboundary of a decomposable quasi-morphism. We also show that in the bounded cohomology of a negatively curved compact Riemannian manifold the Massey triple product is always trivial when the second factor is represented by an exact differential form.
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A vanishing criterion for cup products and Massey products in bounded cohomology
本文提出了有界上同调中积的一致消没准则，据此重新证明并扩展了自由群及群在 CAT(0) 立方复形上作用的等变有界上同调中杯积与 Massey 三重积的消没结果。
Massey products in Galois cohomology and Pythagorean fields
本文证明Minac-Tan的Massey消没猜想的加强版本对极大pro-2伽罗瓦群为初等型且平方类有限的域成立，从而涵盖有限平方类数的毕达哥拉斯域及其有限扩张。
Digraphs, pro-$p$ groups and Massey products in Galois cohomology
本研究依据关联有向图刻画了对所有$n>2$均无本质$n$重Massey积的定向右角Artin pro-$p$群，证明其$\text{F}_p$-上同调代数同构于对应外Stanley-Reisner代数，并为该领域提供群论入门介绍。
Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks …
Translating results due to J. Labute into group cohomological language, A. Schmidt proved that a finitely presented pro-p-group G is mild and hence of cohomological dimension cd G=2 if $H^1(G,\F_p)=U\oplus V$ as $\F_p$-vector space and the cup-product $H^1(G,\F_p)\otimes H^1(G,\F_p)\to H^2(G,\F_p)$ maps $U\otimes V$ surjectively onto $H^2(G,\F_p)$ and is identically zero on $V\otimes V$. This has led to important results in the study of p-extensions of number fields with restricted ramification, in particular in the case of tame ramification. In this paper, we extend Labute's theory of mild pr…
For all primes $p$ and all fields, we find a sufficient and necessary condition of the existence of a unipotent Galois extension of degree $p^6$. The main goal of this paper is to describe an explicit construction of such a Galois extension over fields admitting such a Galois extension. This construction is surprising in its simplicity and generality. The problem of finding such a construction has been left open since 2003. Recently a possible solution of this problem gained urgency because of an effort to extend new advances in Galois theory and its relations with Massey products in Galois co…
We observe that the singular part of the second bounded cohomology group of boundedly simple groups is trivial.
We prove that the cup product of $Δ$-decomposable quasimorphisms, Brooks quasimorphisms or Rolli quasimorphisms with any bounded cohomology class of arbitrary positive degree is trivial.
Cohomology of Trivial Linear Cycle Sets
在H和I均为有限循环p-群（p为奇数，或p=2且H元素个数不超过4）的假设下，给出了平凡线性循环集H被交换群I扩张的完全分类，得到了所有可能扩张的显式参数化，并计算了所得线性循环集的基座和中心。
The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial braces is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed.
We study a class of quasimorphisms of the free group that can be expressed as infinite sums of Brooks quasimorphisms with some nice properties. We then review Heuer's framework of decompositions developed in arXiv:1710.03193, and put these quasimorphisms in that context. This allows us to prove triviality of a number of cup products in the bounded cohomology of the free group with trivial real coefficients. Finally, we introduce free products of decompositions and relate them to Rolli's quasimorphisms of free products. A technical result concerning these, which remains an open question, implie…
The theory of bounded cohomology of groups has many applications. A key open problem is to compute the full bounded cohomology $H_b^n(F, R)$ of a non-abelian free group $F$ with trivial real coefficients. It is known that $H_b^n(F,R)$ is trivial for $n=1$ and uncountable dimensional for $n=2,3$, but remains unknown for any $n \geq 4$. For $n=4$, one may construct classes by taking the cup product $α\cup β\in H_b^4(F, R)$ between two $2$-classes $α, β\in H^2_b(F, R)$. However, we show that all such cup products are trivial if $α$ and $β$ are classes induced by the quasimorphisms defined by Broo…
The computation of the cohomology for finite groups of Lie type in the describing characteristic is a challenging and difficult problem. In earlier work, the authors constructed an induction functor which takes modules over the finite group of Lie type to modules for the ambient algebraic group G. In particular this functor when applied to the trivial module yields a natural G-filtration. This filtration was utilized in the earlier work to determine the first non-trivial cohomology class when the underlying root system is of type A_{n} or C_{n}. In this paper the authors extend these results t…
In this paper we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm{IA}_3$ of a free group of rank three by using combinatorial group theory and representation theory. In particular, we detect non-trivial irreducible component in the second cohomology group of $\mathrm{IA}_3$, which does not contained in the image of the cup product map of the first cohomology groups. We also show that the image of the triple cup product map of the first cohomology groups in the third cohomology group is trivial. As a corollary, we obtain that the fourth term of the low…
We introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the expo…
Finitely generated simple left orderable groups with vanishing second bounded cohomology
本文证明一类有限生成单左可序群在平凡实系数与整系数下第二有界上同调均为零，给出首个非不可指示的此类群例，并解答Navas的ICM问题。
Essential quotients of the cohomology of products of symmetric and alternating groups
本文计算对称群与交错群乘积的模二上同调关于可忽略元素的商，回答Serre提出的问题，并基于其近期结果，用于域的上同调不变量研究。
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