Which are the most powerful topological invariants of toroidal orbifolds?


In particular I am looking for topological invariants of two-dimensional toroidal orbifolds such as $T^{2}/Z\_{k}\times Z\_{k}$ and $T^{2}/Z\_{k}$. 


I'm wondering if the corresponding orbifold Euler characteristics are zero or not and how I can calculate them.



Topological invariants of toroidal orbifolds
There are 10 orbifolds that can be covered by the torus i.e. 10 compact Euclidean 2 orbifolds. However, only 7 of them are quotients of the torus by a cyclic group or Abelian product of cyclic groups. The intuition that Euler characteristics are zero is correct. Formulas for the orbifold Euler characteristic appear throughout the literature. I like Chapter 13 of [Thurston's notes](http://library.msri.org/nonmsri/gt3m/) in terms of a reference. Specifically, it has a table of all of the 2-orbifolds with non-negative Euler characteristic, which is helpful in this context. Genevieve Walsh's survey [Orbifolds and Commensurability](http://www.tufts.edu/~gwalsh01/papers/orbifoldsandcommensurability%5B4%5D.pdf) is also quite relevant.


The seven orbifolds that are quotients of the torus you are interested in are:
$T^2$, the Klein bottle (which can be realized as $T^2/\mathbb{Z}/2\mathbb{Z}$), $S^2(2,2,2,2)$ (which can be also realized as $T^2/\mathbb{Z}/2\mathbb{Z}$ however the group does not act freely in this case), $S^2(2,3,6)$ (which can be realized as $T^2/\mathbb{Z}/6\mathbb{Z}$), $S^2(3,3,3)$ (which can be realized as $T^2/\mathbb{Z}/3\mathbb{Z}$), $S^2(2,4,4)$ (which can be realized as $T^2/\mathbb{Z}/4\mathbb{Z}$), and $RP^2(2,2)$ (which can be realized as $T^2/(\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z})$). The other 3 compact Euclidean 2 orbifolds
are the quotients of the plane by the Euclidean triangle groups. In these cases, the group acting on the torus is dihedral. 


