---
_id: '10220'
abstract:
- lang: eng
text: "We study conditions under which a finite simplicial complex K can be mapped
to ℝd without higher-multiplicity intersections. An almost r-embedding is a map
f: K → ℝd such that the images of any r pairwise disjoint simplices of K do not
have a common point. We show that if r is not a prime power and d ≥ 2r + 1, then
there is a counterexample to the topological Tverberg conjecture, i.e., there
is an almost r-embedding of the (d +1)(r − 1)-simplex in ℝd. This improves on
previous constructions of counterexamples (for d ≥ 3r) based on a series of papers
by M. Özaydin, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, and the second
and fourth present authors.\r\n\r\nThe counterexamples are obtained by proving
the following algebraic criterion in codimension 2: If r ≥ 3 and if K is a finite
2(r − 1)-complex, then there exists an almost r-embedding K → ℝ2r if and only
if there exists a general position PL map f: K → ℝ2r 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 a cohomological obstruction and extends an
analogous codimension 3 criterion by the second and fourth authors. As another
application, we classify ornaments f: S3 ⊔ S3 ⊔ S3 → ℝ5 up to ornament concordance.\r\n\r\nIt
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: Research supported by the Swiss National Science Foundation (Project
SNSF-PP00P2-138948), by the Austrian Science Fund (FWF Project P31312-N35), by the
Russian Foundation for Basic Research (Grants No. 15-01-06302 and 19-01-00169),
by a Simons-IUM Fellowship, and by the D. Zimin Dynasty Foundation Grant. We would
like to thank E. Alkin, A. Klyachko, V. Krushkal, S. Melikhov, M. Tancer, P. Teichner
and anonymous referees for helpful comments and discussions.
article_processing_charge: No
article_type: original
arxiv: 1
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: Arkadiy B.
full_name: Skopenkov, Arkadiy B.
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 AB, Wagner U. Eliminating higher-multiplicity
intersections. III. Codimension 2. Israel Journal of Mathematics. 2021;245:501–534.
doi:10.1007/s11856-021-2216-z
apa: Avvakumov, S., Mabillard, I., Skopenkov, A. B., & Wagner, U. (2021). Eliminating
higher-multiplicity intersections. III. Codimension 2. Israel Journal of Mathematics.
Springer Nature. https://doi.org/10.1007/s11856-021-2216-z
chicago: Avvakumov, Sergey, Isaac Mabillard, Arkadiy B. Skopenkov, and Uli Wagner.
“Eliminating Higher-Multiplicity Intersections. III. Codimension 2.” Israel
Journal of Mathematics. Springer Nature, 2021. https://doi.org/10.1007/s11856-021-2216-z.
ieee: S. Avvakumov, I. Mabillard, A. B. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity
intersections. III. Codimension 2,” Israel Journal of Mathematics, vol.
245. Springer Nature, pp. 501–534, 2021.
ista: Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. 2021. Eliminating higher-multiplicity
intersections. III. Codimension 2. Israel Journal of Mathematics. 245, 501–534.
mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections. III.
Codimension 2.” Israel Journal of Mathematics, vol. 245, Springer Nature,
2021, pp. 501–534, doi:10.1007/s11856-021-2216-z.
short: S. Avvakumov, I. Mabillard, A.B. Skopenkov, U. Wagner, Israel Journal of
Mathematics 245 (2021) 501–534.
date_created: 2021-11-07T23:01:24Z
date_published: 2021-10-30T00:00:00Z
date_updated: 2023-08-14T11:43:55Z
day: '30'
department:
- _id: UlWa
doi: 10.1007/s11856-021-2216-z
external_id:
arxiv:
- '1511.03501'
isi:
- '000712942100013'
intvolume: ' 245'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1511.03501
month: '10'
oa: 1
oa_version: Preprint
page: '501–534 '
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P31312
name: Algorithms for Embeddings and Homotopy Theory
publication: Israel Journal of Mathematics
publication_identifier:
eissn:
- 1565-8511
issn:
- 0021-2172
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '8183'
relation: earlier_version
status: public
- id: '9308'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Eliminating higher-multiplicity intersections. III. Codimension 2
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 245
year: '2021'
...
