## Algebraic & Geometric Topology

### Representation stability for the cohomology of the pure string motion groups

Jennifer Wilson

#### 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 $k≥1$, 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.

#### Article information

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

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

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

#### 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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