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Title: An orbifold approach to black and white crystallographic groups
Author: Frizzelle, Gerald W.
ISNI:       0000 0001 3484 6707
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1984
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Given a crystallographic space group G , Bonahon and Siebenmann show in [B + SI] that it can be thought of as the fundamental group of a closed 3-orbifold Q which, because in most cases G preserves some direction V in 1R , usually admits an S1-fibration over a 2-orbifold B : we write (IR3,V)/G = Q -p → B . Readers familiar with the definitions and notations for orbifolds and crystallographic groups may wish to omit §0, where these ideas are introduced, dipping back into it only when necessary. Using these methods, Bonahon and Siebenmann give a new and entirely topological classification of the crystallographic groups, depicting Q → B by a convenient diagram; their methods are described briefly in §1. However, they make no attempt to link this new classification with the existing one i.e. to determine which orbifolds correspond to which crystallographic groups; this is done here, for the first time, in Table 4. Given an index two subgroup G¹ of a crystallographic group G , the pairs (G,G¹) , classified up to affine homeomorphism of IR , are known as black and white groups. In terms of orbifolds they correspond to fibred double covers Q1 → B¹ of Q → B . Such covers for the local structure of fibred orbifolds are constructed in §2 and summarized in Table 3; §4 and §5 then show how to piece them together to form global covers. In §3 we prove that, whenever there is a direction V in IR which gives a unique fibration, the obvious notion of equivalence for two such covers corresponds exactly to the standard definition of equivalence for black and white groups. In §6 we deal with those groups whose corresponding orbifolds cannot be fibred; only the orientable orbifolds in the list given here have been described before (in [Du 1]). Finally, in §7, we demonstrate, by means of examples, how to classify the black and white groups by constructing double covers of orbifolds.
Supervisor: Not available Sponsor: Science and Engineering Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics