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On sequential versions of various parametrized invariants

Navnath Daundkar, Abhishek Sarkar, Ankur Sarkar

TL;DR

This work develops a unified framework for sequential parametrized invariants in fibrewise homotopy theory, extending classical invariants to a fibrewise, parameterized, and sequential setting to address complex motion-planning problems. The authors define $TC_{B,r}(E)$ and $D_B(f_1,...,f_r)$ as core sequential invariants, establish their fibrewise homotopy invariance, and relate them to fibrewise LS category and cohomological bounds, with both unpointed and pointed variants. They further connect these invariants to relative and subspace versions, derive product, composition, and fibration inequalities, and analyze comparisons between pointed and unpointed theories, including conditions for equality. The resulting framework yields computable lower and upper bounds (e.g., via $H_B^*$ and homotopy dimension) and clarifies how sequential parametrized invariants specialize to classical invariants in suitable limits, with implications for parameterized robot motion planning and topology of fibrations.

Abstract

In this paper, we introduce and study sequential versions of several fibrewise homotopy invariants, including parametrized topological complexity, parametrized (subspace) homotopic distance. We investigate their basic properties, establish relationships among them, and compare them with the corresponding classical homotopical invariants.

On sequential versions of various parametrized invariants

TL;DR

This work develops a unified framework for sequential parametrized invariants in fibrewise homotopy theory, extending classical invariants to a fibrewise, parameterized, and sequential setting to address complex motion-planning problems. The authors define and as core sequential invariants, establish their fibrewise homotopy invariance, and relate them to fibrewise LS category and cohomological bounds, with both unpointed and pointed variants. They further connect these invariants to relative and subspace versions, derive product, composition, and fibration inequalities, and analyze comparisons between pointed and unpointed theories, including conditions for equality. The resulting framework yields computable lower and upper bounds (e.g., via and homotopy dimension) and clarifies how sequential parametrized invariants specialize to classical invariants in suitable limits, with implications for parameterized robot motion planning and topology of fibrations.

Abstract

In this paper, we introduce and study sequential versions of several fibrewise homotopy invariants, including parametrized topological complexity, parametrized (subspace) homotopic distance. We investigate their basic properties, establish relationships among them, and compare them with the corresponding classical homotopical invariants.
Paper Structure (18 sections, 45 theorems, 109 equations)

This paper contains 18 sections, 45 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Background on parametrized invariants
  3. Fibrewise homotopy theory
  4. Fibrewise sectional category and induced invariants
  5. Formal aspects of sequential parametrized topological complexity
  6. Sequential fibrewise topological complexity
  7. Fibrewise pointed version of sequential topological complexity
  8. Sequential parametrized homotopic distance
  9. Sequential parametrized homotopic distance
  10. Relation with fibrewise unpointed LS Category
  11. Properties of the sequential parametrized homotopic distance
  12. Composition of fibrewise maps
  13. Fibrewise maps on normal spaces
  14. Fibrewise maps between fibrewise fibrations
  15. Sequential pointed parametrized homotopic distance and its comparison with unpointed version
  16. ...and 3 more sections

Key Result

Theorem 2.4

Let $\pi\colon E\to X$ be a fibrewise pointed map. Then where $\mathrm{secat}_B^B(\pi)$ denotes the fibrewise pointed sectional category of the map $\pi$.

Theorems & Definitions (90)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 80 more