Peter Scott, Gadde A. Swarup
The authors previously described an algebraic analogue of the JSJ-decomposition of a 3-manifold. This analogue is defined for any finitely presented, one-ended group. We study this analogue in the special case of Poincaré duality pairs.
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Finiteness properties for some rational Poincaré duality groups
将Besthouse-Brady莫尔斯理论与非循环反射群技巧相结合，得到了一个无挠的Q-Poincaré对偶群，它不是非球面闭ANR Q-同调流形的基本群.非循环的构造建议我们问，哪些Q-Poincaré对偶群自由地作用在Q-非循环空间上，即，这些基团是FH（Q）。例如，一个好的Orbifold的Orbifold基本群Γ满足Q-Poincaré对偶，并且如果某些固定集的欧拉示性数为零，我们证明了Γ是FH（Q）.
We prove the analogue of Johannson's Deformation Theorem for PD3 pairs.
Analogues of JSJ decompositions were developed for Poincaré duality pairs in [19]. These decompositions depend only on the group. Our focus will be on describing the edge splittings of these decompositions more precisely. We use our results to compare these decompositions with two other closely related decompositions.
We show that every oriented $n$-dimensional Poincaré duality group over a $*$-ring $R$ is amenable or satisfies a linear homological isoperimetric inequality in dimension $n-1$. As an application, we prove the Tits alternative for such groups when $n=2$. We then deduce a new proof of the fact that when $n=2$ and $R = \mathbb Z$ then the group in question is a surface group.
Property (T) and Poincaré duality in dimension three
本文通过边界扩张理论证明三维庞加莱对偶群不具有性质(T)，并由此得出此类群不可能是Kähler群，同时证明了紧致3流形基本群若具有性质(T)则必为有限群。
Pro-$p$ Poincaré-Duality groups of infinite rank
本文引入了满足Poincaré对偶的无限秩pro-$p$广义Poincaré对偶群类，证明了其基本性质，并揭示了特定正合序列条件下群必为Demushkin群的结论。
Bredon-Poincare Duality Groups
若$G$是一个允许$\mathrm{B}G$有流形模型的群，则$G$是一个Poincaré对偶群.我们研究了推广的庞加莱对偶群，最初介绍了戴维斯和Leary，动机组$G$与共紧流形模型$M$为$\underline{\mathrm{E}}G$其中$M^H$是一个可收缩子流形的所有有限子群$H$的$G$。我们给出了这些Bredon-Poincaré对偶群的例子和构造的几个来源，包括使用Davis和Leary的等变反射群技巧来构造Bredon-Poincaré对偶群的例子，这些对偶群是由流形$M$上的作用产生的，其中指定了子流形$M^H$的维数。我们对低维的Bredon-Poincaré对偶群进行了分类，并讨论了在群扩张和群图下的行为。
Virtual fibring of Poincaré-duality groups
该研究刻画了RFRS庞加莱对偶群存在核为庞加莱对偶群的虚拟满同态的等价条件，建立了代数纤维化群的无穷上同调与纤维的关联，并证明三维及以上非球面流形代数纤维化的纤维具有单端性。
We prove a pro-$p$ version of Sela's theorem stating that a finitely generated group is $k$-acylindrically accessible. This result is then used to prove that $\mathrm{PD}^n$ pro-$p$ groups admit a unique $k$-acylindrical JSJ-decomposition.
Every finite complex is the classifying space for proper bundles of a virtual Poincaré duality group
证明了有限连通的单纯复形同伦等价于可收缩流形的商在虚挠自由群的适当作用下的同伦等价性。作为推论，我们得到了每个有限连通单纯复形同伦等价于某个虚Poincaré对偶群的真丛的分类空间。
A Topological Splitting Theorem for Poincare Duality Groups and High-dimensional Manifolds
本文证明了对一类广泛的流形对N，M满足dim（M）= dim（N）+ 1，每个π 1-内射映射f：N -> M作为嵌入的有限覆盖分解到同伦.这个结果，在精神瓦尔德豪森的环面定理，是来自使用卡佩尔的手术方法从一个新的代数分裂定理庞加莱对偶群。作为应用，我们得到了π_1-内射映射存在性的一个新的障碍.
