Topological crystallography. With a view towards discrete geometric analysis. (English) Zbl 1271.55001

Sunada, Toshikazu

Surveys and Tutorials in the Applied Mathematical Sciences 6. Tokyo: Springer (ISBN 978-4-431-54176-9/pbk; 978-4-431-54177-6/ebook). xii, 229 p. (2013).

The author formulates a mathematical model of physical crystals based on an abstract graph, called a topological crystal, and a notion of standard realization of topological crystals in Euclidean space. The necessary mathematical background is developed in the first part of the book, so that all but the most advanced material is accessible to a scientifically-literate reader without prior knowledge of topology or abstract algebra.
A topological crystal \(X\) is a free abelian cover of a finite graph, or equivalently, a graph \(X\) equipped with a free action of a free abelian group \(L\) of finite rank. A realization of \(X\) is a map of \(X\) to Euclidean space carrying the action of \(L\) to an action by some lattice of translations. A standard realization is one which minimizes potential energy per unit cell, treating each node as a simple harmonic oscillator. Although the model ignores many physical and chemical considerations in favor of mathematical ones, many physical crystals are standard realizations of topological crystals, for \(L\) of rank three.
The theory is developed extensively and in a mathematically elegant way. Definitions and constructions depend on the most elementary algebraic topology; most proofs involve only graph theory and linear algebra. Discrete analogues of classical mathematical objects, particularly the Laplace operator and the Abel-Jacobi map, play important roles in the theory. Examples include many simple and complicated physical crystals, as well as abstract (e.g., higher-dimensional) crystals which are purely of mathematical interest.
The book is divided into three parts. In the first part, the necessary material from algebraic topology is developed in the context of graphs. Topics include group actions and quotient spaces, covering spaces, fundamental group and homology. All of this is developed with sufficient rigor to satisfy mathematicians, while the restriction to one-dimensional cell complexes makes the theory simple and intuitive enough to be accessible to mathematical novices. The second part of the book presents the main objects of study: topological crystals and their standard realizations. A canonical standard realization is constructed, for any topological crystal, and a uniqueness result is proven. Standard realizations are shown to have maximal symmetry. The methods and concepts are illustrated with a large collection of examples. The third part of the book presents more advanced topics, such as random walks, the discrete Abel-Jacobi map, and automorphism groups. There are five appendices containing background material on sets and functions, groups, and linear algebra.

Reviewer: Michael J. Falk (Flagstaff)

MSC:

55-02	Research exposition (monographs, survey articles) pertaining to algebraic topology
52C99	Discrete geometry
57M10	Covering spaces and low-dimensional topology
57M15	Relations of low-dimensional topology with graph theory
05C10	Planar graphs; geometric and topological aspects of graph theory
74N05	Crystals in solids

Keywords:

crystal; covering graph; group action; lattice; homology; crystallographic group