Representation stability for the cohomology of the pure string motion groups

Wilson, Jennifer¹

¹Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago IL 60637, USA

Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 909-931

Résumé

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; ℚ) are uniformly representation stable; that is, the description of the decomposition of Hk(PΣn; ℚ) into irreducible Wn–representations stabilizes for n >> k. We use a characterization of H∗(PΣn; 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 Hk(Σn; ℚ) 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.

Détail

DOI : 10.2140/agt.2012.12.909

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

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