Contents

# Contents

## Definition

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

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

## Properties

### Coarse underlying space

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

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

## References

• Alex Eskin, Andrei Okounkov, Pillowcases and quasimodular forms, In: Ginzburg V. (ed.) Algebraic Geometry and Number Theory, Progress in Mathematics, vol 253. Birkhäuser 2006

• Elise Goujard, Martin Moeller, Pillowcase covers: Counting Feynman-like graphs associated with quadratic differentials (arXiv:1809.05016)

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