Homotopy and Homology of Finite Lattices

Abstract

We exhibit an explicit homotopy equivalence between the geometric realizations of the order complex of a finite lattice and the simplicial complex of coreless sets of atoms whose join is not ${}\hat 1{}$. This result, which extends a theorem of Segev, leads to a description of the homology of a finite lattice, extending a result of Björner for geometric lattices.