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The story of how the name 'orbifold' came about is pretty well-documented, but I can't find any explanation as to why Satake originally named orbifolds 'V-manifolds'. The 'manifold' part is clear enough; it's the 'V' I'm curious about.

Does the 'V' stand for anything, or is it just a random letter of the alphabet? (or is it typographical representation of a cone point?)

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Yes, V represents a cone-point. –  Igor Rivin May 25 '12 at 3:24
    
Could it be "virtual"? –  temp May 25 '12 at 3:33
    
'Virtual' could be right, but I don't see what's so 'Virtual' about V-manifolds. Igor, do you have any further evidence for the cone-point interpretation? –  Alex Amenta May 25 '12 at 3:49
    
Someone at Berkeley should ask Satake. –  S. Carnahan May 25 '12 at 7:32
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up vote 9 down vote accepted

Satake (in his PNAS paper where V-manifolds are introduced and in his Journal of the Mathematical Society of Japan paper where Gauss-Bonnet theorem for V-manifolds is proven) never explains the origin of the name. If I were to guess, I, as temp, would say "V" stands for "virtual" (since, for instance, in topology and group theory the word "virtual" means "up to a finite cover" or "up to a finite-index subgroup", and orbifolds/V-manifolds are "locally" manifolds up to a finite branched cover). But my guess is as good as yours and we are left with a mystery. Maybe, as Igor says, "V" stands for a "cone singularity" because of the shape of the letter V (could have been a $\Lambda$-manifold just as well). Or, maybe "V" does not really stand for anything, like R.H.Bing's initials. (According to wikipedia, once R.H.Bing was applying for a visa and was requested not to use initials. He explained that his name was really "R-only H-only Bing", and ended up receiving a visa made out to "Ronly Honly Bing".)

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I didn't know of this use of the word 'virtual', so this explanation seems plausible. Thanks! –  Alex Amenta May 25 '12 at 7:04
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