---
_id: '9308'
acknowledgement: This research was carried out with the support of the Russian Foundation
for Basic Research(grant no. 19-01-00169)
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: A. B.
full_name: Skopenkov, A. B.
last_name: Skopenkov
citation:
ama: Avvakumov S, Wagner U, Mabillard I, Skopenkov AB. Eliminating higher-multiplicity
intersections, III. Codimension 2. Russian Mathematical Surveys. 2020;75(6):1156-1158.
doi:10.1070/RM9943
apa: Avvakumov, S., Wagner, U., Mabillard, I., & Skopenkov, A. B. (2020). Eliminating
higher-multiplicity intersections, III. Codimension 2. Russian Mathematical
Surveys. IOP Publishing. https://doi.org/10.1070/RM9943
chicago: Avvakumov, Sergey, Uli Wagner, Isaac Mabillard, and A. B. Skopenkov. “Eliminating
Higher-Multiplicity Intersections, III. Codimension 2.” Russian Mathematical
Surveys. IOP Publishing, 2020. https://doi.org/10.1070/RM9943.
ieee: S. Avvakumov, U. Wagner, I. Mabillard, and A. B. Skopenkov, “Eliminating higher-multiplicity
intersections, III. Codimension 2,” Russian Mathematical Surveys, vol.
75, no. 6. IOP Publishing, pp. 1156–1158, 2020.
ista: Avvakumov S, Wagner U, Mabillard I, Skopenkov AB. 2020. Eliminating higher-multiplicity
intersections, III. Codimension 2. Russian Mathematical Surveys. 75(6), 1156–1158.
mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections, III.
Codimension 2.” Russian Mathematical Surveys, vol. 75, no. 6, IOP Publishing,
2020, pp. 1156–58, doi:10.1070/RM9943.
short: S. Avvakumov, U. Wagner, I. Mabillard, A.B. Skopenkov, Russian Mathematical
Surveys 75 (2020) 1156–1158.
date_created: 2021-04-04T22:01:22Z
date_published: 2020-12-01T00:00:00Z
date_updated: 2023-08-14T11:43:54Z
day: '01'
department:
- _id: UlWa
doi: 10.1070/RM9943
external_id:
arxiv:
- '1511.03501'
isi:
- '000625983100001'
intvolume: ' 75'
isi: 1
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1511.03501
month: '12'
oa: 1
oa_version: Preprint
page: 1156-1158
publication: Russian Mathematical Surveys
publication_identifier:
issn:
- 0036-0279
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
related_material:
record:
- id: '8183'
relation: earlier_version
status: public
- id: '10220'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Eliminating higher-multiplicity intersections, III. Codimension 2
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 75
year: '2020'
...
---
_id: '610'
abstract:
- lang: eng
text: 'The fact that the complete graph K5 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 Kn embeds in a closed surface M (other than the Klein bottle) if and only
if (n−3)(n−4) ≤ 6b1(M), where b1(M) is the first Z2-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 Kn+1) embeds in R2k if and only if
n ≤ 2k + 1. Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex
embeds in a compact, (k − 1)-connected 2k-manifold with kth Z2-Betti number bk
only if the following generalized Heawood inequality holds: (k+1 n−k−1) ≤ (k+1
2k+1)bk. This is a common generalization of the case of graphs on surfaces as
well as the van Kampen–Flores theorem. In the spirit of Kühnel’s conjecture, we
prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold
with kth Z2-Betti number bk, then n ≤ 2bk(k 2k+2)+2k+4. This bound is weaker than
the generalized Heawood inequality, but does not require the assumption that M
is (k−1)-connected. Our results generalize to maps without q-covered points, in
the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result
of Volovikov about maps that satisfy a certain homological triviality condition.'
acknowledgement: The work by Z. P. was partially supported by the Israel Science Foundation
grant ISF-768/12. The work by Z. P. and M. T. was partially supported by the project
CE-ITI (GACR P202/12/G061) of the Czech Science Foundation and by the ERC Advanced
Grant No. 267165. Part of the research work of M.T. was conducted at IST Austria,
supported by an IST Fellowship. The research of P. P. was supported by the ERC Advanced
grant no. 320924. The work by I. M. and U. W. was supported by the Swiss National
Science Foundation (grants SNSF-200020-138230 and SNSF-PP00P2-138948). The collaboration
between U. W. and X. G. was partially supported by the LabEx Bézout (ANR-10-LABX-58).
