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Abstract
The cohomology of the pure string motion group
P Σ n
admits a natural action by the hyperoctahedral group
W n .
In recent work, Church and Farb conjectured that for each
k
≥ 1 , the cohomology
groups
H k ( P Σ n ;
ℚ )
are uniformly representation stable; that is, the description of the decomposition of
H k ( P Σ n ;
ℚ ) into irreducible
W n –representations
stabilizes for
n
> >
k . We use
a characterization of
H ∗ ( P Σ n ;
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
H k ( Σ n ;
ℚ ) vanish for
k
≥ 1 . We also prove that
the subgroup of
Σ n +
⊆ Σ n
of orientation-preserving string motions, also known as the braid-permutation group,
is rationally cohomologically stable in the classical sense.
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
Mathematical Subject Classification 2000
Primary: 20J06, 20C15
Secondary: 20F28, 57M25
Publication
Received: 11 August 2011
Accepted: 19 December 2011
Published: 24 April 2012