J 2023

Computing Homotopy Classes for Diagrams

FILAKOVSKÝ, Marek and Lukáš VOKŘÍNEK

Basic information

Original name

Computing Homotopy Classes for Diagrams

Authors

FILAKOVSKÝ, Marek and Lukáš VOKŘÍNEK (203 Czech Republic, belonging to the institution)

Edition

Discrete and Computational Geometry, Springer, 2023, 0179-5376

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

Impact factor

Impact factor: 0.800 in 2022

RIV identification code

RIV/00216224:14310/23:00134379

Organization unit

Faculty of Science

UT WoS

001033657400003

Keywords in English

Equivariant homotopy; Algorithm; Tverberg-type problem

Tags

Tags

International impact, Reviewed
Změněno: 16/1/2024 09:18, Mgr. Marie Šípková, DiS.

Abstract

V originále

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, $$\dim X\le 2\cdot {\text {conn}}Y$$ dim X ≤ 2 · conn Y , $${\text {conn}}Y\ge 1$$ conn Y ≥ 1 , computes the set $$[X,Y]^A$$ [ X , Y ] A of homotopy classes of maps of diagrams $$\ell :X\rightarrow Y$$ ℓ : X → Y extending a given $$f:A\rightarrow Y$$ 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 $$\ell :X\rightarrow Y$$ ℓ : X → Y extending a given equivariant map $$f:A\rightarrow Y$$ f : A → Y under the stability assumption $$\dim X^H\le 2\cdot {\text {conn}}Y^H$$ dim X H ≤ 2 · conn Y H and $${\text {conn}}Y^H\ge 1$$ conn Y H ≥ 1 , for all subgroups $$H\le G$$ 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 $$K\rightarrow {\mathbb {R}}^d$$ K → R d without r-tuple intersection points? In the metastable range of dimensions, $$rd\ge (r+1) k+3$$ r d ≥ ( r + 1 ) k + 3 , the problem is shown algorithmically decidable in polynomial time when k, d, and r are fixed.

Links

GBP201/12/G028, research and development project
Name: Ústav Eduarda Čecha pro algebru, geometrii a matematickou fyziku
Investor: Czech Science Foundation