Tipus de document
Treball de fi de màsterData de publicació
Llicència de publicació
Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/230619
Avoiding collisions: an introduction to the topology of configuration spaces
Títol de la revista
Autors
Director/Tutor
ISSN de la revista
Títol del volum
Recurs relacionat
Resum
This thesis provides a comprehensive study of configuration spaces, tracing their topological properties from classical homogeneous manifolds to the discrete setting of topological graphs. Configuration spaces serve as a fundamental bridge between algebraic topology, geometric group theory, and robotics, capturing the collision-free motion of multiple particles.
We begin by establishing the foundational properties of ordered configuration spaces on connected manifolds, with a rigorous focus on the forgetting map and the Fadell–Neuwirth fibration theorem. This geometric framework transitions naturally into the study of braid groups, where we explore their algebraic presentations and characterisations. Following this, we analyze the specific topology of the planar configuration space $\mathcal{F}_n(\mathbb{R}^2)$ by examining the classical Fox-Neuwirth theorem. Additionally, we introduce the complexities that arise when extending these techniques to compact connected surfaces, where the splitting behavior of the Fadell–Neuwirth sequence is no longer guaranteed.
Finally, we shift our focus to the distinct combinatorial nature of configuration spaces on graphs. We investigate the local structure by analyzing the configuration space of the star of a vertex, and scale these techniques to analyze the global space of a whole graph. To illustrate the concrete applications of this discrete framework, we present a fully worked example modeling a segment graph with three points.
Descripció
Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Any: 2026. Director: Natàlia Castellana Vila
Matèries (anglès)
Citació
Col·leccions
Citació
MARTÍNEZ SANT, Jordi. Avoiding collisions: an introduction to the topology of configuration spaces. [consulted: 25 of July of 2026]. Available at: https://hdl.handle.net/2445/230619