Some surface subgroups survive surgery
D Cooper, D D Long
证明了在单尖点双曲三维流形上通过Dehn手术得到的基本群，除极少数例外，都包含一个闭曲面的基本群.
It is shown that with finitely many exceptions, the fundamental group obtained by Dehn surgery on a one cusped hyperbolic 3-manifold contains the fundamental group of a closed surface.
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The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group $I_{n+1}(π,w)$ and the closed manifold subgroup $C_{n+1}(π,w)$ are equal in dimensions $n+1\geq 6$, for any finitely-presented group $π$ and any orientation character $w\colon π\to \cy 2$.
Surgery in codimension 3 and the Browder--Livesay invariants
手术障碍群$L_n(π)$的惯性子群$I_n(π)$是由作用在具有$π_1(X)=π$的闭拓扑流形$X^{n-1}$的同伦三角剖分集$\Cal S(X)$上的元素生成的.这个群是群$C_n(π)$的一个子群，群由闭流形的法向映射所能实现的元素组成。在所有已知的情况下，这些群体是一致的，它们的计算是外科理论的基本问题之一。组$C_n(π)$的计算等价于组合映射$A:H_{n}(Bπ, \bold L_{\bullet})\to L_{n}(π)$的图像的计算。流形$X$的每个Browder-Livesay滤子都提供了Browder-Livesay不变量的集合，这些不变量是闭流形手术问题中的禁止不变量.在本文中，我们描述了所有可能的禁止不变量，可以给一个Browder-Livesay过滤计算的惯性子群。我们的方法是汉布尔顿和Kharshiladze的方法的自然推广。更准确地说，我们证明了一个给定的流形的Browder-Livesay过滤可以给以下禁止的不变量的元素$x\in L_n(π_1(X))$属于子群$I_n(π)$：非平凡的Browder-Livesay不变量在余维0，1，2和非平凡类的障碍的限制正常映射的子流形在余维3。
Kawauchi defined a group structure on the set of homology $S^1$$\times$$S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. It is natural to ask whether the zero-surgery homomorphism is injective. We show that this question has a negative answer in the smooth category. Indeed, using knot concordance invariants derived from knot Floer homology we show that the kernel of the zero-surgery homomorphism contains a $\mathbb{Z}^\infty$-subgroup.
Surface subgroups and handlebody attachment
本文的主要定理将Dehn手术的最新结果推广到了体部附着的情况。我们考虑了在不可约、边界不可约、环面和非圆柱形三维流形的边界上，分别附着两个实体和两个实环面。我们证明了，对于一个大类的同胚附加这些体，所得到的流形的基本群包含的基本群的闭曲面的亏格至少为2。
A surgery approach to abelian quotients of the level 2 congruence group and the Torelli group
本文通过计算模2同调球和映射环面的Rochlin不变量，建立统一框架研究Torelli群的Birman-Craggs映射和2级同余子群的Sato映射，给出同态性的新证明、Dehn扭转上的计算方法及两者关系。
Immersed surfaces and Dehn surgery
设$F$是双曲三维流形$M$中的真本质浸入曲面，其边界与$M$的环面边界分量$T$不相交.令$α$为$T$上的$F$的共环斜率的集合。本文的主要定理表明，在$T$上存在一个常数$K$和一个有限的斜率集$Λ$，使得如果$β$是$T$上的一个斜率，且对$α$中的所有$α_i$都有$Δ(β, α_i) > K$，且$β$不在$Λ$中，则在沿斜率$β$向$T$上进行Dehn填充后，$F$仍然是不可压缩的.从某种意义上说，这意味着$F$可以在大多数Dehn填充物中存活。证明使用了极小曲面理论、微分形式的积分和几何有限群的性质。作为我们的方法的结果，它也将被证明，弗里德曼管浸没几何有限表面是必不可少的，如果管是足够长的。
Using a recent result of Bartels and Lueck (arXiv:0901.0442) we deduce that the Farrell-Jones Fibered Isomorphism conjecture in L-theory is true for any group which contains a finite index strongly poly-free normal subgroup, in particular, for the Artin full braid groups. As a consequence we explicitly compute the surgery groups of the Artin pure braid groups. This is obtained as a corollary to a computation of the surgery groups of a more general class of groups, namely for the fundamental group of the complement of any fiber-type hyperplane arrangement in the complex n-space.
