Extending continuous maps: polynomiality and undecidability

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F13%3A10172788" target="_blank" >RIV/00216208:11320/13:10172788 - isvavai.cz</a>
Result on the web
<a href="http://dl.acm.org/citation.cfm?id=2488683&CFID=274423984&CFTOKEN=57736595" target="_blank" >http://dl.acm.org/citation.cfm?id=2488683&CFID=274423984&CFTOKEN=57736595</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1145/2488608.2488683" target="_blank" >10.1145/2488608.2488683</a>

Result language
angličtina
Original language name
Extending continuous maps: polynomiality and undecidability
Original language description
We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of computational complexity. The extension problem asks, given topological spaces X,Y, a subspace A SUBSET OF OR EQUAL TO X, and a (continuous) map f:A -> Y, whether f can be extended to a map X -> Y. For computational purposes, we assume that X and Y are represented as finite simplicial complexes, A is a subcomplex of X, and f is given as a simplicial map. In this generality the problem is undecidable, as follows from Novikov's result from the 1950s on uncomputability of the fundamental group ?1(Y). We thus study the problem under the assumption that, for some k GREATER-THAN OR EQUAL TO 2, Y is (k-1)-connected;informally, this means that Y has "no holes up to dimension k-1" i.e., the first k-1 homotopy groups of Y vanish (a basic example of such a Y is the sphere Sk). We prove that, on the one hand, this problem is still undecidable for dim X=2
Czech name
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Czech description
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Project
<a href="/en/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Center of excellence - Institute for theoretical computer science (CE-ITI)</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2013
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection
Proceedings of the 45th annual ACM symposium on Symposium on theory of computing
ISBN
978-1-4503-2029-0
ISSN
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e-ISSN
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Number of pages
10
Pages from-to
595-604
Publisher name
ACM
Place of publication
New York, NY, USA
Event location
Palo Alto, USA
Event date
Jun 1, 2013
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
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