Contents

# 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

$\gamma_P \;\colon\; S^1 \longrightarrow \mathbb{C} \setminus \{0,1\} \,,$

which represents:

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

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

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

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:

• Leo Pochhammer, Zur Theorie der Euler’schen Integrale, Mathematische Annalen 35 (1890) 495–526 $[$doi:10.1007/BF02122658$]$