ƒ-Vectors of acyclic complexes

✍ Scribed by Gil Kalai

• Publisher: Elsevier Science
• Year: 1985
• Tongue: English
• Weight: 128 KB
• Volume: 55
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(85)80024-8

✦ Synopsis

A simplicial complex C is acyclic if C is connected and H,(C) = 0 for all i > 0. (Hi(C) is the ith homology group of C.) l-dimensional acyclic complexes are just trees which were extensively studied in the combinatorial literature. However, the combinatorial properties of high-dimensional acyclic complexes have become a subject of investigation only in recent years (see [3, 41 and for a more general reference, [ 11).

For a finite simplicial complex C and for k a -1 denote fk(C) = {SE C: ISI = k + 1). Th e vector f(C) = u,,(C), fi(C), . . .) is the f-vector of C. The purpose of this note is to present a characterization of f-vectors of (finite) acyclic simplicial complexes. The characterization is by one well-known relation x(C) = 1 and a system of inequalities (see (1.5) below).