author:
- first_name: Xavier
full_name: Goaoc, Xavier
last_name: Goaoc
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: Pavel
full_name: Paták, Pavel
last_name: Paták
- first_name: Zuzana
full_name: Patakova, Zuzana
id: 48B57058-F248-11E8-B48F-1D18A9856A87
last_name: Patakova
orcid: 0000-0002-3975-1683
- first_name: Martin
full_name: Tancer, Martin
id: 38AC689C-F248-11E8-B48F-1D18A9856A87
last_name: Tancer
orcid: 0000-0002-1191-6714
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. On generalized
Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability
result. Israel Journal of Mathematics. 2017;222(2):841-866. doi:10.1007/s11856-017-1607-7'
apa: 'Goaoc, X., Mabillard, I., Paták, P., Patakova, Z., Tancer, M., & Wagner,
U. (2017). On generalized Heawood inequalities for manifolds: A van Kampen–Flores
type nonembeddability result. Israel Journal of Mathematics. Springer.
https://doi.org/10.1007/s11856-017-1607-7'
chicago: 'Goaoc, Xavier, Isaac Mabillard, Pavel Paták, Zuzana Patakova, Martin Tancer,
and Uli Wagner. “On Generalized Heawood Inequalities for Manifolds: A van Kampen–Flores
Type Nonembeddability Result.” Israel Journal of Mathematics. Springer,
2017. https://doi.org/10.1007/s11856-017-1607-7.'
ieee: 'X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, and U. Wagner,
“On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability
result,” Israel Journal of Mathematics, vol. 222, no. 2. Springer, pp.
841–866, 2017.'
ista: 'Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. 2017. On generalized
Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability
result. Israel Journal of Mathematics. 222(2), 841–866.'
mla: 'Goaoc, Xavier, et al. “On Generalized Heawood Inequalities for Manifolds:
A van Kampen–Flores Type Nonembeddability Result.” Israel Journal of Mathematics,
vol. 222, no. 2, Springer, 2017, pp. 841–66, doi:10.1007/s11856-017-1607-7.'
short: X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, U. Wagner, Israel
Journal of Mathematics 222 (2017) 841–866.
date_created: 2018-12-11T11:47:29Z
date_published: 2017-10-01T00:00:00Z
date_updated: 2023-02-23T10:02:13Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s11856-017-1607-7
ec_funded: 1
intvolume: ' 222'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1610.09063
month: '10'
oa: 1
oa_version: Preprint
page: 841 - 866
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication: Israel Journal of Mathematics
publication_status: published
publisher: Springer
publist_id: '7194'
quality_controlled: '1'
related_material:
record:
- id: '1511'
relation: earlier_version
status: public
scopus_import: 1
status: public
title: 'On generalized Heawood inequalities for manifolds: A van Kampen–Flores type
nonembeddability result'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 222
year: '2017'
...
---
_id: '1123'
abstract:
- lang: eng
text: "Motivated by topological Tverberg-type problems in topological combinatorics
and by classical\r\nresults about embeddings (maps without double points), we
study the question whether a finite\r\nsimplicial complex K can be mapped into
Rd without triple, quadruple, or, more generally, r-fold points (image points
with at least r distinct preimages), for a given multiplicity r ≤ 2. In particular,
we are interested in maps f : K → Rd that have no global r -fold intersection
points, i.e., no r -fold points with preimages in r pairwise disjoint simplices
of K , and we seek necessary and sufficient conditions for the existence of such
maps.\r\n\r\nWe present higher-multiplicity analogues of several classical results
for embeddings, in particular of the completeness of the Van Kampen obstruction
\ for embeddability of k -dimensional\r\ncomplexes into R2k , k ≥ 3. Speciffically,
we show that under suitable restrictions on the dimensions(viz., if dimK = (r
≥ 1)k and d = rk \\ for some k ≥ 3), a well-known deleted product criterion
(DPC ) is not only necessary but also sufficient for the existence of maps without
global r -fold points. Our main technical tool is a higher-multiplicity version
of the classical Whitney trick , by which pairs of isolated r -fold points of
opposite sign can be eliminated by local modiffications of the map, assuming
codimension d – dimK ≥ 3.\r\n\r\nAn important guiding idea for our work was that
suffciency of the DPC, together with an old\r\nresult of Özaydin's on the existence
of equivariant maps, might yield an approach to disproving the remaining open
cases of the the long-standing topological Tverberg conjecture , i.e., to construct
maps from the N -simplex σN to Rd without r-Tverberg points when r not a prime
power and\r\nN = (d + 1)(r – 1). Unfortunately, our proof of the sufficiency
