Algebraic & Geometric Topology

Representation stability for the cohomology of the pure string motion groups

Jennifer Wilson

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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}

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


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.

Export citation


  • J C Baez, D K Wise, A S Crans, Exotic statistics for strings in 4D $BF$ theory, Adv. Theor. Math. Phys. 11 (2007) 707–749
  • N Brady, J McCammond, J Meier, A Miller, The pure symmetric automorphisms of a free group form a duality group, J. Algebra 246 (2001) 881–896
  • T Brendle, A Hatcher, Configuration spaces of rings and wickets
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
  • A Brownstein, R Lee, Cohomology of the group of motions of $n$ strings in 3–space, from: “Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991)”, Contemp. Math. 150, Amer. Math. Soc., Providence, RI (1993) 51–61
  • T Church, B Farb, Representation theory and homological stability
  • D J Collins, Cohomological dimension and symmetric automorphisms of a free group, Comment. Math. Helv. 64 (1989) 44–61
  • D M Dahm, A generalization of braid theory, Phd thesis, Princeton University (1962)
  • R Fenn, R Rimányi, C Rourke, Some remarks on the braid-permutation group, from: “Topics in knot theory (Erzurum, 1992)”, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 399, Kluwer Acad. Publ., Dordrecht (1993) 57–68
  • R Fenn, R Rimányi, C Rourke, The braid-permutation group, Topology 36 (1997) 123–135
  • W Fulton, Young tableaux, London Mathematical Society Student Texts 35, Cambridge Univ. Press (1997)
  • M Geck, G Pfeiffer, Characters of finite Coxeter groups and Iwahori–Hecke algebras, London Mathematical Society Monographs. New Series 21, The Clarendon Press Oxford University Press, New York (2000)
  • D L Goldsmith, The theory of motion groups, Michigan Math. J. 28 (1981) 3–17
  • J Griffin, Diagonal complexes and the integral homology of the automorphism group of a free product
  • A Hatcher, N Wahl, Stabilization for mapping class groups of 3–manifolds, Duke Math. J. 155 (2010) 205–269
  • D J Hemmer, Stable decompositions for some symmetric group characters arising in braid group cohomology, J. Combin. Theory Ser. A 118 (2011) 1136–1139
  • C Jensen, J McCammond, J Meier, The integral cohomology of the group of loops, Geom. Topol. 10 (2006) 759–784
  • C A Jensen, N Wahl, Automorphisms of free groups with boundaries, Algebr. Geom. Topol. 4 (2004) 543–569
  • J McCool, On basis-conjugating automorphisms of free groups, Canad. J. Math. 38 (1986) 1525–1529
  • A Pettet, Finiteness properties for a subgroup of the pure symmetric automorphism group, C. R. Math. Acad. Sci. Paris 348 (2010) 127–130
  • R L Rubinsztein, On the group of motions of oriented, unlinked and unknotted circles in $\mathbb{R}^3$ I, preprint (2002)
  • V V Vershinin, On homological properties of singular braids, Trans. Amer. Math. Soc. 350 (1998) 2431–2455
  • F Wattenberg, Differentiable motions of unknotted, unlinked circles in 3–space, Math. Scand. 30 (1972) 107–135