Algebraic & Geometric Topology

Representation stability for the cohomology of the pure string motion groups

Jennifer Wilson

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Abstract

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 k1, 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 k1. 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.

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 2 (2012), 909-931.

Dates
Received: 11 August 2011
Accepted: 19 December 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715374

Digital Object Identifier
doi:10.2140/agt.2012.12.909

Mathematical Reviews number (MathSciNet)
MR2928898

Zentralblatt MATH identifier
1282.20059

Subjects
Primary: 20J06: Cohomology of groups 20C15: Ordinary representations and characters
Secondary: 20F28: Automorphism groups of groups [See also 20E36] 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

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

Citation

Wilson, Jennifer. Representation stability for the cohomology of the pure string motion groups. Algebr. Geom. Topol. 12 (2012), no. 2, 909--931. doi:10.2140/agt.2012.12.909. https://projecteuclid.org/euclid.agt/1513715374


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