Hardness of embedding simplicial complexes in R^d
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F11%3A10099166" target="_blank" >RIV/00216208:11320/11:10099166 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.4171/JEMS/252" target="_blank" >http://dx.doi.org/10.4171/JEMS/252</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4171/JEMS/252" target="_blank" >10.4171/JEMS/252</a>
Alternative languages
Result language
angličtina
Original language name
Hardness of embedding simplicial complexes in R^d
Original language description
Let EMBEDkRIGHTWARDS ARROWd be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R? Known results easily imply polynomiality of EMBEDkRIGHTWARDS ARROW2(k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDkRIGHTWARDS ARROW2k for all k GREATER-THAN OR EQUAL TO 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBEDdRIGHTWARDS ARROWd and EMBED(dMINUS SIGN 1)RIGHTWARDS ARROWd are undecidable for each d GREATER-THAN OR EQUAL TO 5. Our main result is NP-hardness of EMBED2RIGHTWARDS ARROW4 and, more generally, of EMBEDkRIGHTWARDS ARROWd for all k, d with d GREATER-THAN OR EQUAL TO 4 and d GREATER-THAN OR EQUAL TO k GREATER-THAN OR EQUAL TO (2d MINUS SIGN 2)/3. These dimensions fall outside the metastable range of a theorem of Hae?iger and Weber, which characterizes embeddability using the deleted product ob
Czech name
—
Czech description
—
Classification
Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
—
Result continuities
Project
<a href="/en/project/1M0545" target="_blank" >1M0545: Institute for Theoretical Computer Science</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>Z - Vyzkumny zamer (s odkazem do CEZ)
Others
Publication year
2011
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of the European Mathematical Society
ISSN
1435-9855
e-ISSN
—
Volume of the periodical
13
Issue of the periodical within the volume
2
Country of publishing house
CH - SWITZERLAND
Number of pages
37
Pages from-to
259-295
UT code for WoS article
000286014400001
EID of the result in the Scopus database
—