The complex of words and Nakaoka stability

Abstract

We give a new simple proof of the exactness of the complex of injective words and use it to prove Nakaoka's homology stability for symmetric groups. The methods are generalized to show acyclicity in low degrees for the complex of words in position. Hm(§ni1;Z) = Hm(§n;Z) for n=2 > m where §n denotes the permutation group of n elements. An elementary proof of this fact has not been available in the literature. In the first section the complex C⁄(m) of abelian groups is studied which in de- gree n is freely generated by injective words of length n. The alphabet consists of m letters. The complex C⁄(m) has the only non vanishing homology in degree m (Theorem 1). This is a result of F.D. Farmer (3) who connected it to properties of the associated poset of injective words and its CW-complex. Then, considering the action of the permutation group on the alphabet, a hyperho- mology argument is used to deduce Nakaoka stability. In the second section - independent from the first - more general complexes of words are shown to have vanishing homology in low degrees. The proof is analogous to Theorem 1. A very general setting is used but the essential application is to the complex of words consisting of vectors in general position over a finite field. In fact this complex is shown to become exact in some fixed degree and fixed dimension of the vector space if only the base field has enough elements. This could be of interest for example for Suslin's GL-stability (8) which works up to now only over infinite fields.