of the DPC requires codimension d – dimK ≥ 3, which is not satisfied for K =
σN .\r\n\r\nIn 2015, Frick [16] found a very elegant way to overcome this \\codimension
3 obstacle" and\r\nto construct the first counterexamples to the topological
Tverberg conjecture for all parameters(d; r ) with d ≥ 3r + 1 and r not a prime
power, by a reduction1 to a suitable lower-dimensional skeleton, for which the
codimension 3 restriction is satisfied and maps without r -Tverberg points exist
by Özaydin's result and sufficiency of the DPC.\r\n\r\nIn this thesis, we present
a different construction (which does not use the constraint method) that yields
counterexamples for d ≥ 3r , r not a prime power. "
acknowledgement: "Foremost, I would like to thank Uli Wagner for introducing me to
the exciting interface between\r\ntopology and combinatorics, and for our subsequent
years of fruitful collaboration.\r\nIn our creative endeavors to eliminate intersection
points, we had the chance to be joined later\r\nby Sergey Avvakumov and Arkadiy
Skopenkov, which led us to new surprises in dimension 12.\r\nMy stay at EPFL and
IST Austria was made very agreeable thanks to all these wonderful\r\npeople: Cyril
Becker, Marek Filakovsky, Peter Franek, Radoslav Fulek, Peter Gazi, Kristof Huszar,\r\nMarek
Krcal, Zuzana Masarova, Arnaud de Mesmay, Filip Moric, Michal Rybar, Martin Tancer,\r\nand
Stephan Zhechev.\r\nFinally, I would like to thank my thesis committee Herbert Edelsbrunner
and Roman Karasev\r\nfor their careful reading of the present manuscript and for
the many improvements they suggested."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
citation:
ama: 'Mabillard I. Eliminating higher-multiplicity intersections: an r-fold Whitney
trick for the topological Tverberg conjecture. 2016.'
apa: 'Mabillard, I. (2016). Eliminating higher-multiplicity intersections: an
r-fold Whitney trick for the topological Tverberg conjecture. Institute of
Science and Technology Austria.'
chicago: 'Mabillard, Isaac. “Eliminating Higher-Multiplicity Intersections: An r-Fold
Whitney Trick for the Topological Tverberg Conjecture.” Institute of Science and
Technology Austria, 2016.'
ieee: 'I. Mabillard, “Eliminating higher-multiplicity intersections: an r-fold Whitney
trick for the topological Tverberg conjecture,” Institute of Science and Technology
Austria, 2016.'
ista: 'Mabillard I. 2016. Eliminating higher-multiplicity intersections: an r-fold
Whitney trick for the topological Tverberg conjecture. Institute of Science and
Technology Austria.'
mla: 'Mabillard, Isaac. Eliminating Higher-Multiplicity Intersections: An r-Fold
Whitney Trick for the Topological Tverberg Conjecture. Institute of Science
and Technology Austria, 2016.'
short: 'I. Mabillard, Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney
Trick for the Topological Tverberg Conjecture, Institute of Science and Technology
Austria, 2016.'
date_created: 2018-12-11T11:50:16Z
date_published: 2016-08-01T00:00:00Z
date_updated: 2023-09-07T11:56:28Z
day: '01'
ddc:
- '500'
degree_awarded: PhD
department:
- _id: UlWa
file:
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creator: dernst
date_created: 2019-08-13T08:45:27Z
date_updated: 2019-08-13T08:45:27Z
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file_size: 2227916
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language:
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oa: 1
oa_version: Published Version
page: '55'
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
publist_id: '6237'
related_material:
record:
- id: '2159'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
title: 'Eliminating higher-multiplicity intersections: an r-fold Whitney trick for
the topological Tverberg conjecture'
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2016'
...
---
_id: '1381'
abstract:
- lang: eng
text: '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 double-struck Rd without higher-multiplicity
intersections. We focus on conditions for the existence of almost r-embeddings,
i.e., maps f : K → double-struck Rd such that f(σ1) ∩ ⋯ ∩ f(σr) = ∅ whenever σ1,
..., σr 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 rd ≥ (r + 1) dim K + 3, then there exists an almost r-embedding
K → double-struck Rd if and only if there exists an equivariant map (K)Δ r → Sr
Sd(r-1)-1, where (K)Δ r is the deleted r-fold product of K, the target Sd(r-1)-1
is the sphere of dimension d(r - 1) - 1, and Sr is the symmetric group. 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 ≥ 3), and settles an open question
raised there.'
alternative_title:
- LIPIcs
author:
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Mabillard I, Wagner U. Eliminating higher-multiplicity intersections, II.
