Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Face vectors of two-dimensional Buchsbaum complexes. | Face vectors of two-dimensional Buchsbaum complexes Satoshi Murai Department of Mathematics Graduate School of Science Kyoto University Sakyo-ku Kyoto 606-8502 Japan murai@ Submitted Dec 3 2008 Accepted May 21 2009 Published May 29 2009 Mathematics Subject Classifications 13F55 Abstract In this paper we characterize all possible h-vectors of 2-dimensional Buchsbaum simplicial complexes. 1 Introduction Given a class C of simplicial complexes to characterize the face vectors of simplicial complexes in C is one of central problems in combinatorics. In this paper we study face vectors of 2-dimensional Buchsbaum simplicial complexes. We recall the basics of simplicial complexes. A simplicial complex A on n 1 2 . n is a collection of subsets of n satisfying that i i G A for all i G n and ii if F G A and G c F then G G A. An element F of A is called a face of A and maximal faces of A under inclusion are called facets of A. A simplicial complex is said to be pure if all its facets have the same cardinality. Let fk A be the number of faces F G A with F k 1 where F is the cardinality of F. The dimension of A is dimA max k fk A 0 . The vector f A f_i A fo A . fd-1 A is called the f-vector or face vector of A where d dimA 1 and where f_1 A 1. When we study face vectors of simplicial complexes it is sometimes convenient to consider h-vectors. Recall that the h-vector h A ho A h1 A . hd A of A is defined by the relation Xd 0 fi_1 A x 1 d-i Xd 0 hi A xd-i. Thus knowing f A is equivalent to knowing h A . Let H A K be the reduced homology groups of A over a field K. The numbers hi A dimK Hi A K are called the Betti numbers of A over K . The link of A with respect to F G A is the simplicial complex lkA F G c n F G u F G A . In the study of face vectors of simplicial complexes one of important classes of sim-plicial complexes are Cohen-Macaulay complexes which come from commutative algebra theory. A d 1 -dimensional simplicial complex A is said to be Cohen-Macaulay
