On the mapping class groups of $#_r(S^p \times S^p)$ for $p = 3, 7$
Diarmuid Crowley
备注：
Presentation significantly improved following referee's comments; e.g. added Corollary 2.8 to make certain classes in the second cohomology of the symplectic group explicit. To appear in Math. Zeit.. 12 pages
For M_r = #_r(S^p \times S^p), p=3, 7, we calculate the group of isotopy classes of orientation preserving diffeomorphisms of $M_r$ modulo isotopy classes with representatives which are the identity outside a 2p-disc and also the group of homotopy classes of orientation preserving homotopy equivalences of M_r.
The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping class groups does not, in general, surject on to a mapping class group of a punctured sphere, but on to a finite index subgroup. We call this finite index subgroup the liftable mapping class group. In order to initiate the generalization of results on the hyperelliptic mapping c…
We construct a new family of examples of parabolically geometrically finite subgroups of the mapping class group in the sense of Dowdall-Durham-Leininger-Sisto and prove they are undistorted in Mod($S$).
We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N.V. Ivanov, which asserts that any "sufficiently rich" object associated to a surface has automorphism group…
Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free subgroup of the mapping class groups.
We obtain representations of the f-based Mapping Class Group of oriented punctured surfaces from an action of mapping classes on Heisenberg homologies of a circle bundle over surface configurations.
We determine the center of a meta-nilpotent quotient of a mapping-torus group. As a corollary, we introduce two invariants, which are quadratic forms, of knots and of mapping classes.
We determine the first homology group with coefficients in $H_1(N;\mathbb{Z})$ for the mapping class group of a non-orientable surface $N$ of genus three with two boundary components.
We study the TQFT mapping class group representations for surfaces with boundary associated with the $SU(2)$ gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show that at a prime root of unity, these representations are all irreducible. We also examine braid group representations for transcendental values of the quantum parameter, where we show that the image of every mapping class group is Zariski dense.
The present paper are the notes of a mini-course addressed mainly to non-experts. It purpose it to provide a first approach to the theory of mapping class groups of non-orientable surfaces.
We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.
We prove that the mapping class group of a Heegaard splitting with a distance of at least 3 is finite. However, we have constructed a counterexample with a distance of 2 that disproves this assertion. In addition, the fact that the mapping class group of a Heegaard splitting with a distance of at most 1 is infinite, when combined with our results, provides an answer to the question of the finiteness of mapping class groups as viewed from Heegaard distance.
Using the Galatius--Kupers--Randal-Williams framework of cellular $E_2$-algebras, we prove a secondary stability theorem for mapping class groups of nonorientable surfaces. As a corollary, we obtain a new best known stability range for the homology of the mapping class groups of nonorientable surfaces with respect to adding torus holes.
We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds.
In this note we give presentations of all finite subgroups of the mapping class group of a closed surface of genus 2 by the Humphries generators up to conjugacy.
We show that Szepietowski's system of generators for the mapping class group of a non-orientable surface is a minimal generating set by Dehn twists and $Y$-homemorphisms.
We prove the Farrell-Jones Conjecture for mapping class groups. The proof uses the Masur-Minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of Teichmueller space. The proof is presented in an axiomatic setup, extending the projection axioms of Bestvina-Bromberg-Fujiwara. More specifically, we prove that the action of the mapping class group on the Thurston compactification of Teichmueller space is finitely F-amenable for the family F consisting of virtual point stabilizers.
Groupoid Extensions of Mapping Class Representations for Bordered Surfaces
具有一个边界分量的曲面的映射类群允许许多有趣的表示，包括作为自由群的自同构群和作为辛变换群。由于映射类群可以与黎曼模空间的基本群等同，因此在选择基点时，它还可以与基本道路群胚的子群等同。映射类广群（英语：mapping class groupoid）是一个组合模型，它来自于泰希米勒空间的不变胞腔剖分，其基本路径广群被称为托勒密广群。尝试将映射类群的表示扩展到映射类groupoid是很自然的，即，构造一个从映射类广群到同一目标的同态，该同态扩展了由基点的各种选择引起的给定表示。除此之外，我们将上述表示扩展到广群水平，在这个意义上，辛表示被合理地和整体地提升。证明的技术包括几个算法，涉及胖图和弦图。前一种扩张由六种基本情形下的显式公式给出，并计算了广群表示的核和像。此外，这提供了通过其在表面的基本群上的作用而分解的映射类群的任何表示的广群扩展，包括例如Magnus表示和平坦联络的模空间上的表示。
We obtain a simple presentation of the hyperelliptic mapping class group $M^h(N)$ of a nonorientable surface N. As an application we compute the first homology group of $M^h(N)$ with coefficients in $H_1(N;Z)$.
A crosscap transposition is an element of the mapping class group of a nonorientable surface represented by a homeomorphism supported on a one-holed Klein bottle and swapping two crosscaps. We prove that the mapping class group of a compact nonorientable surface of genus $g\ge 7$ is generated by conjugates of one crosscap transposition. In the case when the surface is either closed or has one boundary component, we give an explicit set of $g+2$ crosscap transpositions generating the mapping class group.
Based on the action of the mapping class group on the space of measured foliations, we construct a new boundary of the mapping class group and study the structure of this boundary. As an application, for any point in Teichmuller space, we consider the orbit of this point under the action of the mapping class group and describe the closure of this orbit in the Thurston compactification and the Gardiner-Masur compactification of Teichmuller space. We also construct some new points in the Gardiner-Masur boundary of Teichmuller space.
We prove that many normal subgroups of the extended mapping class group of a surface with punctures are geometric, that is, that their automorphism groups and abstract commensurator groups are isomorphic to the extended mapping class group. In order to apply our theorem to a normal subgroup we require that the "minimal supports" of its elements satisfy a certain complexity condition that is easy to check in practice. The key ingredient is proving that the automorphism groups of many simplicial complexes associated to punctured surfaces are isomorphic to the extended mapping class group. This r…
We prove that each Torelli group of an orientable surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus-Farb-Putman's techniques. Further we show that the distortion of each Torelli group in the level $d$ mapping class group is the same as that of in the mapping class group.
In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative and give a proof that every countable group appears as a sub…
In this article, we give a necessary and sufficient condition for embedding a finite index subgroup of Artin's braid group into the mapping class group of a connected orientable surface.
