Automorphisms of symmetric groups are toy examples for mapping class groups

We give a new proof establishing the automorphism groups of the symmetric groups inspired by the analogous result of Ivanov for the extended mapping class group. As a key tool, we consider the actions on the Kneser graphs.     

On upper bounds on stable commutator lengths in mapping class groups

We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating curve on an oriented closed surface of genus 2 are not equal to each other.     

On the Witten-Reshetikhin-Turaev representations of mapping class groups

We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The Witten-Reshetikhin-Turaev TQFTs associated to and induce linear representations of this group. We show that the denominators of matrices which describe these representations over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus with no colored points, the representations have finite image.

     

Surfaces in 4-manifolds and their mapping class groups

A surface in a smooth 4-manifold is called flexible if, for any diffeomorphism ? on the surface, there is a diffeomorphism on the 4-manifold whose restriction on the surface is ? and which is isotopic to the identity. We investigate a sufficient condition for a smooth 4-manifold M to include flexible knotted surfaces, and introduce a local operation in simply connected 4-manifolds for obtaining a flexible knotted surface from any knotted surface.     

Central extensions of gauge groups revisited

We present an explicit construction for the central extension of the group Map(X, G) where X is a compact manifold and G is a Lie group. If X is a complex curve we obtain a simple construction of the extension by the Picard variety Pic(X). The construction is easily adapted to the extension of Aut(E), the gauge group of automorphisms of a nontrivial vector bundle E.     

Bordism invariants of the mapping class group

We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We introduce a new representation in terms of spin bordism, and we prove that this single representation contains all of the information given by the Johnson homomorphism, the Birman-Craggs homomorphism, and the Morita homomorphism.     

Class groups of dihedral extensions

Let L/F be a dihedral extension of degree 2p, where p is an odd prime. Let K/F and k/F be subextensions of L/F with degrees p and 2, respectively. Then we will study relations between the p‐ranks of the class groups Cl(K) and Cl(k). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algebraic entropy and the action of mapping class groups on character varieties

We extend the definition of algebraic entropy to endomorphisms of affine varieties. We then calculate the algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a pseudo-Anosov mapping class. Our calculations are compatible with all known calculations of the topological entropy of this action.     

Higher genus mapping class group invariants from factorizable Hopf algebras

Lyubashenko?s construction associates representations of mapping class groups _Mapg:n${\mathrm{Map}}_{g:n}$ of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the category of bimodules over a finite-dimensional factorizable ribbon Hopf algebra H  . For any such Hopf algebra we find an invariant of _Mapg:n${\mathrm{Map}}_{g:n}$ for all values of g and n  . More generally, we obtain such invariants for any pair (H,ω)$(H,\omega )$, where ω is a ribbon automorphism of H.     

Dihedral side extensions and class groups

We present a non-abelian mirror-type principle relating the p-ranks of class groups of subfields of a dihedral field of degree 2p for an odd prime p with a limited ramification condition.     

Infinite loop space structure(s) on the stable mapping class group

Tillmann introduced two infinite loop space structures on the plus construction of the classifying space of the stable mapping class group, each with different computational advantages (Invent. Math. 130 (1997) 257; Math. Ann. 317 (2000) 613). The first one uses disjoint union on a suitable cobordism category, whereas the second uses an operad which extends the pair of pants multiplication (i.e. the double loop space structure introduced by Miller, J. Differential Geom. 24 (1986) 1). She conjectured that these two infinite loop space structures were equivalent, and managed to prove that the first delooping are the same. In this paper, we resolve the conjecture by proving that the two structures are indeed equivalent, exhibiting an explicit geometric map.     

Some computations of stable twisted homology for mapping class groups

Abstract

In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this paper.

Communicated by Jason P. Bell     

Mod p homology of the stable mapping class group

We calculate the homology of the mapping class group Γ_g,n in the stable range. The calculation is based on Madsen and Weiss’ proof of the “Generalised Mumford Conjecture”: Γ_g,n has the same homology as a component of the space in the stable range.     

Central quotients of biautomatic groups

The quotient of a biautomatic group by a subgroup of the center is shown to be biautomatic. The main tool used is the Neumann-Shapiro triangulation of S ^n-1, associated to a biautomatic structure on . Among other applications, a question of Gersten and Short is settled by showing that direct factors of biautomatic groups are biautomatic. Received: October 4, 1994     

Group extensions and free actions by finite groups on solvmanifolds

The purpose of this paper is to generalize Yalçin's result [Trans. Amer. Math. Soc. 352 , 2689–2700 (2000)] from tori to solvmanifolds. As a result, we prove that if (?_p)^r acts freely on a solvmanifold M of dimension n, then r ≤ n (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A cohomological characterization of Leibniz central extensions of Lie algebras

Mainly motivated by Pirashvili's spectral sequences on a Leibniz algebra, a cohomological characterization of Leibniz central extensions of Lie algebras is given. In particular, as applications, we obtain the cohomological version of Gao's main theorem for Kac-Moody algebras and answer a question in an earlier paper by Liu and Hu (2004).

     

Computing ray class groups, conductors and discriminants

We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of for an ideal of a number field , as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones.

     

Group actions and group extensions

In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if acts freely on , then .