---
_id: '8183'
abstract:
- lang: eng
text: "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."
acknowledgement: We would like to thank A. Klyachko, V. Krushkal, S. Melikhov, M.
Tancer, P. Teichner and anonymous referees for helpful discussions.
article_number: '1511.03501'
article_processing_charge: No
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: A.
full_name: Skopenkov, A.
last_name: Skopenkov
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity
intersections, III. Codimension 2. *arXiv*.
apa: Avvakumov, S., Mabillard, I., Skopenkov, A., & Wagner, U. (n.d.). Eliminating
higher-multiplicity intersections, III. Codimension 2. *arXiv*.
chicago: Avvakumov, Sergey, Isaac Mabillard, A. Skopenkov, and Uli Wagner. “Eliminating
Higher-Multiplicity Intersections, III. Codimension 2.” *ArXiv*, n.d.
ieee: S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity
intersections, III. Codimension 2,” *arXiv*. .
ista: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity
intersections, III. Codimension 2. arXiv, 1511.03501.
mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections, III.
Codimension 2.” *ArXiv*, 1511.03501.
short: S. Avvakumov, I. Mabillard, A. Skopenkov, U. Wagner, ArXiv (n.d.).
date_created: 2020-07-30T10:45:19Z
date_published: 2015-11-15T00:00:00Z
date_updated: 2021-11-11T15:39:18Z
day: '15'
department:
- _id: UlWa
external_id:
arxiv:
- '1511.03501'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1511.03501
month: '11'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: submitted
related_material:
record:
- id: '8156'
relation: dissertation_contains
status: public
- id: '9308'
relation: later_version
status: public
- id: '10220'
relation: later_version
status: public
status: public
title: Eliminating higher-multiplicity intersections, III. Codimension 2
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2015'
...