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On the sequential topological complexity of group homomorphisms

Nursultan Kuanyshov

Abstract

We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f:X\to Y,$ denoted $TC_{r}(f)$, that interacts with $TC_{r}(X)$ and $TC_{r}(Y)$ in the same way Jamie Scott's topological complexity map $TC(f)$ interacts with $TC(X)$ and $TC(Y).$ Furthermore, we apply $TC_{r}(f)$ to studying group homomorphisms $φ: Γ\to Λ.$ In addition, we prove that the sequential topological complexity of any nonzero homomorphism of a torsion group cannot be finite. Also, we give the characterisation of cohomological dimension of group homomorphisms.

On the sequential topological complexity of group homomorphisms

Abstract

We define and develop a homotopy invariant notion for the sequential topological complexity of a map denoted , that interacts with and in the same way Jamie Scott's topological complexity map interacts with and Furthermore, we apply to studying group homomorphisms In addition, we prove that the sequential topological complexity of any nonzero homomorphism of a torsion group cannot be finite. Also, we give the characterisation of cohomological dimension of group homomorphisms.
Paper Structure (13 sections, 26 theorems, 34 equations)

This paper contains 13 sections, 26 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sectional Category
  4. Ganea-Schwarz's approach to $\operatorname{secat}$
  5. Berstein–Schwarz-Schwarz cohomology class
  6. Lusternik-Schnirelmann category
  7. The sequential topological complexity
  8. The pullback of sequential topological complexity of map
  9. Cohomological dimension of group homomorphisms
  10. The $D-$topological complexity introduced by Farber and Oprea
  11. Group homomorphisms
  12. Reduction to epimorphisms
  13. A group epimorphism

Key Result

Theorem 1.1

Let $\phi:\Gamma\to \Lambda$ be an epimorphism of finitely generated abelian groups. Then In particular, if $\Gamma$ is free abelain groups, then $TC_{r}(\phi)=(r-1)(\operatorname{rank}(\Lambda)+k(T(\Lambda)))$ where $k(T(\lambda))$ is the Smith Normal number for given finite abelian group $T(\Lambda).$

Theorems & Definitions (58)

  • Theorem 1.1: Theorem \ref{['FGAG']}
  • Theorem 1.2: Theorem \ref{['Free']}
  • Theorem 1.3: Proposition \ref{['char. of group hom']}
  • Theorem 1.4: Theorem \ref{['surface groups']}
  • Definition 2.1
  • Theorem 2.4: SchJa
  • Theorem 2.5: Universality DR,Sch
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 48 more