Lower-bound theorems for pseudomanifolds (Q1346138)

Extending results and confirming conjectures of \textit{G. Kalai} [Invent. Math. 88, 125-151 (1987; Zbl 0624.52004)], \textit{V. Klee} [Houston J. Math. 1, 81-86 (1975; Zbl 0316.52004)] and \textit{A. Börner} [Eur. J. Comb. 1, 27-31 (1980; Zbl 0451.52004)], the author gives purely combinatorial proofs of lower-bound theorems on the number of \(k\)- dimensional faces of pseudomanifolds with or without boundary. The proof methods are mainly based on a form of rigidity which was introduced by \textit{M. Gromov} [``Partial differential relations'', Springer, Berlin (1986; Zbl 0651.53001)] and also investigated by Connelly and Whiteley. This so-called \(M_ d\)-rigidity is weaker than the known generic 3- rigidity of 2-manifolds, whose connection to the problems discussed here was pointed out already by Kalai.