We introduce some compact orbifolds on which there is a certain finite group action having a simple convex polytope as the orbit space. We compute the orbifold fundamental group and homology groups of these orbifolds. We compute the cohomology rings of these orbifolds when the dimension of the orbifold is even. These orbifolds are intimately related to the notion of small cover.
The author would like to thank his advisor Mainak Poddar for helpful suggestions and stimulating discussions. The author is thankful to the anonymous referee for helpful suggestions. He would also like to thank I. S. I. for supporting his research grant during the preparation of the work was done.
"Some small orbifolds over polytopes." Tbilisi Math. J. 4 1 - 27, 2011. https://doi.org/10.32513/tbilisi/1528768864