In Thurston's book *The Geometry and Topology of Three-Manifolds* it is proven that the underlying space of a two-dimensional orbifold is always a topological surface. 


Are there any easy examples of higher dimensional orbifolds whose underlying spaces are not topological manifolds?



What is an example of an orbifold which is not a topological manifold?
It is quite easy to give an example in real dimension $4$. 


In fact, it was shown by D. Mumford in the paper 


[**The topology of normal singularities of an algebraic surface and a criterion for simplicity**](http://www.ams.org/mathscinet-getitem?mr=153682), *Inst. Hautes Etudes Sci. Publ. Math.* (1961), no. 9, 5 - 
22 


that a normal algebraic surface over $\mathbb{C}$, whose underlying topological space (in the usual topology) is a topological manifold, must be nonsingular.


Now take the orbifold $X:=\mathbb{C}^2/G$, where $G$ is the group of order $2$ whose generator $\xi$ acts as $\xi \cdot (x, \, y) = (-x, \, -y)$.


The $G$-invariant subalgebra is generated by $A:=x^2, \, B=xy, \, C:=y^2$, so the underlying algebraic surface $X$ is isomorphic to the affine quadric cone $\{B^2-AC=0 \} \subset \mathbb{C}^3$, which has a $A\_1$-singularity at its vertex. Such a singularity is normal, because it is a codimension two hypersurface singularity. 


Then $X$ cannot be a topological manifold by Mumford's result mentioned above. 


