Shellable complexes and homology on $\Pi_n$

Abstract

Shellability and homology of finite poset are closely related. The homotopy of shellable simplicial complex is equivalent to the homotopy of some wedge of spheres. And we can abtain the basis elements of the homology of shellable simplicial complexes. Finally the homology groups of geometric lattice are calculated. We show that the homology group of $∏_n$ which is a geometry lattice of length n-1 is vanish in all dimensions except the top dimension n-3, when n≥2.