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On the topological complexity of non-simply connected spaces

Yuki Minowa

Abstract

Topological complexity is a numerical homotopy invariant that measures the instability of motion planning in a space. To study the topological complexity of non-simply connected spaces, Costa and Farber introduced a cohomology class whose nilpotency gives a lower bound of topological complexity. Farber and Mescher constructed a spectral sequence that evaluates this nilpotency without direct computation. We extend these results with respect to a group homomorphism. As an application, we determine the topological complexity of some 3-manifolds with nonabelian fundamental group.

On the topological complexity of non-simply connected spaces

Abstract

Topological complexity is a numerical homotopy invariant that measures the instability of motion planning in a space. To study the topological complexity of non-simply connected spaces, Costa and Farber introduced a cohomology class whose nilpotency gives a lower bound of topological complexity. Farber and Mescher constructed a spectral sequence that evaluates this nilpotency without direct computation. We extend these results with respect to a group homomorphism. As an application, we determine the topological complexity of some 3-manifolds with nonabelian fundamental group.
Paper Structure (13 sections, 19 theorems, 56 equations)

This paper contains 13 sections, 19 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Topological preliminaries
  3. Sectional category and $\mathsf{TC}$-weight
  4. Canonical class and spectral sequence
  5. Algebraic preliminaries
  6. The induction and restriction functor
  7. Ext group homomorphisms
  8. Representations
  9. Reformulation and extension
  10. Diagonal inclusion
  11. Centralizer inclucion
  12. On the topological complexity of $S^3/Q_{8m}$
  13. Final remarks

Key Result

Theorem 1.1

For $m\ge 1$, the topological complexity of $S^3/Q_{8m}$ is $6$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3: CF
  • Theorem 2.4: CF
  • Theorem 2.5: FM
  • Proposition 2.6: FM
  • Lemma 2.7
  • proof
  • Lemma 3.1
  • ...and 25 more