  We show that the decision problem of determining whether a given (abstract
simplicial) $k$-complex has a geometric embedding in $\mathbb R^d$ is complete
for the Existential Theory of the Reals for all $d\geq 3$ and $k\in\{d-1,d\}$.
This implies that the problem is polynomial time equivalent to determining
whether a polynomial equation system has a real solution. Moreover, this
implies NP-hardness and constitutes the first hardness results for the
algorithmic problem of geometric embedding (abstract simplicial) complexes.
