Sofia Amontova, Michelle Bucher
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.
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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…
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…
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。
Cup product of inhomogeneous Tate cochains, and Galois cohomology of tori over local fields that split over cyclic extensions
本文给出Tate上同调中杯积的非齐次上链公式，并用于计算分裂于循环扩张的局部域上环面的一阶上同调类的显式上循环。
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…
Median quasimorphisms on CAT(0) cube complexes and their cup products
本文将自由群 Brooks 拟态射杯积的消没定理推广至 CAT(0) 立方体复形上的中位数拟态射，并应用于树作用群与直角 Artin 群。
We prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.
Two results on cohomology of groups adapted to cochains
本文证明群上同调中 $G$ 作用平凡性与杯积交换性两个经典结论的上链层级版本，并给出实现相应同伦的显式映射。
Let $G$ be an elementary abelian $p$-group, $G\cong{\mathbb F}_p^r$ and let $s_1,\ldots,s_r$ be a basis of $G$ over ${\mathbb F}_p$. Let $V$ be the dual of $G$, $V={\rm Hom}(G,{\mathbb F}_p)=H^1(G,{\mathbb F}_p)$. Let $x_1,\ldots,x_r$ be the basis of $V$ over ${\mathbb F}_p$ which is dual to the basis $s_1,\ldots,s_r$ of $G$. For $1\leq i\leq r$ we denote by $y_i=β(x_i)\in H^2(G,{\mathbb F}_p)$, where $β:H^1(G,{\mathbb F}_p)\to H^2(G,{\mathbb F}_p)$ is the connecting Bockstein map. The ring $(H^*(G,{\mathbb F}_p),+,\cup )$ satisfies $$H^*(G,{\mathbb F}_p)\cong\begin{cases}{\mathbb F}_p[x_1,\ld…
In this paper we prove a family of results connecting the problem of computing cup products in surface bundles to various other objects that appear in the theory of the cohomology of the mapping class group $\operatorname{Mod}_g$ and the Torelli group $\mathcal{I}_g$. We show that N. Kawazumi's twisted MMM class $m_{0,k}$ can be used to compute $k$-fold cup products in surface bundles, and that $m_{0,k}$ provides an extension of the higher Johnson invariant $τ_{k-2}$ to $H^{k-2}(\operatorname{Mod}_{g,*}, \wedge^k H_1)$. These results are used to show that the behavior of the restriction of the…
On a Completion of Cohomological Functors Generalising Tate Cohomology II
本文研究了Mislin完成的性质，证明Ext函子的完成能检测域内对象的有限射影维数，并建立了维度移位、Eckmann-Shapiro结果及上同调乘积等。
A discrete group G has periodic cohomology over R if there is an element in a cohomology group, cup product with which induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed if R = Z, then this condition is equivalent to the existence of a finite dimensional free-G-CW- complex homotopy equivalent to a sphere. It has been conjectured by Olympia Talelli, that if G is also torsion-free then it must have finite cohomological di- mension. In this paper we use the implied condition of jump cohomology over R to prove the conjecture for HF-groups and solvable groups. We …
We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for the cup-product in the cohomology of a finite 2-complex, and an algorithm for computing the corresponding holonomy Lie algebra, using a Magnus expansion method. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as link groups, one-relator groups, and fundamental groups of orientable Seifert fibere…
In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer-Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the …
The purpose of this article is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs $\{Γ_i\}_{i\in\mathbb{N}}$ forms a family of expander graphs if and only if a certain natural mini-max invariant arising from the cup product in the cohomology rings of the groups $\{A(Γ_i)\}_{i\in\mathbb{N}}$ agrees with the Cheeger constant of the sequence of graphs, thus allowing us to characterize expander graphs via cohomology. This result is proved in the more general framework of \emph{vector space expanders}, a novel struc…
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