The deleted product criterion in the r-metastable range. In: Vol 51. Schloss Dagstuhl-
Leibniz-Zentrum fur Informatik GmbH; 2016:51.1-51.12. doi:10.4230/LIPIcs.SoCG.2016.51'
apa: 'Mabillard, I., & Wagner, U. (2016). Eliminating higher-multiplicity intersections,
II. The deleted product criterion in the r-metastable range (Vol. 51, p. 51.1-51.12).
Presented at the SoCG: Symposium on Computational Geometry, Medford, MA, USA:
Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH. https://doi.org/10.4230/LIPIcs.SoCG.2016.51'
chicago: Mabillard, Isaac, and Uli Wagner. “Eliminating Higher-Multiplicity Intersections,
II. The Deleted Product Criterion in the r-Metastable Range,” 51:51.1-51.12. Schloss
Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016. https://doi.org/10.4230/LIPIcs.SoCG.2016.51.
ieee: 'I. Mabillard and U. Wagner, “Eliminating higher-multiplicity intersections,
II. The deleted product criterion in the r-metastable range,” presented at the
SoCG: Symposium on Computational Geometry, Medford, MA, USA, 2016, vol. 51, p.
51.1-51.12.'
ista: 'Mabillard I, Wagner U. 2016. Eliminating higher-multiplicity intersections,
II. The deleted product criterion in the r-metastable range. SoCG: Symposium on
Computational Geometry, LIPIcs, vol. 51, 51.1-51.12.'
mla: Mabillard, Isaac, and Uli Wagner. Eliminating Higher-Multiplicity Intersections,
II. The Deleted Product Criterion in the r-Metastable Range. Vol. 51, Schloss
Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016, p. 51.1-51.12, doi:10.4230/LIPIcs.SoCG.2016.51.
short: I. Mabillard, U. Wagner, in:, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik
GmbH, 2016, p. 51.1-51.12.
conference:
end_date: 2016-06-17
location: Medford, MA, USA
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2016-06-14
date_created: 2018-12-11T11:51:41Z
date_published: 2016-06-01T00:00:00Z
date_updated: 2021-01-12T06:50:17Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SoCG.2016.51
file:
- access_level: open_access
checksum: 92c0c3735fe908f8ded6e484005cb3b1
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:10:06Z
date_updated: 2020-07-14T12:44:47Z
file_id: '4791'
file_name: IST-2016-621-v1+1_LIPIcs-SoCG-2016-51.pdf
file_size: 622969
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file_date_updated: 2020-07-14T12:44:47Z
has_accepted_license: '1'
intvolume: ' 51'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 51.1 - 51.12
project:
- _id: 25FA3206-B435-11E9-9278-68D0E5697425
grant_number: PP00P2_138948
name: 'Embeddings in Higher Dimensions: Algorithms and Combinatorics'
publication_status: published
publisher: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH
publist_id: '5830'
pubrep_id: '621'
quality_controlled: '1'
scopus_import: 1
status: public
title: Eliminating higher-multiplicity intersections, II. The deleted product criterion
in the r-metastable range
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 51
year: '2016'
...
---
_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
arxiv: 1
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: 2023-09-07T13:12:17Z
day: '15'
department:
- _id: UlWa
external_id:
arxiv:
- '1511.03501'
language:
- iso: eng
main_file_link:
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url: https://arxiv.org/abs/1511.03501
month: '11'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: submitted
related_material:
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relation: later_version
status: public
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relation: later_version
status: public
- id: '8156'
relation: dissertation_contains
status: public
status: public
title: Eliminating higher-multiplicity intersections, III. Codimension 2
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2015'
...