Peripheral fillings of relatively hyperbolic groups
德恩手术的一组理论版本进行了研究。本文从任意相对双曲群$G$出发，定义了一个外围填充过程，它通过模拟完全有限体积双曲3-流形$M$的Dehn填充对基本群$π_1(M)$的影响，生成$G$的等价物.本文的主要结果是Thurston的双曲Dehn手术定理的代数对应。我们还证明了$G$的“几乎”的外围子群具有同余扩张性质，并且群$G$可由外围填充得到的子群近似（在代数意义上）.这些结果的各种应用进行了讨论。
Torus Surgery Constraints and Intermediate Covers of the Cartwright--Steger Surface
本文研究 Cartwright–Steger 曲面上的环面手术与二次覆盖构造，确定了节点曲线子群和基本群，得到具有特定符号数与基本群的极小辛流形。
Extendable mapping classes of knotted surfaces obtained by rim surgery in $S^4$
研究在 $S^4$ 中通过沿非分离曲线进行标准无扭边缘手术得到的打结曲面的可扩展映射类子群，并利用结群的子午线-经线刚性条件精确计算该子群。
By examining the homology groups of a 4-manifold associated to an integral surgery on a knot $K$ in a rational homology 3-sphere $Y$ yielding a rational homology 3-sphere $Y^*$ with surgery dual knot $K^*$, we show that the subgroups generated by $[K]$ and $[K^*]$ in $H_1(Y)$ and $H_1(Y^*)$, respectively, have co-prime orders. We obtain an immediate corollary that, in conjunction with an argument of Lidman and Moore, proves the cosmetic crossing conjecture for knots whose branched double covers are Heegaard Floer L-spaces.
Dehn filling and the knot group II: Ubiquity of persistent elements
研究非平凡结的Dehn填充下持久元素的存在性，证明每非平凡结群中存在无限多个持久元素，且这些元素在群结构中广泛分布。
Adding high powered relations to large groups
一个群称为“大”群，如果某个有限指数子群允许一个满同态到一个非交换的自由群上。本文的主要定理如下。设G是一个n阶生成的大群，且设g_1，...，g_r是G的元素的集合。则G/<<g_1 ^n，.，g_r^n>>也很大，对于无穷多个整数n。此外，当G是自由的时，这对除n之外的所有n都成立。这些结果在Dehn手术中有以下应用。设M是紧可定向3-带边流形的环面流形.假设Dehn在边界上填充一定斜率得到的三维流形具有大的基本群。那么对于无限多个填方边坡，这是正确的。
Given a mapping class f of an oriented surface Sigma and a lagrangian lambda in the first homology of Sigma, we define an integer n_{lambda}(f). We use n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping class group of Sigma as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are bas…
Fold maps are higher dimensional versions of Morse functions and fundamental and important tools in studying algebraic and differential topological properties of manifolds: as the theory established by Morse and the higher dimensional version, started by Thom and Whitney, later actively studied by Eliashberg, Levine etc. and recently studied by Kobayashi, Saeki, Sakuma etc., explicitly show this. One of fundamental, important and difficult studies on this field is, constructing explicit fold maps and investigating their source manifolds. As fundamental and strong tools for systematic construct…
This paper studies properly embedded surfaces in the 4-ball that are exotically knotted (i.e., topologically but not smoothly isotopic), and leverages this local phenomenon to study surfaces in larger 4-manifolds. The main results provide a new construction of exotically knotted surfaces, including exotic slice surfaces of all genera in the 4-ball and exotic closed surfaces in larger 4-manifolds. The construction is well-suited to the complex and symplectic settings, providing the first examples of exotically knotted complex curves and symplectic 2-spheres. Along the way, we articulate some di…
Let $Σ$ be a compact oriented surface. The Dehn twist along every simple closed curve $γ\subset Σ$ induces an automorphism of the fundamental group $π$ of $Σ$. There are two possible ways to generalize such automorphisms if the curve $γ$ is allowed to have self-intersections. One way is to consider the `generalized Dehn twist' along $γ$: an automorphism of the Malcev completion of $π$ whose definition involves intersection operations and only depends on the homotopy class $[γ]\in π$ of $γ$. Another way is to choose in the usual cylinder $U:=Σ\times [-1,+1]$ a knot $L$ projecting onto $γ$, to p…
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