The Shape of Space

Citation

Weeks, J. R. (2020). The Shape of Space (3rd ed.). CRC Press (Chapman & Hall).

Chapter Summary

The third edition runs to about 350 pages with 194 colour illustrations, in four parts and twenty short chapters, followed by appendices with answers to the exercises, a bibliography and Conway's ZIP proof. The publisher describes it as a lighthearted textbook on the basic geometry and topology of two- and three-dimensional spaces, aimed at helping readers visualise possible shapes for our universe. New in this edition, according to the publisher, are full-colour figures throughout, "picture proofs" in place of algebraic ones, a simpler handles-and-crosscaps approach to surfaces and an updated discussion of cosmology. The author is a freelance geometer and MacArthur Fellow whose work spans pure mathematics, cosmology and popular exposition.

Part I: Surfaces and Three-Manifolds

Eight chapters: Flatland; Gluing; Vocabulary; Orientability; Classification of Surfaces; Products; Flat Manifolds; Orientability vs. Two-Sidedness. The part starts from the idea of a two-dimensional world and from building spaces by gluing the edges of polygons, then sets up the vocabulary used for the rest of the book. It treats orientability and the classification of surfaces (through the handles-and-crosscaps approach the publisher mentions, with Conway's ZIP proof in an appendix), moves to three dimensions through product spaces and flat three-manifolds, and closes by separating orientability from two-sidedness.

Part II: Geometries on Surfaces

Four chapters: The Sphere; The Hyperbolic Plane; Geometries on Surfaces; Gauss-Bonnet Formula and Euler Number. Topology now gains geometry. After the sphere and the hyperbolic plane, the part asks which geometry each surface can carry and ends with the Gauss–Bonnet formula, which links the curvature of a surface to its Euler number.

Part III: Geometries on Three-Manifolds

Six chapters: Four-Dimensional Space; The Hypersphere; Hyperbolic Space; Geometries on Three-Manifolds I; Bundles; Geometries on Three-Manifolds II. The same questions are asked one dimension up. The part prepares the reader with four-dimensional space, introduces the hypersphere and hyperbolic space as the curved three-dimensional models, and surveys the geometries that three-manifolds can carry in two chapters separated by one on bundles.

Part IV: The Universe

Two chapters: The Universe; The History of Space. The book ends by applying the preceding ideas to cosmology, the question of what shape our own universe may have, in the discussion the publisher says has been updated for this edition.

Key Concepts

Part I: Surfaces and Three-Manifolds

• Gluing: building a space by identifying edges of a polygon, or faces of a solid, so that what leaves one side returns on another.
• Orientability: whether a space contains a path that reverses left and right, as on a Möbius strip.
• Classification of surfaces: the result that every closed surface is a sphere with handles or crosscaps attached.
• Three-manifolds: spaces that look like ordinary three-dimensional space near each point, including products and flat manifolds.

Part II: Geometries on Surfaces

• Spherical and hyperbolic geometry: the geometries of constant positive and constant negative curvature.
• Euler number: a count of vertices, edges and faces that distinguishes surfaces.
• Gauss–Bonnet formula: the link between the total curvature of a surface and its Euler number, which restricts the geometries a surface can have.

Parts III–IV: Three-dimensional geometry and the universe

• Four-dimensional space and the hypersphere: higher-dimensional settings used to picture curved three-dimensional spaces.
• Hyperbolic space: the three-dimensional geometry of constant negative curvature.
• Bundles: spaces assembled from a base and fibres, used in describing geometries on three-manifolds.
• Shape of the universe: the application of these ideas to the possible topology and geometry of space itself.

Critical Analysis

Strengths

• Visual intuition before formalism. The publisher emphasises picture proofs, full-colour figures throughout and an informal exposition. The book builds the core ideas of topology (gluing, orientability, classification) and geometry (curvature, Gauss–Bonnet) through pictures, and the publisher points to intuitive examples that are missing from many college and graduate curricula.
• Reaches three-manifolds. The book does not stop at surfaces: two of its four parts go beyond them, to the hypersphere, hyperbolic space and the geometries of three-manifolds, and the last part ties these to cosmology.
• Exercises with answers. The publisher stresses the exercises, and Appendix A gives the answers, which makes the book usable for self-study.
• Software from the author. The author distributes free programs that let readers explore the same ideas: Torus Games, aimed at children and other newcomers to multiconnected spaces, and Curved Spaces, a flight simulator through multiconnected three-dimensional universes intended for students studying three-manifolds.

Limitations

• Not a rigorous course. With picture proofs in place of algebraic ones, the book gives intuition, not the definitions and proofs of point-set or algebraic topology; the publisher presents it as a supplement to formal texts for mathematics students.
• A cosmology section that needs updating. The publisher notes that the cosmological discussion had to be updated for this edition; this part depends on current observations, and readers should check recent results.
• Little for the algebraic side. The table of contents has no chapter on the fundamental group, homology or other algebraic invariants, so the book prepares the intuition for algebraic topology but not its methods.

Where It Fits in the Course

The book sits at the start of the topology and geometry course, alongside Richard Earl's Topology: A Very Short Introduction. Earl is a much shorter overview of the subject's ideas, from graphs and knots to surfaces; Weeks goes further into surfaces and three-dimensional spaces and comes with exercises and answers, so it can take a student from curiosity to real understanding of the examples.

The formal books follow. Wilson Sutherland's Introduction to Metric and Topological Spaces builds topology from metric spaces, and James Munkres's Topology covers point-set topology and then the fundamental group; Allen Hatcher's Algebraic Topology, freely available from the author, develops the fundamental group, homology and cohomology. Weeks supplies the pictures behind their definitions: gluing diagrams for quotient spaces, the classification of surfaces, and many of the standard examples.

How to Use This Book

The publisher names a broad audience: liberal arts students, undergraduate and graduate mathematics students who want an intuitive understanding to supplement formal texts, and general readers studying on their own. School geometry and algebra are a reasonable starting point.

Read Part I in order, slowly, with paper and scissors at hand: making gluing diagrams and checking orientability by hand is the best way into the ideas. Part II follows naturally; take time over the Gauss–Bonnet chapter, since it connects topology with geometry. Part III, which climbs from four-dimensional space to the geometries of three-manifolds, is the most demanding and rewards rereading. Part IV can be read as a conclusion, and those interested only in cosmology should still read Parts I and II first.

Do the exercises as you go and check them against Appendix A. Use Torus Games while reading Part I and Curved Spaces while reading Parts III and IV, to see from the inside what it is like to live in the spaces the book describes.