Algebraic & Geometric Topology 12 (2012) 909–931

DOI: 10.2140/agt.2012.12.909

Abstract

The cohomology of the pure string motion group PΣ_n admits a natural action by the hyperoctahedral group W_n. In recent work, Church and Farb conjectured that for each k ≥ 1, the cohomology groups H^k(PΣ_n; Q) are uniformly representation stable; that is, the description of the decomposition of H^k(PΣ_n; Q) into irreducible W_n–representations stabilizes for n >> k. We use a characterization of H^*(PΣ_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 ≥ 1. We also prove that the subgroup of Σ_n⁺ ⊆ 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

Primary: 20C15, 20J06

Secondary: 20F28, 57M25

References

Publication

Received: 11 August 2011
Accepted: 19 December 2011
Published: 24 April 2012

Authors