Cell-Like Maps and Surgery

Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds

This paper proves a theorem about Dehn surgery using a new theorem about PSL(2, C) character varieties. Confirming a conjecture of Boyer and Zhang, this paper shows that a small hyperbolic knot in a homotopy sphere having a non-trivial cyclic slope r has an incompressible surface with non-integer boundary slope strictly between r-1 and r+1. A corollary is that any small knot which has only integer boundary slopes has Property P. The proof uses connections between the topology of the complement of the knot, M, and the PSL(2, C) character variety of M that were discovered by Culler and Shalen. T…

Moduli Spaces of Planar Pentagonal Linkages: Combinatorial Description

Moduli spaces of planar polygonal linkages admit a cell structure which can be realized as a surgery on the permutohedron. We present a 3D visualization of the result of the surgery for all types of non-degenerate pentagonal linkages.

Notes on fold maps obtained by surgery operations and algebraic information of their Reeb spaces

The theory of Morse functions and their higher dimensional versions or fold maps on manifolds and its application to geometric theory of manifolds is one of important branches of geometry and mathematics. Studies related to this was started in 1950s by differential topologists such as Thom and Whitney and they have been studied actively. In this paper, we study fold maps obtained by surgery operations to fundamental fold maps, and especially Reeb spaces, defined as the spaces of all connected components of preimages and in suitable situations inheriting fundamental and important algebraic inva…

On the splitting problem for manifold pairs with boundaries

The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the $ rel \partial$ case the surgery on a manifold pair is considered with the given fixed manifold structure on the boundary. In the relative case the surgery on the manifold with boundary is considered without fixing maps on the boundary. Consider a normal map to a manifold pair $(Y, \partial Y)\subset (X, \partial …

Combinatorial Calabi flow with surgery on surfaces

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Yamabe flow introduced by Luo. To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu and Gu-Luo-Sun-Wu, we prove that for any initial Euclidean or hyperbolic polyhe…

拉伸叶状结构与双曲 Dehn 手术

Stretch laminations and hyperbolic Dehn surgery

研究双曲3流形Dehn手术族中最佳Lipschitz圆值映射的最大拉伸叶状结构，基于Thurston范数和Dehn填充斜率长度给出准则判断其何时为填充核心曲线并集，并证明存在无穷多例子使叶状结构仅含闭叶，为无限循环覆盖中的轨道闭包提供信息。

Surgery spectral sequence and stratified manifolds

Cappell and Shaneson pointed out in 1978 interesting properties of Browder - Livesay invariants which are similar to differentials in some spectral sequence. Such spectral sequence was constructed in 1991 by Hambleton and Kharshiladze. This spectral sequence is closely related to a problem of realization of elements of Wall groups by normal maps of closed manifolds. The main step of construction of the spectral sequence is an infinite filtration of spectra in which only the first two, as is well-known, have clear geometric sense. The first one is a spectrum $\Bbb L(π_1(X))$ for surgery obstruc…

On $π- π$ theorem for manifold pairs with boundaries

Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(π_1(Y)\toπ_1(X))$. A well known result of Wall, the so called $π$-$π$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $π_1(X)\congπ_1(Y)$ is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced surgery obstruction group for manifold pairs $LP_*$ and splitting obstruction groups $LS_*$. In the present paper we formulate and …

Surgery in codimension 3 and the Browder--Livesay invariants

The inertia subgroup $I_n(π)$ of a surgery obstruction group $L_n(π)$ is generated by elements which act trivially on the set of homotopy triangulations $\Cal S(X)$ for some closed topological manifold $X^{n-1}$ with $π_1(X)=π$. This group is a subgroup of the group $C_n(π)$ which consists of the elements which can be realized by normal maps of closed manifolds. In all known cases these groups coincide and the computation of them is one of the basic problems of surgery theory. The computation of the group $C_n(π)$ is equivalent to the computation the image of the assembly map $A:H_{n}(Bπ, \bol…

Splitting along a submanifold pair

The paper introduces a group $LSP$ of obstructions for splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the $LSP$-groups with various surgery obstruction groups for manifold triple and structure sets arising from triples of manifolds. The natural map from the surgery obstruction group of the ambient manifold to the $LSP$-group provides an invariant when elements of the Wall group are not realized by normal maps of closed manifolds. Some $LSP$-groups are computed precisely.

无定向链环Floer同调中的一个无定向skein正合三角形

An unoriented skein exact triangle in unoriented link Floer homology

本文在无定向链环Floer同调中定义了带映射并证明其构成无定向skein正合三角形，同时通过引入瞬子Floer同调中2-手术正合三角形的Heegaard Floer类比作为关键工具，揭示了无定向纽结Floer同调与$I^{\sharp}$的对应关系。

Degree one maps between small 3-manifolds and Heegaard genus

We prove a rigidity theorem for degree one maps between small 3-manifolds using Heegaard genus, and provide some applications and connections to Heegaard genus and Dehn surgery problems.

瞬子弗勒尔同调的纽结手术公式 I：主定理

Knot surgery formulae for instanton Floer homology I: the main theorem

本文针对3-流形Y中满足[K]=0∈H₁(Y;ℚ)且m≠0的纽结K，证明了带边瞬子同调Iᵛ(Yₘ(K))的整手术公式，其证明基于缝合瞬子同调SHI和导出范畴的八面体引理，还得到相关正合三角形及独立价值的新结果。

Extending Topological Surgery to Natural Processes and Dynamical Systems

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where a sphere of dimension 0 or 1 is selected, forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of tornadoes, in the phenomenon of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we introduce new theoreti…

关于嵌入与无环映射

On embeddings and acyclic maps

本文研究闭流形无环映射下嵌入关系，先解庞加莱对偶空间变体，再用手术机器得流形结果，并推导光滑同调球块嵌入空间的同伦类型。

Non-zero degree maps between $2n$-manifolds

Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism $ϕ: H^n(L;Z)\to H^n(M;Z)$ can be realized by a map $f:M\to L$ of degree $k$ for closed $(n-1)$-connected $2n$-manifolds $M$ and $L$, $n>1$. A corollary is that each $(n-1)$-connected $2n$-manifold admits selfmaps of degree larger than 1, $n>1$. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree $k$ map from a closed orientable 4-manifold $M$ to a closed simply connected 4-man…

Generalized Markoff Maps and McShane's Identity

We study general representations of the free group on two generators into $SL(2,C)$, and the connection with generalized Markoff maps, following Bowditch. We show that Bowditch's Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations and that the subset of representations satisfying these conditions is the largest open subset in the relative character variety on which the mapping class group acts properly discontinuously. Moreover we generalize Bowditch's results on variations of McShane's identity for complet…