Topological Complexity is a Fibrewise L-S Category

TL;DR

This paper establishes equalities relating topological complexity and fibrewise L-S category for spaces, introduces a stronger version of topological complexity, and discusses issues and corrections in the proofs of related theorems.

Contribution

It proves new equalities connecting topological complexity with fibrewise L-S category and advances the understanding of these invariants in fibrewise spaces, including those with singular fibres.

Findings

Established that $ ext{TC}(B) = ext{cat}_B^{ ext{b}}( ext{double}(B)) + 1$

Proved that $ ext{TC}_M(B) = ext{cat}_B^{ ext{BB}}( ext{double}(B)) + 1$

Identified issues in previous proofs and provided corrections for related theorems.

Abstract

Topological complexity $\TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version $\TCM{B}$ of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities $\TC{B} = \catBb{\double{B}}+1$ and $\TCM{B} = \catBB{\double{B}}+1$, where $\double{B}=B{\times}B$ is a fibrewise pointed space over $B$ whose projection and section are given by $p_{\double{B}}=\proj_{2} : B{\times}B \to B$ the canonical projection to the second factor and $s_{\double{B}}=\Delta_{B} : B \to B{\times}B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres. Recently, we found a problem with the proof of Theorem 1.13 which states that for a fibrewise well-pointed space $X$ over $B$, we have $\catBB{X}$ = \catBb{X}$ and that for a locally finite simplicial complex $B$, we have $\TC{B} = \TCM{B}$. While we still conjecture that Theorem 1.13 is true, this problem means that, at present, no proof is given to exist. Alternatively, we show the difference between two invariants $\catBb{X}$ and $\catBB{X}$ is at most 1 and the conjecture is true for some cases. We give further corrections mainly in the proof of Theorem 1.12.