Self-mapping Degrees of 3-Manifolds

Finiteness of 3-manifolds associated with non-zero degree mappings

We prove a finiteness result for the $\partial$-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and $\partial$-irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a given compact 3-manifold belong, up to homeomorphism, to a finite collection of compact 3-manifolds. We show also that any closed orientable 3-manifold dominates only finitely many integral homology spheres and any compact 3-manifolds orientable 3-manifold dominates only finitel…

The mapping class group of manifolds which are like projective planes

In this paper we compute the mapping class group of simply-connected closed smooth manifolds $M$ with integral homology $H_{*}(M) \cong \mathbb Z \oplus \mathbb Z \oplus \mathbb Z$ provided that $\dim M \ne 4$.

On Self-mapping Degrees of $S^3$-geometry manifolds

In this paper we determined all of the possible self mapping degrees of the manifolds with $S^3$-geometry, which are supposed to be all 3-manifolds with finite fundamental groups. This is a part of a project to determine all possible self mapping degrees of all closed orientable 3-manifold in Thurston's picture.

Finiteness of mapping degree sets for 3-manifolds

By constructing certain maps, this note completes the answer of the Question: For which closed orientable 3-manifold $N$, the set of mapping degrees $cD(M,N)$ is finite for any closed orientable 3-manifold $M$?

The stable mapping class group of simply connected 4-manifolds

We consider mapping class groups Γ(M) = pi_0 Diff(M fix \partial M) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3-spheres. We show that if M contains CP^2 (with either orientation) as a connected summand then Γ(M) is independent of the number of boundary components. By repackaging classical results of Wall, Kreck and Quinn, we show that the natural homomorphism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with CP^2 # \bar{CP^2}. We next consid…

Non-density of stable mappings on non-compact manifolds

Around 1970, Mather established a significant theory on the stability of $C^\infty$ mappings and gave a characterization of the density of proper stable mappings in the set of all proper mappings. The result yields a characterization of the density of stable mappings in the set of all mappings in the case where the source manifold is compact. The aim of this paper is to complement Mather's result. Namely, we show that the set of stable mappings in the set of all mappings is never dense if the source manifold is non-compact. Moreover, as a corollary of Mather's result and the main theorem of th…

The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold

This article is the author's contribution to the volume "Problems on mapping class groups and related topics" which will be published in December 2006, with Benson Farb as Editor. Various individuals were invited by Farb to submit open problems which seemed interesting to them about surface mapping class groups. We singled out a particular recent development, Hempel's concept of the `Heegaard distance' of an element in the mapping class group, and used it to describe a number of open questions which relate to the correspondence between Heegaard surfaces in 3-manifolds and the mapping class gro…

Profinite rigidity of graph manifolds, II: knots and mapping classes

In this paper we study some consequences of the author's classification of graph manifolds by their profinite fundamental groups. In particular we study commensurability, the behaviour of knots, and relation to mapping classes. We prove that the exteriors of graph knots are distinguished among all 3-manifold groups by their profinite fundamental groups. We also prove a strong conjugacy separability result for certain mapping classes of surfaces.

Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic

It is well known that Sullivan showed that the mapping class group of a simply connected high-dimensional manifold is commensurable with an arithmetic group, but the meaning of "commensurable" in this statement seems to be less well known. We explain why this result fails with the now standard definition of commensurability by exhibiting a manifold whose mapping class group is not residually finite. We do not suggest any problem with Sullivan's result: rather we provide a gloss for it.

Virtual Betti numbers of mapping tori of 3-manifolds

Given a reducible $3$-manifold $M$ with an aspherical summand in its prime decomposition and a homeomorphism $f\colon M\to M$, we construct a map of degree one from a finite cover of $M\rtimes_f S^1$ to a mapping torus of a certain aspherical $3$-manifold. We deduce that $M\rtimes_f S^1$ has virtually infinite first Betti number, except when all aspherical summands of $M$ are virtual $T^2$-bundles. This verifies all cases of a conjecture of T.-J. Li and Y. Ni, that any mapping torus of a reducible $3$-manifold $M$ not covered by $S^2\times S^1$ has virtually infinite first Betti number, except…

将加法幺半群实现为映射度集

Realizing additive monoids as mapping degree sets

该研究证明映射度集在特定运算下保持稳定，实现了由ℤ的加法子半群经有限次和与积得到的集合作为映射度集，拓展了实现问题研究并关联相关学者提出的问题。

The simplicial volume of mapping tori of 3-manifolds

We prove that any mapping torus of a closed 3-manifold has zero simplicial volume. When the fiber is a prime 3-manifold, classification results can be applied to show vanishing of the simplicial volume, however the case of reducible fibers is by far more subtle. We thus analyse the possible self-homeomorphisms of reducible 3-manifolds, and use this analysis to produce an explicit representative of the fundamental class of the corresponding mapping tori. To this end, we introduce a new technique for understanding self-homeomorphisms of connected sums in arbitrary dimensions on the level of clas…

三维流形的映射类群综述

A survey on mapping class groups of 3-manifolds

该研究是对紧致定向连通三维流形映射类群的综述，涵盖相关计算工具、文献指南、特定流形证明及分解对映射类群的推论。

带边界的单连通4-流形的映射类群

Mapping class groups of simply connected 4-manifolds with boundary

该研究计算了所有紧致单连通拓扑4-流形的拓扑映射类群，填补了带边界情形的研究空白，并对光滑情形推导了稳定光滑映射类群的描述，拓展了已有研究成果。

Countable groups are mapping class groups of hyperbolic 3-manifolds

We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G.

