Suppose I have a three-dimensional cube (I tend to think of it as a regular ideal cube in $\mathbb{H}^3,$ but you don't have to). I glue up its sides in some way to obtain topological spaces. The question is: has someone made a list of such? (if you think of it as an ideal cube, then the vertices, or their images after gluing, are not really in the glued up object). How many of these are manifolds/orbifolds (again, I am interested in both cases with and without punctures at the vertices).
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asked Dec 4, 2013 at 21:02
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$\begingroup$ "Sides" means $2$-dimensional faces? "In some way" means some pairing of the faces with some orientation or some more arbitrary thing? $\endgroup$– Qiaochu YuanCommented Dec 4, 2013 at 21:11
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$\begingroup$ @QiaochuYuan Yes, sides means "2-dimensional faces", and the "in some way", means identifying faces in pair (I did not say that I wanted the quotient to be orientable -- I do, but obviously non-orientable guys are of interest as well). $\endgroup$– Igor RivinCommented Dec 4, 2013 at 21:13
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$\begingroup$ Some appear in this paper: ams.org/mathscinet-getitem?mr=1231694 $\endgroup$– Ian AgolCommented Dec 5, 2013 at 5:31
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$\begingroup$ Sorry, I meant here: books.google.com/… $\endgroup$– Ian AgolCommented Dec 5, 2013 at 5:38
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1$\begingroup$ It's been done for all the platonic solids in the case you want a manifold quotient: arxiv.org/abs/math/0104182. $\endgroup$– Ryan BudneyCommented Dec 5, 2013 at 10:47
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