Back on December 11, Marc LeBrun asked:
If we can have "fractional dimension", can we somehow have "fractional genii"?
Dan Asimov replied:
There are uses for fractional Euler characteristics in describing orbifolds, but as far as I know ordinary surfaces have integer genera.
There's also a sense in which certain sorts of infinite-dimensional objects can be sensibly said to have fractional Euler characteristic; e.g., the set of functions from [0,1] to {0,1} such that the pre-images of both 0 and 1 consist of finitely many points and intervals is something like an infinite dimensional polytope, and it seems to be trying to have Euler characteristic 1/2. See "Exponentiation and Euler Characteristic" (math.CO/0204009). I also recall that there's a sense in which the classifying space of a finite group G has 1/|G| elements. See http://www.math.ucr.edu/home/baez/counting/ for relevant references. Jim Propp