nLab Pochhammer loop

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$]$

See also

• Wikipedia, Pochhammer contour

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

• Alexander Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics 21, World Scientific (1995) (doi:10.1142/2467)

• Pavel Etingof, Igor Frenkel, Alexander Kirillov, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998) $[$ISBN:978-1-4704-1285-2, review pdf$]$