用$K$ -理论实现指定阿贝尔群的同调

Prescribed virtual homological torsion of 3-manifolds

We prove that given any finite abelian group $A$ and any irreducible $3$-manifold $M$ with empty or toroidal boundary which is not a graph manifold there exists a finite cover $M' \to M$ so that $A$ is a direct factor in $H_1(M',\mathbb{Z})$. This generalizes results of Sun and of Friedl-Herrmann.

Non-commutative groups as prescribed polytopal symmetries

We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.

受控粗同调与等周不等式

Controlled coarse homology and isoperimetric inequalities

我们研究一个粗糙的同调论与规定的增长条件。对于具有字长度量的有限生成群G，这个同调论与G的顺从性有关。我们刻画消失的某个基本类在我们的同调在G上的等周不等式，并表明，在任何组至多线性控制是需要这个类消失。后者是一个同调版本的经典伯恩赛德问题的无限群，正解。作为应用，我们描述了存在的原始的体积形式与规定的增长，并表明粗同源类阻碍加权庞加莱不等式。