# Invariants of the $\mathbb Z_2$ orbifolds of the Podle\'s' two spheres

## TL;DR

This paper computes the Hochschild and cyclic homology and cohomology groups, along with Chern-Connes indices, for the two $ ext{Z}_2$ orbifolds of Podle's quantum two-sphere, specifically the quantum two-disc and quantum real projective space.

## Contribution

It provides the first detailed homological and index calculations for these specific quantum orbifolds, expanding understanding of their noncommutative geometric properties.

## Key findings

- Hochschild and cyclic homology groups are explicitly calculated.
- Chern-Connes indices are determined for the orbifolds.
- Results reveal new insights into the structure of quantum orbifolds.

## Abstract

There are two $\mathbb Z_2$ orbifolds of the Podle\'s quantum two-sphere, one being the quantum two-disc $D_q$ and other the quantum two-dimensional real projective space $\mathbb RP^2_q$ . In this article we calculate the Hochschild and cyclic homology and cohomology groups of these orbifolds and also the corresponding Chern-Connes indices.