Some questions on subgroups of 3-dimensional Poincaré duality groups
本文基于三维流形拓扑学提出关于三维庞加莱对偶群及其子群的若干开放性问题，探讨其结构与性质。
Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. We construct groups $H_*^\varkappa(G,M)$ for which $H_k^\varkappa (G,M^{tw}) \cong H^{n-k-1}_λ(G,M),$ where $M^{tw}$ is a twisting of a $G$-module $M$ defined in Section $5$ and $H^{*}_λ(G,M)$ is a variation of the group cohomology introduced by Zarelua, which in many cases is isomorphic to the symmetric cohomology of groups defined by Staic. The groups $H_*^\varkappa(G,M)$ come together with transformations from Tate cohomology. We find conditions under which these transformations are isomorphisms.
Quasi-Isometry Invariance of Group Splittings over Coarse Poincaré Duality Groups
我们表明，如果$G$是一个组的类型$FP_{n+1}^{\mathbb{Z}_2}$，是粗略地分为三个基本的，粗糙的不相交，粗糙的互补成分由一个粗糙的$PD_n^{\mathbb{Z}_2}$空间$W,$，然后$W$是在有限的Hausdorff距离从一个子群$H$的$G$，此外，$G$分裂的一个子群的$H$。我们使用这一点来推导出$G=A*_HB$形式的分裂，其中$G$是类型$FP_{n+1}^{\mathbb{Z}_2}$和$H$是一个粗糙的$PD_n^{\mathbb{Z}_2}$组，使得$|\mathrm{Comm}_A(H): H|$和$|\mathrm{Comm}_B(H): H|$都大于2，是不变的准等距。
On Equivariant Poincaré Duality, Gysin Morphisms and Euler Classes
这些笔记的目的，最初打算作为一本书的附录，在等变上同调的基础，是建立形式主义的$G$ -等变庞加莱对偶定向$G$ -流形，任何连接的紧李群$G$，以下工作J。L. Brylinski导致光谱序列$$\mathop{\rm Extgr}\nolimits_{H_G}(H_{G,\rm c} (M),H_G)\Rightarrow H_{G}(M)[d_{M}]\,.$$等变Gysin函子$(\_)_!:=Ω_{G}(\_)\in\mathcal D^{+}(\mathord{\rm DGM}(H_{G}))$（resp. $(\_)_{*}:=Ω_{G,\rm c}(\_)$）定义在定向$G$ -流形和真映射（分别为.无限制映射），作为等变微分形式函子$Ω_{G,\rm c}(\_)$的Cartan复形的合成（分别为. $Ω_{G}(\_)$）与对偶函子$I\mkern-4.5muR\,{\rm Hom}_{H_{G}}^{\bullet}(\_,H_{G})$和等变Poincaré附接$I\mkern-4.5muD_{G} (M):Ω_{G} (M)[d_{M}]\to I\mkern-4.5muR\,{\rm Hom}_{H_{G}}^{\bullet}(Ω_{G,\rm c} (M),H_{G} )$（resp. $I\mkern-4.5muD_{G}' (M):Ω_{G,\r…
We study the finite generation of homotopy groups of closed manifolds and finite CW-complexes by relating it to the cohomology of their fundamental groups. Our main theorems are as follows: when $X$ is a finite CW-complex of dimension $n$ and $π_1(X)$ is virtually a Poincaré duality group of dimension $\geq n-1$, then $π_i(X)$ is not finitely generated for some $i$ unless $X$ is homotopy equivalent to the Eilenberg--MacLane space $K(π_1(X),1)$; when $M$ is an $n$-dimensional closed manifold and $π_1(M)$ is virtually a Poincaré duality group of dimension $\ge n-1$, then for some $i\leq [n/2]$, …
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