I\n 1 (p)®Sn2(Q) is generated by tensors w E p®nl ® Sn2(Q) satisfying w(ij) = w for some transposition (ij), it is naturally an Sn(B)-submodule.
M} and k-dimensional faces of the standard (m - 1)simplex. It follows that ILinkF (( VI, ... , V m » I is the barycentric subdivision of the boundary of the standard (m - 1)-simplex and is hence an (m - 2)-sphere. Thus we have ILink:F((Vl,"" vm»1 ~m-lILinkt((vl"'" * ILinkt((Vl,"" vm»1 vm»I, where E denotes the suspension functor. To complete the proof, it remains to lihow that Linkt (( VI, ... , V m » is (d - m)- acyclic. This is proved by induction 011 m. Denote by L the poset Linkt((Vl,'" ,vm ».