Embeddings of k-Complexes into 2k-Manifolds

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10491054" target="_blank" >RIV/00216208:11320/24:10491054 - isvavai.cz</a>
Alternative codes found
RIV/68407700:21240/24:00371172
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=OyyHD-Md8u" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=OyyHD-Md8u</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00454-023-00595-w" target="_blank" >10.1007/s00454-023-00595-w</a>

Result language
angličtina
Original language name
Embeddings of k-Complexes into 2k-Manifolds
Original language description
We improve the bound on K & uacute;hnel's problem to determine the smallest n such that the k-skeleton of ann-simplex Delta((k) )(n)does not embed into a compact PL 2k-manifoldMby showing that if Delta((k))(n)nembeds intoM, thenn <=(2k+1)+(k+1)beta k(M;Z2). Asaconsequence we obtain improved Radon and Helly type results for set systems in suchmanifolds. Our main tool is a new description of an obstruction for embeddability of ak-complex K into a compact PL 2k-manifoldMvia the intersection form on M. Inour approach we need that for every map f:K -> M the restriction to the(k-1)-skeleton of Kis nullhomotopic. In particular, this condition is satisfied in interestingcases if K is(k-1)-connected, for example a k-skeleton of n-simplex, or ifMis(k-1)-connected. In addition, if M is(k-1)-connected andk >= 3, the obstruction is complete, meaning that a k-complex K embeds into M if and only if the obstruction vanishes. For trivial intersection forms, our obstruction coincides with the standard van Kampen obstruction. However, if the form is non-trivial, the obstruction is not linear but rather 'quadratic' in a sense that it vanishes if and only if certain system ofquadratic diophantine equations is solvable. This may potentially be useful in attacking algorithmic decidability of embedd ability of k-complexes into PL 2k-manifolds.
Czech name
—
Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA22-19073S" target="_blank" >GA22-19073S: Combinatorial and computational complexity in topology and geometry</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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