Global Rigidity of Triangulated Manifolds

We prove that if $G$ is the graph of a connected triangulated $(d-1)$-manifold, for $d\geq 3$, then $G$ is generically globally rigid in $\mathbb R^d$ if and only if it is $(d+1)$-connected and, if $d=3$, $G$ is not planar. The special case $d=3$ resolves a conjecture of Connelly. Our results actually apply to a much larger class of simplicial complexes, namely the circuits of the simplicial matroid. We also give two significant applications of our main theorems. We show that that the characterisation of pseudomanifolds with extremal edge numbers given by the Lower Bound Theorem extends to circuits of the simplicial matroid. We also prove the generic case of a conjecture of Kalai concerning the reconstructability of a polytope from its space of stresses. The proofs of our main results adapt earlier ideas of Fogelsanger and Whiteley to the setting of global rigidity. In particular we resolve a special case of Whiteley's vertex splitting conjecture for global rigidity.