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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

The cohomology of the pure string motion group PΣn admits a natural action by the hyperoctahedral group Wn. In recent work, Church and Farb conjectured that for each k 1, the cohomology groups Hk(PΣn; ) are uniformly representation stable; that is, the description of the decomposition of Hk(PΣn; ) into irreducible Wn–representations stabilizes for n >> k. We use a characterization of H(PΣn; ) 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 Hk(Σn; ) vanish for k 1. We also prove that the subgroup of Σn+ Σn of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.

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
Received: 11 August 2011
Accepted: 19 December 2011
Published: 24 April 2012
Jennifer C H Wilson
Department of Mathematics
University of Chicago
5734 University Avenue
Chicago IL 60637