Manifolds with non-stable fundamental groups, at infinity

Connected sum at infinity and 4-manifolds

We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m at least 3, we construct an infinite family of pairs of m-manifolds on which the connected sum at infinity operation yields distinct manifolds for certain ray choices. We use cohomology algebras at infinity to distinguish these manifolds.

Fuchsian轨迹附近拟Fuchsian流形的无穷远测度叶状结构与常平均曲率曲面

Measured foliations at infinity and constant mean curvature surface in quasi-Fuchsian manifolds close to the Fuchsian locus

本文研究接近Fuchsian轨迹的拟Fuchsian流形，证明小尺度填满型无理测度叶状结构可唯一实现为无穷远测度叶状结构，并证明近Fuchsian流形可由CMC曲面单调叶状化。

On uniqueness of end sums and 1-handles at infinity

For oriented manifolds of dimension at least 4 that are simply connected at infinity, it is known that end summing is a uniquely defined operation. Calcut and Haggerty showed that more complicated fundamental group behavior at infinity can lead to nonuniqueness. The present paper examines how and when uniqueness fails. Examples are given, in the categories TOP, PL and DIFF, of nonuniqueness that cannot be detected in a weaker category (including the homotopy category). In contrast, uniqueness is proved for Mittag-Leffler ends, and generalized to allow slides and cancellation of (possibly infin…

拟Fuchsian流形无穷远处的可测叶状结构

Measured foliations at infinity of quasi-Fuchsian manifolds

本文研究拟Fuchsian流形无穷远边界上的可测叶状结构对，证明了接近Fuchsian情形时该叶状结构对具有填充性，并进一步证明任意填充对在充分小的尺度下均可实现为某拟Fuchsian流形的无穷远叶状结构，从而回答了Schlenker在Fuchsian轨迹附近提出的问题。

无穷远处厚且具有薄叶状结构的流形中极小超曲面的通用疤痕现象

Generic Scarring for Minimal Hypersurfaces in Manifolds Thick at Infinity with a Thin Foliation at Infinity

本文研究了一类完备非紧流形中极小超曲面的通用疤痕现象，证明了在特定度量条件下，任意稳定极小超曲面附近均存在面积发散且重正化后收敛于它的极小超曲面序列。

无穷远处的链接与非紧致流形上的标量曲率衰减

Linking at Infinity and Scalar Curvature Decay on Non-Compact Manifolds

研究非紧致正标量曲率流形上曲率衰减与无穷远处拓扑结构的关系，证明拓扑链接迫使弱有界几何流形上标量曲率多项式衰减，并发展局部于非紧致流形端点的障碍理论。

Infinity Links L, infinity-4-Manifolds M_L and Kirby Categories

We construct what we call a Kirby category, a monoidal category whose morphisms are smooth 4-manifolds, projecting down to another monoidal category whose morphisms are orientable 3-manifolds, the projection being induced by the boundary map on manifolds. We construct a higher categorical generalization of such concepts and introduce the notion of ribbon $\infty$-categories, a generalization of braided monoidal $\infty$-categories (\cite{Lu1}), which gives rise to the concepts of $\infty$-links, $\infty$-4-manifolds as well as the more general notion of walled $\infty$-4-manifolds if one focus…

从无穷远重构 Anosov 流

Reconstruction of Anosov flows from infinity

本文从带不变几乎 laminations 的圆周群作用出发，重构出闭 3-流形中的可定向传递 pseudo-Anosov 流及其流形，并给出基于 Poincaré 圆盘乘积的几何模型，同时将结果应用于 Cannon 猜想。

Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds

The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length $\ext(f)$ of $f$, and an upper bound on $\ext(f)$. We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poinca…

Noncompact manifolds that are inward tame

We continue our study of ends of non-compact manifolds, with a focus on the inward tameness condition. For manifolds with compact boundary, inward tameness, has significant implications. For example, such manifolds have stable homology at infinity in all dimensions. We show that these manifolds have 'almost perfectly semistable' fundamental group at each end. That observation leads to further analysis of group theoretic conditions at infinity, and to the notion of a 'near pseudo-collar' structure. We obtain a complete characterization of n-manifolds (n>5) admitting such a structure, thereby ge…

魔幻 3-流形的球面 CR 均匀化

Spherical CR uniformization of the magic 3-manifold

本文证明复双曲三角形群Δ_{3,∞,∞;∞}与Δ_{3,4,∞;∞}的无穷远3-流形分别为魔幻流形6_1^3和m295，从而表明二者均具有球面CR均匀化，并支持关于一般Δ_{3,n,m;∞}无穷远流形可由6_1^3经Dehn填充得到的猜想。

Topological properties of spaces admitting free group actions

In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is not stably Z, then M does not cover any manifold (except itself). In the non-manifold case, Wright's method showed that when a one-ended, simply connected, locally compact ANR X with pro-monomorphic fundamental group at infinity admits an action of Z by covering transformations then the fundamental group at infinity of X is (up to pro-isomorphism) an inverse …

BO 上的空间是加厚流形

Spaces over BO are thickened manifolds

本文证明加厚光滑流形的无穷范畴等价于 BO 上有限空间的无穷范畴，通过几何构造推出作为核心工具，为精确辛几何铺路且不依赖光滑逼近或 h-原理。

Characterization of pseudo-collarable manifolds with boundary

In this paper we obtain a complete characterization of pseudo-collarable $n$-manifolds for $n\geq 6$. This extends earlier work by Guilbault and Tinsley to allow for manifolds with noncompact boundary. In the same way that their work can be viewed as an extension of Siebenmann's dissertation that can be applied to manifolds with non-stable fundamental group at infinity, our main theorem can also be viewed as an extension of the recent Gu-Guilbault characterization of completable $n$-manifolds in a manner that is applicable to manifolds whose fundamental group at infinity is not peripherally st…