Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Abstract We exhibit an explicit homotopy equivalence | Homotopy and homology of finite lattices Andreas Blass Mathematics Department University of Michigan Ann Arbor MI 48109-1109 . ablass@ Submitted Aug 28 2001 Accepted Aug 2 2003 Published Aug 21 2003 MR Subject Classifications 06A11 05E25 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 1. This result which extends a theorem of Segev leads to a description of the homology of a finite lattice extending a result of Bjorner for geometric lattices. 1 Introduction The purpose of this paper is to unify and extend three directions of work that originated from Rota s broken-circuit formula 4 for the Mobius function of a geometric lattice. In this introduction we shall present the necessary terminology state Rota s theorem outline the three developments that are relevant for our purposes and then describe our results and how they are related to the previous ones. Throughout this paper L is a finite lattice with lattice operations written V and A and with ordering written . Its smallest and largest elements are 0 and 1 and the least upper bound of a subset X is written V X. We always assume that L is non-degenerate . that 0 1. The set of atoms . minimal non-0 elements is called A. The Mobius function y x y is defined for all x y in L and is uniquely characterized by the equations y x x 1 and Vx y y x z 0. x z y We shall be interested primarily in the special case y 0 1 . The general values y x y can be obtained by applying this special case to the intervals z x z y of L. Partially supported by NSF grant DMS-0070723. THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R30 1 In the special case that L is a geometric lattice . the lattice of flats of a matroid up to isomorphism a circuit of L is defined to be a subset of A that is minimal with respect to the property that for some a E A V A V A a . This agrees .
