Representation stability for the cohomology of the pure string motion groups

Wilson, Jennifer

doi:10.2140/agt.2012.12.909

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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)