Discrete & Computational Geometry

, Volume 61, Issue 2, pp 452–463 | Cite as

Hardness of almost embedding simplicial complexes in \(\mathbb {R}^d\)

  • Arkadiy SkopenkovEmail author
  • Martin Tancer


A map \(f:K\rightarrow \mathbb {R}^d\) of a simplicial complex is an almost embedding if \(f(\sigma )\cap f(\tau )=\emptyset \) whenever \(\sigma ,\tau \) are disjoint simplices of K.

TheoremFix integers\(d,k\ge 2\)such that\(d={3k}/2+1\).
  1. (a)

    Assume that\(\ne \) NP. Then there exists a finitek-dimensional complexKthat does not admit an almost embedding in\(\mathbb {R}^d\)but for which there exists an equivariant map from the deleted product\(\widetilde{K}\) to \(S^{d-1}\).

  2. (b)

    The algorithmic problem of recognition of almost embeddability of finitek-dimensional complexes in\(\mathbb {R}^d\)is NP-hard.

The proof is based on the technique from the Matoušek–Tancer–Wagner paper (proving an analogous result for embeddings), and on singular versions of the higher-dimensional Borromean rings lemma. The new part of our argument is a stronger ‘almost embeddings’ version of the generalized van Kampen–Flores theorem.


Almost embedding NP-hard Equivariant map PL embedding 

Mathematics Subject Classification

Primary 57Q35 Secondary 68Q17 05E45 



We are thankful to Slava Krushkal and the anonymous referees for helpful remarks. Arkadiy Skopenkov: Research supported by the Russian Foundation for Basic Research Grant No. 15-01-06302, by Simons-IUM Fellowship and by the D. Zimin’s Dynasty Foundation Grant. Martin Tancer: Supported by the GAČR project 16-01602Y and by Charles University project UNCE/SCI/004.


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Authors and Affiliations

  1. 1.Institutskiy per.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  2. 2.Independent University of MoscowMoscowRussia
  3. 3.Department of Applied MathematicsCharles University in PraguePraha 1Czech Republic

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