对的粗亚历山大对偶及其应用

补空间的粗上同调

Coarse cohomology of the complement

该研究定义了度量空间子空间补集的粗（上）同调以推广Roe的理论，构建补集粗几何结构的模型空间，提出粗庞加莱对偶空间新研究方法，证明粗亚历山大对偶版本并给出粗PD($n$)空间的同调判别准则。

三维中的拟同态与庞加莱对偶

Quasimorphisms and Poincaré duality in dimension 3

研究具有粗连通拟核的无界实值拟同态的PD^3群，证明其要么是环面或克莱因瓶丛的基本群，要么拟等距于黎曼流形(R^3,g)，且拟核粗等价于双曲平面；若群双曲，则在S^1上有忠实拟对称作用。

Quasi-actions on trees II: Finite depth Bass-Serre trees

This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poincare duality groups. The main theorem says that, under certain hypotheses, if G is a finite graph of coarse Poincare duality groups then any finitely generated group quasi-isometric to the fundamental group of G is also the fundamental group of a finite graph of coarse Poincare dua…

Noncommutative knot theory

The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S^3-K, considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From the perspective of the knot group, G, these invariants reflect the structure of G^(1)/G^(2) as a module over G/G^(1) (here G^(n) is the n-th term of the derived series of G). Hence any phenomenon associated to G^(2) is invisible to abelian invariants. This paper begins the systematic study of invarian…

On Gromov's positive scalar curvature conjecture for virtual duality groups

We prove the inequality $$ \dim_{mc}\Wi M\le n-2$$ for the macroscopic dimension of the universal covers $\Wi M$ of almost spin $n$-manifolds $M$ with positive scalar curvature whose fundamental group $π_1(M)$ is a virtual duality group that satisfies the coarse Baum-Connes conjecture.

Generating families and Legendrian contact homology in the standard contact space

We show that if a Legendrian knot in standard contact ${\bb R}^3$ possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting we show Sabloff's duality result for LCH may be viewed as Alexander duality. In addition, we provide an explicit construction of a generating family for a front diagram with graded normal ruling and give a new approach to augmentation $\Rightarrow$ normal ruling.

Symplectic $(-2)$ spheres and the symplectomorphism group of small rational 4-manifolds, I

Let $(X,ω)$ be a symplectic rational 4 manifold. We study the space of tamed almost complex structures $\mathcal{J}_ω$ using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional Alexander duality. This decomposition provides new understandings of both the variation and stability of the symplectomorphism group $Symp(X,ω)$ when deforming $ω$. In particular, we compute the rank of $π_1(Symp(X,ω))$ with Euler number $χ(X)\leq 7$ in terms of the number $N$ of -2 symplectic sphere classes.

稳定混合换位子长度的粗群理论研究

Coarse group theoretic study on stable mixed commutator length

本文利用粗群理论研究群及其正规子群上稳定混合换位子长度的大尺度行为，通过双不变度量将其与拟态商空间联系，证明了粗核同构于整数格，并给出了在有限生成群中的应用。