Extending continuous maps: polynomiality and undecidability

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Extending continuous maps: polynomiality and undecidability

Popis výsledku v původním jazyce

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

Název v anglickém jazyce

Extending continuous maps: polynomiality and undecidability

Popis výsledku anglicky

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

Druh

D - Stať ve sborníku

CEP obor

IN - Informatika

OECD FORD obor

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Projekt

<a href="/cs/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Centrum excelence - Institut teoretické informatiky (CE-ITI)</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2013

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

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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Počet stran výsledku

10

Strana od-do

595-604

Název nakladatele

ACM

Místo vydání

New York, NY, USA

Místo konání akce

Palo Alto, USA

Datum konání akce

1. 6. 2013

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

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