形如3维完全交的流形的映射类群

Mapping class group of manifolds which look like $3$-dimensional complete intersections

本文计算了形如3维完全交的闭单连通6维流形的映射类群，确定了其阿贝尔化、中心、剩余有限性等代数性质，并给出了生成元及在π_3(M)上平凡作用的子群的生成元与关系。

非紧流形上 $C^0$-稳定映射的非稠密性

Non-density of $C^0$-stable mappings on non-compact manifolds

本文利用拓扑临界点证明非紧源流形上 $C^0$-稳定映射不稠密，并结合Mather定理给出其稠密当且仅当源流形紧的刻画。

Mapping class groups of compression bodies and 3-manifolds

We analyze the mapping class group of extendible automorphisms of the exterior boundary W of a compression body of dimension 3 or 4, which extend over the compression body (Q,V), where V is the interior boundary. Those that extend as automorphisms of (Q,V) rel V are called discrepant automorphisms, forming the mapping class group of discrepant automorphisms of W in Q. We describe a short exact sequence of these mapping class groups. For an orientable, compact, reducible 3 manifold W, there is a canonical "maximal" 4-dimensional compression body Q whose exterior boundary is W and whose interior…

Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds

Mapping class groups of Haken 3-manifolds enjoy many of the homological finiteness properties of mapping class groups of 2-manifolds of finite type. For example, H(M) has a torsionfree subgroup of finite index, which is geometrically finite (i. e. is the fundamental group of a finite aspherical complex). This was proven by J. Harer for 2-manifolds and by the second author for Haken 3-manifolds. In this paper we prove that H(M) acts properly discontinuously on a contractible simplicial complex, with compact quotient. This implies that every torsionfree subgroup of finite index in H(M) is geomet…

The rational stable homology of mapping class groups of universal nil-manifolds

We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute the rational stable homology of the outer automorphism groups and of the mapping class groups of the associated aspherical nil-manifolds in the TOP, PL, and DIFF categories.

solenoidal 流形的映射类群

Mapping class groups of solenoidal manifolds

本研究计算 solenoidal 流形的映射类群，引入动态可公度群并开发相关计算方法，推广了 Odden、Bering 和 Studenmund 的先前结果，还包含相关示例。

三维流形上映射环面的最小体积熵

Minimal volume entropy of mapping tori over 3-manifolds

本文证明定向闭光滑3-流形上映射环面的最小体积熵为零，方法基于可容范畴变体及Babenko-Sabourau的同调类最小体积熵版本。

带1柄的4流形的映射类群

On the mapping class groups of 4-manifolds with 1-handles

本文推广Budney-Gabai的$W_3$不变量到带1柄的4流形，并证明此类流形的映射类群中心具有无限秩。

Examples of non-trivial contact mapping classes for overtwisted contact manifolds in all dimensions

We construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity.

单连通4维流形中具有平凡可延拓映射类群的嵌入曲面

Embedded surfaces with trivial extendable mapping class groups in simply connected $4$-manifolds

该研究在单连通4维流形中构造无穷多类满足特定映射类群与Alexander模条件的嵌入曲面，借助补集群结构实现区分与刚性检测。

The action of the mapping class group on metrics of positive scalar curvature

We present a rigidity theorem for the action of the mapping class group $π_0(\mathrm{Diff}(M))$ on the space $\mathcal{R}^+(M)$ of metrics of positive scalar curvature for high dimensional manifolds $M$. This result is applicable to a great number of cases, for example to simply connected $6$-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the $2$-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected $7$-dimensional $\mathr…

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$. In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold is virtually symplectic unless its virtual first Betti number is 1.

Cantor流形的渐近映射类群及其有限性性质

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

该研究证明了特定曲面渐近映射类群的$F_\infty$型有限性，提出更一般的Cantor流形渐近映射类群$F_\infty$型判定定理，给出多类重要示例并验证了部分流形下其同调与稳定同调的一致性。

4-流形中的曲面与可扩展映射类

Surfaces in 4-manifolds and extendible mapping classes

本文研究4-流形中曲面的嵌入与可扩展映射类，证明多数曲面在4-球或4-球面同调型流形中无灵活嵌入，并给出可扩展映射类的新生成集，改进先前结果。

Mapping the surgery exact sequence for topological manifolds to analysis

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also o…

Immersing almost geodesic surfaces in a closed hyperbolic three manifold

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.

Finiteness of nonzero degree maps between three-manifolds

In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that degree onto only finitely many homeomorphically distinct orientable closed 3-manifolds.

