Abstract
We prove that certain sequences of Artin monoids containing the braid monoid as a submonoid satisfy homological stability. When the $K(\pi,1)$ conjecture holds for the associated family of Artin groups this establishes homological stability for these groups. In particular, this recovers and extends Arnol'd's proof of stability for the Artin groups of type $A$, $B$ and $D$.
Highlights
A sequence of groups or monoids with maps between themG1 → G2 → · · · → Gn → · · · is said to satisfy homological stability if the induced maps on homologyHi(Gn) → Hi(Gn+1)are isomorphisms for n sufficiently large compared to i
In the case of finite Coxeter groups, Corollary C recovers the few known cases of stability for families of Artin groups of the form studied in this paper, that is, homological stability holds for the sequences of Artin groups {An}n 1 of type A, B and D, given by the following diagrams: These three sequences consist of Artin groups which relate to finite Coxeter groups
∂kp : as the projection φp : (A+n) \\ Cpn → A+n \\ Cpn−1, where ∂kp is induced by the face maps of Cn, which we recall are a composite of right multiplication of the representative for the equivalence class in Cpn = A+(n; n − p − 1) by, before the inclusion to the equivalence class in Cpn−1
Summary
G1 → G2 → · · · → Gn → · · · is said to satisfy homological stability if the induced maps on homology. The classifying space of an Artin monoid BA+ is homotopy equivalent to some interesting spaces that arise naturally in mathematics In the case of finite Coxeter groups, Corollary C recovers the few known cases of stability for families of Artin groups of the form studied in this paper, that is, homological stability holds for the sequences of Artin groups {An}n 1 of type A, B and D, given by the following diagrams: These three sequences consist of Artin groups which relate to finite Coxeter groups. Combining the results of [14] with the high connectivity results established in this paper, our homological stability result with constant coefficients (Theorem A) can most likely be enhanced to one with abelian coefficients and coefficient systems of finite degree in the sense of [14, Section 4]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
