Finite Abelian Subgroups of the Mapping Class Group

Higher generation by abelian subgroups in Lie groups

To a compact Lie group $G$ one can associate a space $E(2,G)$ akin to the poset of cosets of abelian subgroups of a discrete group. The space $E(2,G)$ was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that $G$ is abelian if and only if $π_i(E(2,G))=0$ for $i=1,2,4$. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply--connected if and only if the group is abelian.

On the dimension of classifying spaces for families of abelian subgroups

We show that a finitely generated abelian group $G$ of torsion-free rank $n\geq 1$ admits a $n+r$ dimensional model for the classifying space with isotropy in the family of subgroups of torsion-free rank less than or equal to $r\geq 0$.

柄体群有限指数子群的交换化

Abelianizations of finite-index subgroups of the handlebody group

研究亏格≥4时柄体映射类群是否有非平凡有理交换化有限指数子群这一问题，通过证明子午线多重扭转在特定子群的\(H_1(\Gamma;\mathbb{Q})\)中消失及某些子群的\(H_1(\Gamma;\mathbb{Q}) = 0\)，提供不存在此类子群的证据。

Fusion systems and group actions with abelian isotropy subgroups

We prove that if a finite group $G$ acts smoothly on a manifold $M$ so that all the isotropy subgroups are abelian groups with rank $\leq k$, then $G$ acts freely and smoothly on $M \times \bbS^{n_1} \times...\times \bbS^{n_k}$ for some positive integers $n_1,...n_k$. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres with trivial action on homology.

Integral Riemann-Roch formulae for cyclic subgroups of mapping class groups

The first author conjectured certain relations for Morita-Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann-Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.

Reduction of UNil for finite groups with normal abelian Sylow 2-subgroup

Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell's unitary nilpotent groups UNil_*(Z[F];Z[F],Z[F]) have an induced isomorphism to the quotient of UNil_*(Z[S];Z[S],Z[S]) by the action of the group F/S. In particular, any finite group F of odd order has the same UNil-groups as the trivial group. The broader scope is the study of the L-theory of virtually cyclic groups, based on the Farrell--Jones isomorphism conjecture. We obtain partial information on these UNil when S is a finite abelian 2…