Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Foundations of Computational Mathematics
Published in: Foundations of Computational Mathematics 5/2023

01-08-2022

Rational Homotopy Type and Computability

Author: Fedor Manin

Published in: Foundations of Computational Mathematics | Issue 5/2023

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Given a simplicial pair (X, A), a simplicial complex Y, and a map \(f:A \rightarrow Y\), does f have an extension to X? We show that for a fixed Y, this question is algorithmically decidable for all X, A, and f if Y has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable. Conversely, for all other Y, the question is at least as hard as certain special cases of Hilbert’s tenth problem which are known or suspected to be undecidable.
previous article Causal Structure Learning: A Combinatorial Perspective
next article Sparse Multi-Reference Alignment: Phase Retrieval, Uniform Uncertainty Principles and the Beltway Problem

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Footnotes
1
This is the triviality problem for group presentations, translated into topological language. This work was extended by Adian and others to show that many other properties of nonabelian group presentations are likewise undecidable.
 
2
The results can plausibly be extended to nilpotent spaces.
 
3
It’s worth pointing out that this fits into a larger family of localizations of spaces, another of which is used in the proof of Lemma 3.3.
 
Literature
1.
P. Andrews and M. Arkowitz, Sullivan’s minimal models and higher order Whitehead products, Canadian Journal of Mathematics 30 (1978), no. 5, 961–982.MathSciNetCrossRefMATH
2.
E. H. Brown, Finite computability of Postnikov complexes, Annals of Mathematics (1957), 1–20.
3.
M. Čadek, M. Krčál, J. Matoušek, F. Sergeraert, L. Vokřínek, and U. Wagner, Computing all maps into a sphere, J. ACM 61 (2014), no. 3, Art. 17, 44.
4.
M. Čadek, M. Krčál, J. Matoušek, L. Vokřínek, and U. Wagner, Extendability of continuous maps is undecidable, Discrete & Computational Geometry 51 (2014), no. 1, 24–66.MathSciNetCrossRefMATH
5.
M. Čadek, M. Krčál, J. Matoušek, L. Vokřínek, and U. Wagner, Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension, SIAM Journal on Computing 43 (2014), no. 5, 1728–1780.MathSciNetCrossRefMATH
6.
M. Čadek, M. Krčál, and L. Vokřínek, Algorithmic solvability of the lifting-extension problem, Discrete & Computational Geometry 57 (2017), no. 4, 915–965.MathSciNetCrossRefMATH
7.
J. Denef, Hilbert’s tenth problem for quadratic rings, Proc. Amer. Math. Soc. 48 (1975), 214–220.MathSciNetMATH
8.
J. Denef and L. Lipshitz, Diophantine sets over some rings of algebraic integers, J. London Math. Soc. (2) 18 (1978), no. 3, 385–391.
9.
Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer (2012).
10.
M. Filakovský and L. Vokřínek, Are two given maps homotopic? An algorithmic viewpoint, Foundations of Computational Mathematics 20 (2020), no. 2, 311–330.MathSciNetCrossRefMATH
11.
P. A. Griffiths and J. W. Morgan, Rational homotopy theory and differential forms, Birkhäuser, (1981).
12.
13.
14.
B. Mazur and K. Rubin, Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math. 181 (2010), no. 3, 541–575.MathSciNetCrossRefMATH
15.
16.
17.
G. Mukherjee and P. Sankaran, Minimal models of oriented Grassmannians and applications, Math. Slovaca 50 (2000), no. 5, 567–579.MathSciNetMATH
18.
B. Poonen, Using elliptic curves of rank one towards the undecidability of Hilbert’s tenth problem over rings of algebraic integers, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 33–42.
19.
B. Poonen, Hilbert’s tenth problem over rings of number-theoretic interest, notes for a lecture series at the Arizona Winter School on “Number Theory and Logic”, (2003).
20.
21.
D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295.
22.
A. Romero, G. Ellis, and J. Rubio, Interoperating between computer algebra systems: computing homology of groups with Kenzo and GAP, ISSAC 2009—Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, ACM, New York, (2009), pp. 303–310. MR2742718
23.
24.
25.
26.
A. Shlapentokh, Hilbert’s tenth problem over number fields, a survey, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Contemp. Math., vol. 270, Amer. Math. Soc., Providence, RI, 2000, pp. 107–137.
27.
S. B. Smith, Rational type of classifying spaces for fibrations, Groups of homotopy self-equivalences and related topics (Gargnano, 1999), Contemp. Math., vol. 274, Amer. Math. Soc., Providence, RI, 2001, pp. 299–307.
28.
E. H. Spanier, Algebraic topology, McGraw-Hill, (1966).
29.
J. Stasheff, H-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970.
30.
D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269–331 (1978).
31.
D. Sullivan, Geometric topology: localization, periodicity and Galois symmetry, \(K\)-Monographs in Mathematics, vol. 8, Springer, (2005).
32.
L. Vokřínek, Decidability of the extension problem for maps into odd-dimensional spheres, Discrete & Computational Geometry 57 (2017), no. 1, 1–11.MathSciNetCrossRefMATH
Metadata
Title
Rational Homotopy Type and Computability
Author
Fedor Manin
Publication date
01-08-2022
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 5/2023
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-022-09582-8

Other articles of this Issue 5/2023

Foundations of Computational Mathematics 5/2023 Go to the issue

OriginalPaper

Causal Structure Learning: A Combinatorial Perspective

OriginalPaper

Detection Thresholds in Very Sparse Matrix Completion

OriginalPaper

Spectral Graph Matching and Regularized Quadratic Relaxations I Algorithm and Gaussian Analysis

OriginalPaper

Sparse Multi-Reference Alignment: Phase Retrieval, Uniform Uncertainty Principles and the Beltway Problem

OriginalPaper

Spectral Graph Matching and Regularized Quadratic Relaxations II

OriginalPaper

Kähler Geometry of Framed Quiver Moduli and Machine Learning

Premium Partner

    Image Credits