紧致3-流形自同构的一个分类

压缩体的映射类群与三维流形

Mapping class groups of compression bodies and 3-manifolds

我们分析了三维或四维压缩体的外边界W的可扩自同构的映射类群，这些可扩自同构在压缩体（Q，V）上延伸，其中V是内边界。扩展为（Q，V）rel V的自同构的那些称为不一致自同构，形成W在Q中的不一致自同构的映射类群.我们描述了这些映射类群的一个短的精确序列。对于一个可定向的、紧致的、可约的3维流形W，存在一个规范的“极大”四维压缩体Q，其外边界是W，其内边界是W的不可约和的不交并。得到了三维流形的映射类群的一个短的正合序列，它给出了不可约和的不交并的映射类群作为整个映射类群与调节自同构群的商.不一致自同构群用生成元来描述。所得结果可用于紧3-流形自同构的Nielsen-Thurston分类。

Automorphisms of Hopf manifolds

We prove that the automorphism groups of Hopf manifolds are Jordan.

Geometric automorphism groups of symplectic 4-manifolds

Let $M$ be a closed, oriented, smooth $4-$manifold with intersection form $Γ$, $A(Γ)$ the automorphism group of $Γ$ and $D(M)$ the subgroup induced by orientation-preserving diffeomorphisms of $M$. In this note we study the question when $D(M)$ is of infinite index in $A(Γ)$ for a symplectic 4-manifold.

Symmetries of Spatial Graphs in $3$-manifolds

We consider when automorphisms of a graph can be induced by homeomorphisms of embeddings of the graph in a $3$-manifold. In particular, we prove that every automorphism of a graph is induced by a homeomorphism of some embedding of the graph in a connected sum of one or more copies of $S^2\times S^1$, yet there exist automorphisms which are not induced by a homeomorphism of any embedding of the graph in any orientable, closed, connected, irreducible $3$-manifold. We also prove that for any $3$-connected graph $G$, if an automorphism $σ$ is induced by a homeomorphism of an embedding of $G$ in an…

非线性体的不可约自同构的例子

Examples of irreducible automorphisms of handlebodies

在三维流形的自同构的分类中，自然地产生了非线性体的自同构。在曲面体的自同构中，有一些称为不可约（或一般）的自同构，它们类似于曲面的伪Anosov自同构。我们表明，不可约的自同构体的存在和发展的方法来构建一系列的例子。

Some finiteness results for groups of automorphisms of manifolds

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of $R^n$ and various types of automorphisms of 2-connected manifolds.

Actions of automorphism groups of free groups on homology spheres and acyclic manifolds

For n at least 3, let SAut(F_n) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,Z) on R^n induces non-trivial actions of SAut(F_n) on R^n and on S^{n-1}. We prove that SAut(F_n) admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, SAut(F_n) cannot act non-trivially on any generalized Z_2-homology sphere of dimension less than n-1, nor on any Z_2-acyclic Z_2-homology manifold of dimension less than n. It follows that SL(n,Z) cannot act non-trivially on such spaces eithe…

关于抛物结构与闭非球面流形的自同构群

On the automorphism group of parabolic structures and closed aspherical manifolds

本文研究闭非球面抛物 ${\sfG}$-流形的自同构群，证明某些抛物结构会强烈限制流形拓扑，并刻画了具有几乎可解基本群的标准 $CR$-流形与四元数接触流形的商结构。

Topological symmetries of simply-connected four-manifolds and actions of automorphism groups of free groups

Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications, let $\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $\mathrm{Aut}% (F_{n})$ $(n\geq 4)$ on $M\neq S^{4}$ by homologically trivial homeomorphisms factors through $\mathbb{Z}/2.$ Moreover, any action of $% \mathrm{SL}_{n}(\mathbb{Q})$ $(n\geq 4)$ on $M\neq S^{4}$ by homeomorphisms is trivial.

关于某些八维分段线性可微分流形的分类和$#_{i=1}^b(S^2\times S^5)$的自同构

On the classification of certain piecewise linear and differentiable manifolds in dimension eight and automorphisms of $#_{i=1}^b(S^2\times S^5)$

在这篇文章中，我们将考虑在奇数维中具有零上同调的8维单连通自旋流形的显式分类。这种流形的研究是由斯特凡·穆勒开始的。为了理解这些流形的结构，我们将分析它们的最小句柄表示，并明确地描述这些句柄表示在多大程度上是由上同调环和特征类决定的。证明了上同调环和特征类不足以完全重构上述类型的流形。事实上，$#_{i=1}^b(S^2\times S^5)$的自同构群${\rm Aut_0}\bigl(#_{i=1}^b(S^2\times S^5)\bigr)/{\rm Aut}_0\bigl(#_{i=1}^b (S^2\times D^6)\bigr)$诱导模上同调恒等式，这些模上同调恒等式延伸到$#_{i=1}^b(S^2\times D^6)$作用在具有固定上同调环和特征类的流形的定向同胚类的集合上，我们也将关注描述这个群和关于上述作用的一些事实。

A note on rational homological stability for automorphisms of manifolds

By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}\sharp(S^n\times S^n)^{\sharp g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $*\le (g-6)/2$. In this note, we explain how this range can be improved to $*\le g-2$ using cohomological vanishing results due to Borel and classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.

3-允许纽结流形作为吸引子的流形

3-manifolds that admit knotted solenoids as attractors

受莫尔斯理论研究和Smale动力学工作的启发，本文研究并回答了以下问题：（1）当一个3-流形允许一个自同构以一个打结Smale螺线管为吸引子时，它的吸引子是什么？(2)当一个3-流形允许一个非徘徊集由Smale流形组成的自同构存在时，它的非游荡集是什么？结果给出了一类3-流形的某些内在对称性。

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.