nLab Pochhammer loop

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Integration theory

Contents

Idea

A Pochhammer loop or Pochhammer contour (Pochhammer 1890) is a loop in the complement of a pair of points in side the plane, hence a map

γ P:S 1{0,1}, \gamma_P \;\colon\; S^1 \longrightarrow \mathbb{C} \setminus \{0,1\} \,,

which represents:

  1. a non-trivial element in the fundamental group, 0[γ P]π 1({0,1})0 \neq [\gamma_P] \,\in\,\pi_1\big( \mathbb{C}\setminus \{0,1\}\big),

    but such that the winding number around each point separately vanishes;

  2. a trivial cycle in homology 0=[γ P]π 1({0,1})0 = [\gamma_P] \,\in\, \pi_1\big( \mathbb{C}\setminus \{0,1\}\big).

Due to this property, Pochhammer loops may underlie non-trivial cycles in twisted homology? (e.g. Varchenko 1995, Fig. 1.1, Etingof, Frenkel & Kirillov 1998, Fig. 4.1).

References

Original articles:

See also

Discussion in the context of the hypergeometric integral construction of solutions to the Knizhnik-Zamolodchikov equation:

Last revised on June 9, 2022 at 09:35:24. See the history of this page for a list of all contributions to it.