Recall that an abstract simplicial complex is a family $AS$ of non-empty finite sets over a ground set $V = \bigcup AS$ which is closed under taking non-empty subsets. We say $AS$ is pure when the inclusion-wise maximal sets of $AS$ all have the same number of elements.
I want to ask two very related questions about realizations. For this I need the following definition: A crossing-free drawing of $AS$ is a point $p_i \in \mathbb{R}^3$ for each $i \in V$ such that the following holds. For any pair of sets $\sigma_1,\sigma_2 \in AS$ there is an affine functional $f(x) = \left< a,x \right> +b$ such that $f(p_i) \leq 0$ for all $i \in \sigma_1$ and $f(p_j) \geq 0$ for all $j \in \sigma_2$. Furthermore, each tetrahedra(simplex of 4 points) should have positive volume.
QUESTION1: What is the algorithmic complexity of deciding if a given abstract simplicial complex can be drawn in $3$-space?
QUESTION2: What is the complexity of deciding if an abstract simplicial complex can be realized as a simplicial complex in 3-space?
In the second question, we really want each face to be in general position and two faces are either disjoint or they share a face.
thanks Till