Marc wrote: << This is doubtless crankery, but... In computing the Euler characteristic I notice that handles get twice the weight of cross-caps.

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Lessee. Suppose we have a surface N and "add a handle" -- remove (the interior of) a disk from each of N and a torus T, then identify the resulting two circular edges -- to get a new surface that is the "connected sum" of N & T, denoted N#T. If X (chi) stands for Euler characteristic, then we have X(N#T) = (X(N) - 1) + (X(T) - 1) = X(N) + X(T) - 2 = X(N) - 2, because circles and tori have X = 0, but a disk D has X(D) = 1. By contrast, "adding a crosscap" to N means likewise creating the connected sum of N and a projective plane P. The calculation is the same except that while X(T) = 0, we have X(P) = 1 (since P is double-covered by the sphere S, well-known to have X(S) = 2). So there is an increase of 1: X(N#T) = X(N)-2 but X(N#P) = X(N)-1. But this because of this calculation, not because of a crosscap getting "twice the weight" of a handle. (In fact, a handle may be thought of as a torus minus a disk, and a crosscap as a projective plane minus a disk (aka a Moebius band), with X(T-D) = -1, while X(P-D) = 0.) << Plugging p=1/2 in for the "genus" in the chromatic formula floor((7+sqrt(1+48*p))/2) gives 6, the number for the Mobius strip. Does that figure, or is it just an accident? Is there some general chromatic expression that covers non-orientable surfaces? If we can have "fractional dimension", can we somehow have "fractional genii"? Happy Holey-days!

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There are uses for fractional Euler characteristics in describing orbifolds, but as far as I know ordinary surfaces have integer genera. The formula for the Euler char of an orientable surface of genus g (connected sum of a sphere and g tori for g >= 0) is X = 2-2g. The Euler char of a nonorientable surface of nonorientable genus q (connected sum of a sphere and q projective planes (q >= 1) is X = 2-q. --Dan