ORDER ANALOGS AND BETTI POLYNOMIALS

We exhibit an order-preserving surjection from the lattice of subgroup s of a finite abelian p-group of type lambda onto the product of chain s of lengths the parts of the partition lambda. Thereby, we establish the subgroup lattice as an order-theoretic, not just enumerative, p-an alogue of the chain product. This insight underlies our study of the s implicial complexes Delta(S)(p), whose simplices are chains of subgrou ps of orders p(k), some k is an element of S. Each of these subgroup c omplexes is homotopy equivalent to a wedge of spheres of dimension \S\ - 1. The number of spheres in the wedge, beta(S)(p), is known to have nonnegative coefficients as a polynomial in p. Our main result provid es a topological explanation of this enumerative result. We use our or der-preserving surjection to find beta(S)(p) maximal simplices in Delt a(S)(p) whose deletion leaves a contractible subcomplex. This work sug gests a definition of order analogue; our main result holds for any se mimodular lattices that are order analogues of a semimodular lattice. (C) 1996 Academic Press, Inc.