Enumeration of Q-acyclic simplicial complexes

Let[Figure not available: see fulltext.] (n, k) be the class of all simplicial complexes C over a fixed set of n vertices (2≦k≦n) such that: (1)C has a complete (k-1)-skeleton, (2)C has precisely ( _k ^n-1 )k-faces, (3)H _k (C)=0. We prove that for[Figure not available: see fulltext.], H _k-1(C) is a finite group, and our main result is:[Figure not available: see fulltext.]. This formula extends to high dimensions Cayley's formula for the number of trees on n labelled vertices. Its proof is based on a generalization of the matrix tree theorem.