Representation stability for the cohomology of the pure string motion groups

Wilson, Jennifer

doi:10.2140/agt.2012.12.909

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Editorial Interests

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author Index

To Appear

Other MSP Journals

Representation stability for the cohomology of the pure string motion groups

Jennifer C H Wilson

Algebraic & Geometric Topology 12 (2012) 909–931

DOI: 10.2140/agt.2012.12.909

Abstract

The cohomology of the pure string motion group $P Σ_{n}$ admits a natural action by the hyperoctahedral group $W_{n}$ . In recent work, Church and Farb conjectured that for each $k \geq 1$ , the cohomology groups $H^{k} (P Σ_{n}; ℚ)$ are uniformly representation stable; that is, the description of the decomposition of $H^{k} (P Σ_{n}; ℚ)$ into irreducible $W_{n}$ –representations stabilizes for $n > > k$ . We use a characterization of $H^{*} (P Σ_{n}; ℚ)$ given by Jensen, McCammond and Meier to prove this conjecture. Using a transfer argument, we further deduce that the rational cohomology groups of the string motion group $H^{k} (Σ_{n}; ℚ)$ vanish for $k \geq 1$ . We also prove that the subgroup of $Σ_{n}^{+} \subseteq Σ_{n}$ of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.

Keywords

representation stability, homological stability, motion group, string motion group, circle-braid group, symmetric automorphism, basis-conjugating automorphism, braid-permutation group, hyperoctahedral group, signed permutation group

Mathematical Subject Classification 2000

Primary: 20J06, 20C15

Secondary: 20F28, 57M25

References

Publication

Received: 11 August 2011

Accepted: 19 December 2011

Published: 24 April 2012

Authors

Jennifer C H Wilson
Department of Mathematics University of Chicago 5734 University Avenue Chicago IL 60637 USA
http://math.uchicago.edu/~wilsonj/

Volume 12, issue 2 (2012)