Frizzelle, Gerald W. (1984) An orbifold approach to black and white crystallographic groups. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Crystallography, Mathematical, Group theory, Geometry, Algebraic, Representations of groups.
Official Date:	October 1984
Dates:	Date Event October 1984 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Epstein, D. B. A.
Sponsors:	Science and Engineering Research Council (Great Britain), Warwick supervision
Extent:	iv, 155 leaves : illustrations
Language:	eng

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