  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 $d=3$ verifies 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 verify a special
case of Whiteley's vertex splitting conjecture for global rigidity.
