Cross-sections of Milnor fibrations and Motion planning

TL;DR

This paper investigates the existence of cross-sections in Milnor fibrations, explores their applications to motion planning problems, and introduces algorithms for tasking within this context.

Contribution

It provides conditions for the existence of cross-sections in Milnor fibrations and applies these results to develop motion planning algorithms.

Findings

Conditions for cross-section existence in Milnor fibrations

Application of fibrations to motion planning algorithms

Development of tasking algorithms for Milnor fibrations

Abstract

Consider a fibration \[p:E\to S^{p-1}\] with fiber $F$. We have the following natural question: Under what conditions does this fibration admit a cross-section? Our purpose is to discuss this problem when the fibration $p$ is the Milnor fibration $f_{\mid}:f^{-1}(S^{p-1}_{\delta})\cap D^{n}_{\epsilon}\to S^{p-1}_{\delta}$ with Milnor fiber $F_f$ and the Milnor fibration of arrangements $Q:\mathcal{M}(\mathcal{A})\to \mathbb{C}^\ast$ with fiber $F=F(\mathcal{A})$. Furthermore, we use our results to study the tasking planning problem for the Milnor fibration as a work map and we give the tasking algorithms.