---
OA_place: publisher
_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.'
corr_author: '1'
date_created: 2018-12-11T11:50:16Z
date_published: 2016-08-01T00:00:00Z
date_updated: 2026-07-29T11:26:07Z
day: '01'
ddc:
- '500'
degree_awarded: PhD
department:
- _id: UlWa
- _id: GradSch
doi_confirm: '1'
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language:
- iso: eng
month: '08'
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: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2016'
...