Order Analogues and Betti Polynomials

AbstractWe exhibit an order-preserving surjection from the lattice of subgroups of a finite abelianp-group of typeλonto the product of chains of lengths the parts of the partitionλ. 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 complexesΔS(p), whose simplices are chains of subgroups of orderspk, somek∈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 inp. Our main result provides a topological explanation of this enumerative result. We use our order-preserving surjection to findβS(pmaximal simplices inΔS(pwhose 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