---
_id: '1511'
abstract:
- lang: eng
text: '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.'
acknowledgement: "The work by Z. P. was partially supported by the Charles University
Grant SVV-2014-260103. The\r\nwork by Z. P. and M. T. was partially supported by
the project CE-ITI (GACR P202/12/G061) of\r\nthe Czech Science Foundation and by
the ERC Advanced Grant No. 267165. Part of the research\r\nwork of M. T. was conducted
at IST Austria, supported by an IST Fellowship. The work by U.W.\r\nwas partially
supported by the Swiss National Science Foundation (grants SNSF-200020-138230 and\r\nSNSF-PP00P2-138948)."
alternative_title:
- LIPIcs
author:
- first_name: Xavier
full_name: Goaoc, Xavier
last_name: Goaoc
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: Pavel
full_name: Paták, Pavel
last_name: Paták
- first_name: Zuzana
full_name: Patakova, Zuzana
id: 48B57058-F248-11E8-B48F-1D18A9856A87
last_name: Patakova
orcid: 0000-0002-3975-1683
- first_name: Martin
full_name: Tancer, Martin
id: 38AC689C-F248-11E8-B48F-1D18A9856A87
last_name: Tancer
orcid: 0000-0002-1191-6714
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. On generalized
Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability
result. In: Vol 34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2015:476-490.
doi:10.4230/LIPIcs.SOCG.2015.476'
apa: 'Goaoc, X., Mabillard, I., Paták, P., Patakova, Z., Tancer, M., & Wagner,
U. (2015). On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type
nonembeddability result (Vol. 34, pp. 476–490). Presented at the SoCG: Symposium
on Computational Geometry, Eindhoven, Netherlands: Schloss Dagstuhl - Leibniz-Zentrum
für Informatik. https://doi.org/10.4230/LIPIcs.SOCG.2015.476'
chicago: 'Goaoc, Xavier, Isaac Mabillard, Pavel Paták, Zuzana Patakova, Martin Tancer,
and Uli Wagner. “On Generalized Heawood Inequalities for Manifolds: A Van Kampen–Flores-Type
Nonembeddability Result,” 34:476–90. Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2015. https://doi.org/10.4230/LIPIcs.SOCG.2015.476.'
ieee: 'X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, and U. Wagner,
“On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability
result,” presented at the SoCG: Symposium on Computational Geometry, Eindhoven,
Netherlands, 2015, vol. 34, pp. 476–490.'
ista: 'Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. 2015. On generalized
Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability
result. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34, 476–490.'
mla: 'Goaoc, Xavier, et al. On Generalized Heawood Inequalities for Manifolds:
A Van Kampen–Flores-Type Nonembeddability Result. Vol. 34, Schloss Dagstuhl
- Leibniz-Zentrum für Informatik, 2015, pp. 476–90, doi:10.4230/LIPIcs.SOCG.2015.476.'
short: X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, U. Wagner, in:,
Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 476–490.
conference:
end_date: 2015-06-25
location: Eindhoven, Netherlands
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2015-06-22
date_created: 2018-12-11T11:52:27Z
date_published: 2015-06-11T00:00:00Z
date_updated: 2023-02-23T12:38:00Z
day: '11'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SOCG.2015.476
ec_funded: 1
file:
- access_level: open_access
checksum: 0945811875351796324189312ca29e9e
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:11:18Z
date_updated: 2020-07-14T12:44:59Z
file_id: '4871'
file_name: IST-2016-502-v1+1_42.pdf
file_size: 636735
relation: main_file
file_date_updated: 2020-07-14T12:44:59Z
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language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 476 - 490
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
publist_id: '5666'
pubrep_id: '502'
quality_controlled: '1'
related_material:
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relation: later_version
status: public
scopus_import: 1
status: public
title: 'On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type
nonembeddability result'
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: '34 '
year: '2015'
...
---
_id: '2159'
abstract:
- lang: eng
text: 'Motivated by topological Tverberg-type problems, we consider multiple (double,
triple, and higher multiplicity) selfintersection points of maps from finite simplicial
complexes (compact polyhedra) into ℝd and study conditions under which such multiple
points can be eliminated. The most classical case is that of embeddings (i.e.,
maps without double points) of a κ-dimensional complex K into ℝ2κ. For this problem,
the work of van Kampen, Shapiro, and Wu provides an efficiently testable necessary
condition for embeddability (namely, vanishing of the van Kampen ob-struction).
