Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The Fundamental Group of Balanced Simplicial Complexes and Posets. | The Fundamental Group of Balanced Simplicial Complexes and Posets Steven Klee Department of Mathematics Box 354350 University of Washington Seattle WA 98195-4350 USA klees@ Submitted Sep 29 2008 Accepted Apr 18 2009 Published Apr 27 2009 Mathematics Subject Classifications 05E25 06A07 55U10 Dedicated to Anders Bjorner on the occasion of his 60th birthday Abstract We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected balanced simplicial complexes and more generally simplicial posets. 1 Introduction One commonly studied combinatorial invariant of a finite d 1 -dimensional simplicial complex A is its f-vector f f0 . fd-1 where fi denotes the number of i-dimensional faces of A. This leads to the study of the h-numbers of A defined by the relation xd o hiXd-i xd o fi-1 A 1 d-i. A great deal of work has been done to relate the f-numbers and h-numbers of A to the dimensions of the singular homology groups of A with coefficients in a certain field see for example the work of Bjorner and Kalai in 2 and 3 and Chapters 2 and 3 of Stanley 13 . In comparison very little seems to be known about the relationship between the f -numbers of a simplicial complex and various invariants of its homotopy groups. In this paper we bound the minimal number of generators of the fundamental group of a balanced simplicial complex in terms of h2. More generally we bound the minimal number of generators of the fundamental group of a balanced simplicial poset in terms of h2. It was conjectured by Kalai 7 and proved by Novik and Swartz in 8 that if A is a d 1 -dimensional manifold that is orientable over the field k then h2 hi Ị 1 1 where p1 is the dimension of the singular homology group H1 A k . The Hurewicz Theorem see Spanier 10 says that H1 X Z is isomorphic to the abelianization of n1 X THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R7 1 for a connected space X. We will see below that n1 A is
