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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections.
We focus on conditions for the existence of almost r-embeddings, i.e., maps f: K -> R^d such that the intersection of f(sigma_1), ..., f(sigma_r) is empty whenever sigma_1,...,sigma_r are pairwise disjoint simplices of K.
Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions: If r d > (r+1) dim K + 2 then there exists an almost r-embedding K-> R^d if and only if there exists an equivariant map of the r-fold deleted product of K to the sphere S^(d(r-1)-1).
This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r-1)k, for some k> 2), and settles an open question raised there.

Isaac Mabillard and Uli Wagner. Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mabillard_et_al:LIPIcs.SoCG.2016.51, author = {Mabillard, Isaac and Wagner, Uli}, title = {{Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {51:1--51:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.51}, URN = {urn:nbn:de:0030-drops-59438}, doi = {10.4230/LIPIcs.SoCG.2016.51}, annote = {Keywords: Topological Combinatorics, Tverberg-Type Problems, Simplicial Complexes, Piecewise-Linear Topology, Haefliger-Weber Theorem} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

The fact that the complete graph K_5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M), where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2.
Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van Kampen--Flores theorem.
In the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.

Xavier Goaoc, Isaac Mabillard, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner. On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 476-490, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goaoc_et_al:LIPIcs.SOCG.2015.476, author = {Goaoc, Xavier and Mabillard, Isaac and Pat\'{a}k, Pavel and Pat\'{a}kov\'{a}, Zuzana and Tancer, Martin and Wagner, Uli}, title = {{On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {476--490}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.476}, URN = {urn:nbn:de:0030-drops-51256}, doi = {10.4230/LIPIcs.SOCG.2015.476}, annote = {Keywords: Heawood Inequality, Embeddings, Van Kampen–Flores, Manifolds} }

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