Computing Homotopy Classes for Diagrams

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F23%3A00134379" target="_blank" >RIV/00216224:14310/23:00134379 - isvavai.cz</a>

Alternative codes found

RIV/00216208:11320/23:10471406

Result on the web

<a href="https://doi.org/10.1007/s00454-023-00513-0" target="_blank" >https://doi.org/10.1007/s00454-023-00513-0</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00454-023-00513-0" target="_blank" >10.1007/s00454-023-00513-0</a>

Result language

angličtina

Original language name

Computing Homotopy Classes for Diagrams

Original language description

We present an algorithm that, given finite diagrams of simplicial sets X, A, Y, i.e., functors $${mathcal {I}}^textrm{op}rightarrow {textsf {s}} {textsf {Set}}$$ I op → s Set , such that (X, A) is a cellular pair, $$dim Xle 2cdot {text {conn}}Y$$ dim X ≤ 2 · conn Y , $${text {conn}}Yge 1$$ conn Y ≥ 1 , computes the set $$[X,Y]^A$$ [ X , Y ] A of homotopy classes of maps of diagrams $$ell :Xrightarrow Y$$ ℓ : X → Y extending a given $$f:Arightarrow Y$$ f : A → Y . For fixed $$n=dim X$$ n = dim X , the running time of the algorithm is polynomial. When the stability condition is dropped, the problem is known to be undecidable. Using Elmendorf’s theorem, we deduce an algorithm that, given finite simplicial sets X, A, Y with an action of a finite group G, computes the set $$[X,Y]^A_G$$ [ X , Y ] G A of homotopy classes of equivariant maps $$ell :Xrightarrow Y$$ ℓ : X → Y extending a given equivariant map $$f:Arightarrow Y$$ f : A → Y under the stability assumption $$dim X^Hle 2cdot {text {conn}}Y^H$$ dim X H ≤ 2 · conn Y H and $${text {conn}}Y^Hge 1$$ conn Y H ≥ 1 , for all subgroups $$Hle G$$ H ≤ G . Again, for fixed $$n=dim X$$ n = dim X , the algorithm runs in polynomial time. We further apply our results to Tverberg-type problem in computational topology: Given a k-dimensional simplicial complex K, is there a map $$Krightarrow {mathbb {R}}^d$$ K → R d without r-tuple intersection points? In the metastable range of dimensions, $$rdge (r+1) k+3$$ r d ≥ ( r + 1 ) k + 3 , the problem is shown algorithmically decidable in polynomial time when k, d, and r are fixed.

Czech name

—

Czech description

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Type

J_imp - 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/GBP201%2F12%2FG028" target="_blank" >GBP201/12/G028: Eduard Čech Institute for algebra, geometry and mathematical physics</a><br>

Continuities

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

Publication year

2023

Confidentiality

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

Name of the periodical

Discrete and Computational Geometry

ISSN

0179-5376

e-ISSN

1432-0444

Volume of the periodical

70

Issue of the periodical within the volume

3

Country of publishing house

US - UNITED STATES

Number of pages

55

Pages from-to

866-920

UT code for WoS article

001033657400003

EID of the result in the Scopus database

2-s2.0-85165252808