nLab pillowcase orbifold




The pillowcase orbifold is the 2-dimensional flat (complex) orbifold which is the global quotient of the torus /[i]\mathbb{C}/\mathbb{Z}[i] by the reflection involution zzz \mapsto -z:

Pillow(/[i]) refl 2 Pillow \;\coloneqq \; (\mathbb{C}/\mathbb{Z}[i])\sslash_{refl} \mathbb{Z}_2


Coarse underlying space

The Weierstrass elliptic function \wp, regarded as a holomorphic function with values in the Riemann sphere P 1\mathbb{C}P^1, exhibits the coarse underlying topological space of the pillowcase orbifold as the 2-sphere:

:/[i](/[i])/ refl 2homeoP 1 \wp \;\colon\; \mathbb{C}/\mathbb{Z}[i] \longrightarrow (\mathbb{C}/\mathbb{Z}[i])/_{refl} \mathbb{Z}_2 \overset{homeo}{\longrightarrow} \mathbb{C}P^1

(e.g Mukase 04, end of 1.4, Eskin-Okounkov 05, p. 1, Goujard-Moeller 18, 2.3)


Created on July 28, 2020 at 11:58:17. See the history of this page for a list of all contributions to it.