--- res: bibo_abstract: - "We study conditions under which a finite simplicial complex $K$ can be mapped to $\\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding is a map $f: K\\to \\mathbb R^d$ such that the images of any $r$\r\npairwise disjoint simplices of $K$ do not have a common point. We show that if $r$ is not a prime power and $d\\geq 2r+1$, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost $r$-embedding of\r\nthe $(d+1)(r-1)$-simplex in $\\mathbb R^d$. This improves on previous constructions of counterexamples (for $d\\geq 3r$) based on a series of papers by M. \\\"Ozaydin, M. Gromov, P. Blagojevi\\'c, F. Frick, G. Ziegler, and the second and fourth present authors. The counterexamples are obtained by proving the following algebraic criterion in codimension 2: If $r\\ge3$ and if $K$ is a finite $2(r-1)$-complex then there exists an almost $r$-embedding $K\\to \\mathbb R^{2r}$ if and only if there exists a general position PL map $f:K\\to \\mathbb R^{2r}$ such that the algebraic intersection number of the $f$-images of any $r$ pairwise disjoint simplices of $K$ is zero. This result can be restated in terms of cohomological obstructions or equivariant maps, and extends an analogous codimension 3 criterion by the second and fourth authors. As another application we classify ornaments $f:S^3 \\sqcup S^3\\sqcup S^3\\to \\mathbb R^5$ up to ornament\r\nconcordance. It follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous criterion for $r=2$ is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.@eng" bibo_authorlist: - foaf_Person: foaf_givenName: Sergey foaf_name: Avvakumov, Sergey foaf_surname: Avvakumov foaf_workInfoHomepage: http://www.librecat.org/personId=3827DAC8-F248-11E8-B48F-1D18A9856A87 - foaf_Person: foaf_givenName: Isaac foaf_name: Mabillard, Isaac foaf_surname: Mabillard foaf_workInfoHomepage: http://www.librecat.org/personId=32BF9DAA-F248-11E8-B48F-1D18A9856A87 - foaf_Person: foaf_givenName: A. foaf_name: Skopenkov, A. foaf_surname: Skopenkov - foaf_Person: foaf_givenName: Uli foaf_name: Wagner, Uli foaf_surname: Wagner foaf_workInfoHomepage: http://www.librecat.org/personId=36690CA2-F248-11E8-B48F-1D18A9856A87 orcid: 0000-0002-1494-0568 dct_date: 2015^xs_gYear dct_language: eng dct_title: Eliminating higher-multiplicity intersections, III. Codimension 2@ ...