Order analogues and betti polynomials

Abstract

We exhibit an order-preserving surjection from the lattice of subgroups of a finite abelian p-group of type λ onto the product of chains of lengths the parts of the partition λ. Thereby, we establish the subgroup lattice as an order-theoretic, not just enumerative, p-analogue of the chain product. This insight underlies our study of the simplicial complexes ΔS(p), whose simplices are chains of subgroups of orders pk, some k ∈ S. Each of these subgroup complexes is homotopy equivalent to a wedge of spheres of dimension |S| - 1. The number of spheres in the wedge, βS(p), is known to have nonnegative coefficients as a polynomial in p. Our main result provides a topological explanation of this enumerative result. We use our order-preserving surjection to find βS(p) maximal simplices in ΔS(p) whose deletion leaves a contractible subcomplex. This work suggests a definition of order analogue; our main result holds for any semimodular lattices that are order analogues of a semimodular lattice. © 1996 Academic Press, Inc.