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Parametrized topological complexity of spherical fibrations over spheres

Yuki Minowa

Abstract

Parametrized topological complexity is a homotopy invariant that represents the degree of instability of motion planning problem that involves external constraints. We consider the parametrized topological complexity in the case of spherical fibrations over spheres. We explicitly compute a lower bound in terms of weak category and determine the parametrized topological complexity of some spherical fibrations.

Parametrized topological complexity of spherical fibrations over spheres

Abstract

Parametrized topological complexity is a homotopy invariant that represents the degree of instability of motion planning problem that involves external constraints. We consider the parametrized topological complexity in the case of spherical fibrations over spheres. We explicitly compute a lower bound in terms of weak category and determine the parametrized topological complexity of some spherical fibrations.

Paper Structure

This paper contains 10 sections, 54 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Parametrized topological complexity
  3. The complex $W(\alpha)$
  4. Crude Hopf invariant
  5. Proof of Theorem \ref{['main 0']} (1)
  6. Proofs of Theorem \ref{['main 0']} (2) and Theorem \ref{['main3']}
  7. Proof of Theorem \ref{['main 0']} (2)
  8. The $n=3,7$ cases
  9. The unit tangent bundle
  10. Open problems

Key Result

Theorem \oldthetheorem

Let $2n+1\le m\le 4n-1$ with $n\ge 1$. Suppose that there are $\alpha\in\pi_{m+2n+1}(S^{2n+1})$ and $\beta\in\pi_m(S^{2n+1})$ such that $\overline{H}_\nabla(\alpha)\neq0$, $\Sigma(\eta\circ\beta)=0$ and $[\beta,1_{S^{2n+1}}]=0$. Then there exists a homotopy fibration $S^{2n+1}\to X\to S^{m+1}$ with

Theorems & Definitions (101)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Example \oldthetheorem
  • Proposition \oldthetheorem
  • Example \oldthetheorem
  • ...and 91 more