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The shape of space. 2nd, revised and expanded ed. (English) Zbl 1030.57001
Pure and Applied Mathematics, Marcel Dekker. 249. New York, NY: Marcel Dekker. xii, 382 p. (2002).
For the review of the first edition of this beautiful book, see [Pure and Applied Mathematics, Marcel Dekker. 96 (1985; Zbl 0571.57001)]. The first 18 chapters of the present book are taken from the 1985 edition. A fourth part has been added, concering contemporary methods in cosmology to determine topological properties of the universe. Also, an appendix has been added presenting Conway’s Zip-proof for the classification of surfaces.
Visualization in low-dimensional topology and geometry can be both helpful and misleading. So it is of importance to learn how to visualize right. The book under review gives guidelines and methods for a correct visualization of low-dimensional problems.
Part I of the book treats surfaces and three manifolds. It begins by the by now almost classical reference to Abbot’s “Flatland” in order to show how higher dimensional phenomena can be captured in fewer dimensions. Basic techniques are introduced: gluing squares and cubes to obtain tori. Further topics in this part: The difference between topological and geometrical notions; Embeddings and isotopy; Local and global phenomena; Closedness and Openess, orientability, connected sums, products and flat manifolds.
Part II treats geometries on surfaces. The sphere and the hypberbolic plane are discussed, and an ingenious method due to W. Thurston to build a hyperbolic plane from sheets of paper is presented. The part closes with a chapter on the Gauss-Bonnet formula and the Euler number.
Part III treats geometries on three-manifolds. The three-sphere, projective three-space and hyperbolic three-space are discussed and finally, all homogeneous geometries of three-manifolds are presented.
Part IV of the book is entitled “The universe” and treats topological and geometrical questions of cosmology. In the newly added last two chapters of the book two research programs are presented concerning the multiconnectivity of our universe. One is the program of “cosmic crystallography” which looks for patterns in the arrangement of the galaxies or rather superclusters of galaxies. The second is the so-called “circles in the sky” method measuring small deviations in the uniformity of the cosmic microwave background radiation. This is not a mathematical textbook in the usual sense, but it is beautiful reading for anyone interested in the subject. Many suggestive illustrations make things more clear and numerous “exercises” (rather: suggestions for further thought) help the reader to get along.

57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
57K20 2-dimensional topology (including mapping class groups of surfaces, Teichm├╝ller theory, curve complexes, etc.)
57K30 General topology of 3-manifolds
51M10 Hyperbolic and elliptic geometries (general) and generalizations
57M99 General low-dimensional topology
53A99 Classical differential geometry