负曲率四流形的映射类群

Mapping class groups of negatively curved four-manifolds

本文证明所有具严格负截面曲率度量的闭定向光滑四流形的 Torelli 群（光滑与拓扑版本）的第一有理同调群均为无限维，推广了 Budney--Gabai 的实双曲流形结果。

Invariant incompressible surfaces in reducible 3-manifolds

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$. As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures: A reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-id$, or the mapping torus of a hyperbolic automorphism.

Intrinsic Differential Geometry and the Existence of Quasimeromorphic Mappings

We give a new proof of the existence of nontrivial quasimeromorphic mappings on a smooth Riemannian manifold, using solely the intrinsic geometry of the manifold.

Automatic continuity for homeomorphism groups of noncompact manifolds

We extend the proof of automatic continuity for homeomorphism groups of manifolds to non-compact manifolds and manifolds with marked points and their mapping class groups. Specifically, we show that, for any manifold $M$ homeomorphic to the interior of a compact manifold, and a set $X \subset M$ homeomorphic to the union of a Cantor set and finite set, the relative homeomorphism group $\mathrm{Homeo}(M, X)$ and the mapping class group $\mathrm{Homeo}(M, X)/\mathrm{Homeo}_0(M,X)$ have the property that any homomorphism from such a group to any separable topological group is necessarily continuo…

同调中无2-挠和3-挠的单连通闭自旋5维流形的映射类群

Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology

该研究确定了同调中无2-挠和3-挠的单连通闭自旋5维流形的Torelli群与配边群的同构关系，明确了#^g(S²×S³)的映射类群及其相关代数性质，还研究了S³在S²×S³中的嵌入。

The action of mapping classes on nilpotent covers of surfaces

Let $Σ$ be a surface whose interior admits a hyperbolic structure of finite volume. In this paper, we show that any infinite order mapping class acts with infinite order on the homology of some universal $k$--step nilpotent cover of $Σ$. We show that a Torelli mapping class either acts with infinite order on the homology of a finite abelian cover, or the suspension of the mapping class is a 3--manifold whose fundamental group has positive homology gradient. In the latter case, it follows that the suspended 3--manifold has a large fundamental group. It follows that every element of the Magnus k…

单连通凯勒流形的映射类群

Mapping Class Groups of Simply Connected Kähler Manifolds

本文综述并补充单连通紧凯勒流形映射类群与Torelli群的结果，重点分析P^4中超曲面情形，证明模空间基本群到映射类群的同态具有大核和无限指数像，并提出若干开放问题。

Statistics of Random 3-Manifolds occasionally fibering over the circle

We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume (whenever the manifold is hyperbolic), the dilatation of the monodromy, the injectivity radius, and the bottom eigenvalue of the Laplacian on these mapping tori. We also study mapping tori of punctured surface bundles, and various invariants of their Dehn fillings. We also study …

Homological eigenvalues of mapping classes and torsion homology growth for fibered 3--manifolds

Let S be an orientable surface with negative Euler characteristic, let ψ\in\Mod(S) be a mapping class of S, and let T_ψ be the mapping torus of ψ. We study the action of lifts of ψon the homology of finite covers of S via the torsion homology growth of towers of finite covers of T_ψ. We show that ψadmits a lift to a finite cover with a homological eigenvalue of length greater than one if and only if the mapping torus T_ψ admits a finite cover X and a certain tower of abelian covers which have exponential torsion homology growth. We show that the existence of such a lift of ψis intrinsic to T_ψ…

hyperkähler流形的光滑Birman-Hilden理论

A smooth Birman-Hilden theory for hyperkähler manifolds

本文为容许hyperkähler结构的流形证明了Birman-Hilden定理的光滑类比，借此探究具有非平凡基本群的4维流形的光滑映射类群，并应用于Enriques曲面的Nielsen实现。

Broken Lefschetz fibrations and mapping class groups

The purpose of this note is to explain a combinatorial description of closed smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of surface mapping classes, and further explore these connections. This is obtained via monodromy representations of simplified broken Lefschetz fibrations on 4-manifolds, for which we provide an extension of Hurwitz moves that allows us to uniquely determine the isomorphism class of a broken Lefschetz fibration. We furthermore discuss broken Lefschetz fibrations whose monodromies are contained in special subgroups of the mapping class group; n…

Computing HF^ by factoring mapping classes

Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these bimodules for arc slides explicitly, and then use them to give a combinatorial description of HF^ of a closed three-manifold, as well as the bordered Floer homology of any 3-manifold with boundary.

不可分解的准共形映射 manifold

Indecomposable Quasiconformal Maps of Manifolds

本文研究了闭合manifold上不可分解的准共形映射的存在性，核心方法是构造具有任意小共形扭曲的映射组合，主要贡献是证明了此类映射的不可分解性。

Weil-Petersson translation length and manifolds with many fibered fillings

We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson translation length contains a finite set of transverse and level closed curves, and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3-manifolds. The number of manifolds in the finite list depends only on the bound for normalized translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil-Petersson translation length of compositions of pseudo-Anosov mapping classes …

Hyperbolic 3-manifolds and Cluster Algebras

We advocate the use of cluster algebras and their y-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster y-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedra. We also comment on the completeness of hyperbolic structures.