For κ ≥ 3, the condition is also sufficient, and yields a polynomial-time algorithm
for deciding embeddability: One starts with an arbitrary map f : K→ℝ2κ, which
generically has finitely many double points; if k ≥ 3 and if the obstruction vanishes
then one can successively remove these double points by local modifications of
the map f. One of the main tools is the famous Whitney trick that permits eliminating
pairs of double points of opposite intersection sign. We are interested in generalizing
this approach to intersection points of higher multiplicity. We call a point y
2 ℝd an r-fold Tverberg point of a map f : Kκ →ℝd if y lies in the intersection
f(σ1)∩. ∩f(σr) of the images of r pairwise disjoint simplices of K. The analogue
of (non-)embeddability that we study is the problem Tverbergκ r→d: Given a κ-dimensional
complex K, does it satisfy a Tverberg-type theorem with parameters r and d, i.e.,
does every map f : K κ → ℝd have an r-fold Tverberg point? Here, we show that
for fixed r, κ and d of the form d = rm and k = (r-1)m, m ≥ 3, there is a polynomial-time
algorithm for deciding this (based on the vanishing of a cohomological obstruction,
as in the case of embeddings). Our main tool is an r-fold analogue of the Whitney
trick: Given r pairwise disjoint simplices of K such that the intersection of
their images contains two r-fold Tverberg points y+ and y- of opposite intersection
sign, we can eliminate y+ and y- by a local isotopy of f. In a subsequent paper,
we plan to develop this further and present a generalization of the classical
Haeiger-Weber Theorem (which yields a necessary and sufficient condition for embeddability
of κ-complexes into ℝd for a wider range of dimensions) to intersection points
of higher multiplicity.'
acknowledgement: Swiss National Science Foundation (Project SNSF-PP00P2-138948)
author:
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Mabillard I, Wagner U. Eliminating Tverberg points, I. An analogue of the
Whitney trick. In: Proceedings of the Annual Symposium on Computational Geometry.
ACM; 2014:171-180. doi:10.1145/2582112.2582134'
apa: 'Mabillard, I., & Wagner, U. (2014). Eliminating Tverberg points, I. An
analogue of the Whitney trick. In Proceedings of the Annual Symposium on Computational
Geometry (pp. 171–180). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582134'
chicago: Mabillard, Isaac, and Uli Wagner. “Eliminating Tverberg Points, I. An Analogue
of the Whitney Trick.” In Proceedings of the Annual Symposium on Computational
Geometry, 171–80. ACM, 2014. https://doi.org/10.1145/2582112.2582134.
ieee: I. Mabillard and U. Wagner, “Eliminating Tverberg points, I. An analogue of
the Whitney trick,” in Proceedings of the Annual Symposium on Computational
Geometry, Kyoto, Japan, 2014, pp. 171–180.
ista: 'Mabillard I, Wagner U. 2014. Eliminating Tverberg points, I. An analogue
of the Whitney trick. Proceedings of the Annual Symposium on Computational Geometry.
SoCG: Symposium on Computational Geometry, 171–180.'
mla: Mabillard, Isaac, and Uli Wagner. “Eliminating Tverberg Points, I. An Analogue
of the Whitney Trick.” Proceedings of the Annual Symposium on Computational
Geometry, ACM, 2014, pp. 171–80, doi:10.1145/2582112.2582134.
short: I. Mabillard, U. Wagner, in:, Proceedings of the Annual Symposium on Computational
Geometry, ACM, 2014, pp. 171–180.
conference:
end_date: 2014-06-11
location: Kyoto, Japan
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2014-06-08
date_created: 2018-12-11T11:56:03Z
date_published: 2014-06-08T00:00:00Z
date_updated: 2023-09-07T11:56:27Z
day: '08'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1145/2582112.2582134
file:
- access_level: open_access
checksum: 2aae223fee8ffeaf57bbabd8d92b6a2c
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:09:12Z
date_updated: 2020-07-14T12:45:30Z
file_id: '4735'
file_name: IST-2016-534-v1+1_Eliminating_Tverberg_points_I._An_analogue_of_the_Whitney_trick.pdf
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language:
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month: '06'
oa: 1
oa_version: Submitted Version
page: 171 - 180
publication: Proceedings of the Annual Symposium on Computational Geometry
publication_status: published
publisher: ACM
publist_id: '4847'
pubrep_id: '534'
quality_controlled: '1'
related_material:
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status: public
scopus_import: 1
status: public
title: Eliminating Tverberg points, I. An analogue of the Whitney trick
type: conference
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
year: '2014'
...