Geometric Embeddability of Complexes is ∃ℝ-complete

We show that the decision problem of determining whether a given (abstract simplicial) k -complex has a geometric embedding in ℝ d is complete for the Existential Theory of the Reals for all d ≥ 3 and k ∈ { d -1, d } by reducing from pseudoline stretchability. Consequently, 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 result for the algorithmic problem of geometrically embedding (abstract simplicial) complexes. This complements recent breakthroughs for the computational complexity of piece-wise linear embeddability [Matoušek, Sedgwick, Tancer, and Wagner, J. ACM 2018, and de Mesmay, Rieck, Sedgwick and Tancer, J. ACM 2020] and establishes connections to computational topology.