TITLE: Is every rational realized as the Euler characteristic of some manifold or orbifold?
QUESTION [23 upvotes]: Let me first ask the question for two-dimensional compact, connected manifolds and orbifolds.
Then, if the answer is No, one can remove various conditions on the dimension,
and allow non-compact examples and disconnected examples, to realize a (perhaps) wider range of rationals.
This came up after a class I'm teaching and I didn't know the answer.
Related: 

MO question "Euler characteristic of orbifolds."
Wikipedia table for 2-dim orbifolds

REPLY [34 votes]: Products of 2-orbifolds with manifolds will do the trick. There are 2-orbifolds of Euler characteristic $1/n$ (take a quotient of $S^2$ by a rotation of order $2n$). Then take a product with a manifold of Euler characteristic $m \in \mathbb{Z}$ to get all rationals.