[论文解读] Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensions

Robert D. Gray, Benjamin Steinberg|arXiv (Cornell University)|May 9, 2018

Geometric and Algebraic Topology参考文献 48被引用 4

一句话总结

本文利用等变分类空间与Bass–Serre理论，建立了单群的拓扑有限性性质，证明了特殊单群的F_n与FP_n性质可由其单位群继承。该文解决了长期悬而未决的问题，表明单 relator 单群⟨A∣r=1⟩属于F_∞类型，且当r不是真幂时，其几何与上同调维数均不超过2。

ABSTRACT

We show how topological methods developed in a previous article can be applied to prove new results about topological and homological finiteness properties of monoids. A monoid presentation is called special if the right-hand side of each relation is equal to $1$. We prove results which relate the finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid inherits the finiteness properties $F_n$ and $FP_n$ from its group of units. We also obtain results which relate the geometric and cohomological dimensions of such a monoid to those of its group of units. We apply these results to prove a Lyndon's Identity Theorem for one-relator monoids of the form $\langle A \mid r=1 angle$. In particular we show that all such monoids are of type $F_{\infty}$ (and $FP_{\infty}$), and that when $r$ is not a proper power, then the monoid has geometric and cohomological dimension at most $2$. The first of these results resolves an important case of a question of Kobayashi from 2000 on homological finiteness properties of one-relator monoids. We also show how our topological approach can be used to prove results about the closure properties of various homological and topological finiteness properties for amalgamated free products and HNN-extensions of monoids. To prove these results we introduce new methods for constructing equivariant classifying spaces for monoids, as well as developing a Bass-Serre theory for free constructions of monoids.

研究动机与目标

• 开发用于研究单群有限性性质的拓扑框架，扩展自群论的方法。
• 通过证明单 relator 单群属于F_∞类型，解决Kobayashi于2000年提出的关于单 relator 单群同调有限性的问题。
• 在自由构造（如单群的自由乘积并和HNN扩张）下，建立拓扑与同调有限性的封闭性质。
• 将Bass–Serre理论推广至单群，并为单群作用构造等变分类空间。
• 将特殊单群的几何与上同调维数与其单位群的维数联系起来。

提出的方法

• 将群论中的拓扑方法（特别是Eilenberg–Mac Lane空间与K(G,1)复形）适配至单群。
• 利用离散群作用与单群在单纯复形上的作用，构造单群的等变分类空间。
• 引入基于树与森林的单群版本Bass–Serre理论，以建模自由构造。
• 利用群与单群环的平坦性与投射性条件，将有限性性质传递至扩张。
• 应用Anick–Groves–Squier定理与Hochschild上同调，关联重写系统与FP_n性质。
• 基于其单位群结构与子单群作用，建立单群属于F_n或FP_n类型的判别准则。

实验结果

研究问题

• RQ1 特殊单群是否能从其单位群继承F_n与FP_n性质？
• RQ2 形如⟨A∣r=1⟩的单 relator 单群是否属于F_∞类型？其几何与上同调维数是多少？
• RQ3 自由乘积并与HNN扩张的理论能否推广至单群，并保持类似的有限性性质？
• RQ4 何种条件可确保单群扩张保持拓扑或同调有限性性质？
• RQ5 特殊单群的几何与上同调维数如何与其中单位群的维数相关？

主要发现

• 特殊单群可从其单位群继承F_n与FP_n性质，建立了单群与群之间有限性性质的强关联。
• 形如⟨A∣r=1⟩的单 relator 单群属于F_∞（及FP_∞）类型，解决了Kobayashi于2000年提出的关键开放问题。
• 当关系式r不是真幂时，此类单群的几何与上同调维数均不超过2。
• 在单群及其子单群作用满足特定条件下，单群的自由乘积并可保持F_n与FP_n性质。
• 当基单群与子单群满足平坦性与有限性条件时，单群的HNN型扩张可保持双FP_n与双F_n性质。
• 在自由性假设下，HNN扩张的几何维数受基单群几何维数与子单群几何维数加一的最大值所限制。