Abstract. An (n -1, 2)-framework in n-space is a structure consisting of a finite set of(n -2)-dimensional panels and a set of rigid bars each joining a pair of panels using ball joints. A body and hinge (or (n + 1, n -1)-) framework in n-space consists of a finite set of n-dimensional bodies articulated by a set of (n -2)-dimensional hinges, i.e., joints in 2-space, line hinges in 3-space, plane-hinges in 4-space, etc. In this paper we characterize the graphs of all rigid (n -1,2)-and (n + 1, n -1)-frameworks in n-space. Rigidity here is statical rigidity or equivalently infinitesimal rigidity.
We give purely combinatorial proofs of the lower-bound theorems for pseudomanifolds with or without boundary. Theorem 1.1. Let A be a (d -1)-pseudomanifold with v vertices. Then: (i) fk(A) >_ q~k (v, d) if 1 < k < d -1. (ii) If equality holds for any k, 1 < k < d -1, then A is a stacked (d -1)-sphere for d > 4 and a triangulated 2-sphere for d = 3. Theorem 1.2. Let A be a (d -1)-pseudomanifold whose nonempty boundary is the disjoint union of normal pseudomanifolds. Suppose A has v i vertices in the interior and 204 Tiong-Seng Tay v b vertices in the boundary, then: b (i) fk(A) >_ ~Ok(Ui , Ub , d) if 1 < k <_ d -1. (ii) fffk(A)= q~b(vi, Cb, d) for some k, 1 <_k <_d-1 then A is a stacked (d -1)-baU if d >_ 4 and a triangulateddisk if d = 3.Kalai [8] proved these two theorems for manifolds and conjectured them to be true for pseudomanifolds and pseudomanifolds with arbitrary boundary (see also [1] and [2]). Previously, Klee [9] had proved the case k = d -1 of Theorem 1.1(i) while Bj6rner [3] had conjectured Theorem 1.2(i). Kalai also pointed out, without giving details, that they would follow from the generic 3-rigidity of triangulated 2-manifolds. Later Fogelsanger [5] proved that triangulated 2-manifolds are generically 3-rigid. As a result of our investigation we think that Theorem 1.2 for the case of arbitrary boundary remains open.In 1986 Gromov [7] defined a weaker form of rigidity, M a-rigidity, and proved that triangulated 2-manifolds are M3-rigid. He then used this result to prove part (i) of Theorem 1.1. However, his proof of the M3-rigidity of triangulated 2-manifolds had some gaps. These were subsequently corrected by Connelly and Whiteley [14]. We give a simple proof of this result. Then we use the techniques developed by Kalai [8] and the idea of M d -rigidity to give a proof of Theorem 1.1. M d -rigidity is a weaker form of generic rigidity as defined in [14]. The techniques used here have their roots in theory of generic rigidity.Recently, Tay et al. [12] introduced the idea of skeletal rigidity of cell complexes, which generalizes infinitesimal rigidity of graphs. Connections between skeletal rigidity and the g-theorem of pl-spheres (piecewise linear spheres) have been made. It is conceivable that a weaker form of skeletal rigidity can be defined that generalizes Md-rigidity. The techniques in this paper could then be generalized to yield a combinatorial proof of the g-theorem.In writing up this paper we benefited greatly from the works of Kalai [8] and Gromov [7] and from continuing discussion with Walter Whiteley. For further historical note on the lower-bound theorems, readers are referred to [8].The paper is organized as follows. In Section 2 we give the basic definitions and state the MPW (McMullen-Perles-Walkup) reduction which reduces the proof of the lower-bound theorem for certain simplicial complexes to the case k = 1. In Section 3 we define normal pseudomanifolds and show that the lower-bound theorem for pseudomanifolds can be reduced to the case where the pseudomanifolds are normal. In...
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