Abstract. Let EMBED k→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R d ?Known results easily imply the polynomiality of EMBED k→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBED k→2k for all k ≥ 3.We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBED d→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is the NP-hardness of EMBED 2→4 and, more generally, of EMBED k→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d − 2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spież, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.

The task of this survey is to present various results on intersection patterns of convex sets. One of main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so called d-representable, d-collapsible and d-Leray simplicial complexes which are very useful for this study. We study the differences among these notions and we also focus on computational complexity for recognizing them. A list of Helly-type theorems is presented in the survey and it is also discussed how (important) role play the above mentioned notions for the theorems. We also consider intersection patterns of good covers which generalize collections of convex sets (the sets may be 'curvy'; however their intersections cannot be too complicated). We mainly focus on new results.

We prove that it is NP-complete to decide whether a given (3-dimensional) simplicial complex is collapsible. This work extends a result of Malgouyres and Francés showing that it is NP-complete to decide whether a given simplicial complex collapses to a 1-complex.

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