On \(f\)-vectors and Betti numbers of multicomplexes
The authors consider the \(f\)-vectors and Betti numbers of multicomplexes. Multicomplexes are generalizations of simplicial complexes (they are collections of monomials closed under divisibility). The pairs \((f,b)\) of \(f\)-vectors and Betti numbers arising from finite simplicial complexes as well as those arising from regular cell complexes with the intersection property are already characterized by the first author and \textit{G. Kalai} [Acta Math. 161, No. 3/4, 279-303 (1988; Zbl 0667.52008) and Applied geometry and discrete mathematics, Festschr. 65th Birthday Victor Klee, DIMACS, Ser. Discret. Math. Theor. Comput. Sci. 4, 81-89 (1991; Zbl 0733.55002)]. The authors construct for any multicomplex \(M\) a cell complex \(\Delta_M\) which specializes for squarefree monomials to the standard realization of a simplicial complex; and they define the Betti number of \(M\) to be the Betti number of \(\Delta_M\). The authors give a sufficient condition and a necessary condition for a pair \((f,b)\) of an \(f\)-vector and a Betti number to arise from a multicomplex. Alternative approaches to characterize the \((f,b)\)-pairs are discussed.
scientific article; zbMATH DE number 1134624