# Orbit spaces of crystallographic groups

In their paper "On Three-Dimensional Space Groups", Conway et al. write

Although this paper was inspired by the orbifold concept, we did not need to consider the 219 orbifolds of space groups individually. We hope to discuss their topology in a later paper.

Did this discussion ever materialize? Or did somebody else classify the orbit spaces for all three-dimensional crystallographic groups?

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## 1 Answer

The orientable Euclidean orbifolds were described in section 7 of this paper of Dunbar.

To complete the list, note that every crystallographic group has an index 2 orientation-preserving subgroup. The corresponding orientable orbifold quotients will be realized as 2-fold orbifold covers of the non-orientable orbifolds. So one needs to look over Dunbar's list, and find all of the orientation reversion involutions of the corresponding orientable orbifolds to find their non-orientable quotients.

For the orbifolds with underlying space $S^3$, there are two possible orientation-reversing quotients: a reflection, and a suspension of an antipodal map. They admit quotients the ball and the suspension of $\mathbb{RP}^2$ respectively. So one needs to go through the list and determine which of these admit reflection or antipodal symmetries. For example, due to the well-known fact that the figure-8 knot is invertible, the label 3 figure-eight orbifold in Table 4 admits a quotient by a suspension antipodal map. For the first 4 examples in table 4, there is a reflection fixing the 1-skeleton of the simplex with quotient a Coxeter group. Some of these also admit further reflection symmetries.

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Thank you very much for this extremely helpful answer, Ian! –  eins6180 Jul 19 '14 at 7:51