The goal of this book is to explore an area of cosmology that might be called “cosmic topology” that endeavors to understand the shape and size of the universe. This has been something of an issue in cosmology for roughly the last twenty-five centuries. Now, just perhaps, we have tools and data available to get a more definite answer.

*The Wraparound Universe* explores cosmology and cosmic topology from the perspective of a physicist writing for a general, scientifically literate audience. There is a story here without a definite conclusion yet, a story in which mathematics has a significant role in addressing broad questions in cosmology and observational astronomy. Cosmologists’ current standard model assumes an essentially infinite Euclidean space created by inflation and containing density fluctuations on all scales. On small scales NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) observed the fluctuations as predicted, but on scales larger than about 60º across the sky, the fluctuations essentially disappear. One possible explanation for the missing fluctuations is that the universe is simply not big enough to support them. If the real universe is a closed 3-manifold, it can support no waves longer than its “circumference”.

What 3-manifolds should be considered as models of the physical space we inhabit? Observational evidence suggests that the observable universe is homogeneous and isotropic to a precision of about one part in 10^{4.} We therefore consider manifolds that look locally like the three-sphere S^{3}, Euclidean space E^{3}, or hyperbolic space H^{3}. To get to a finite universe, we consider also the quotients X/Γ of the simply connected space X (S^{3}, E^{3}, or H^{3}) under the action of a discrete fixed-point-free group Γ of isometries. This is how we get to the “wraparound”, or multiply-connected, universe.

Could we detect the wraparound? If the universe is small enough relative to its age, it might be possible to detect multiple images galaxies or quasars in different parts of the sky. From the locations and number of these images we could essentially detect the topology of the universe.

It is a grand quest, and the author is perfectly suited to lead us in approaching it. He has even structured the book as a kind of wraparound: the reader can approach it linearly, or follow the author’s arrows in the margins to link forwards or backwards to other parts of the book. In linear sequence, the first part of the book provides a logical sequence of developments from the first naive questions about the shape of space to models of the wraparound universe and their observational consequences. This includes, for example, explorations of the curvature of space, general relativity, black holes and the four scales of geometry of interest in physics.

The second part of the book, titled “Folds in the Universe”, explores in greater detail issues briefly discussed in the first part. The chapters in this part include an abundance of topics in cosmology and topology. For example, the author discusses the expansion of the universe, the age of the universe, dark matter, the dark night paradox, polyhedra, the classification of surfaces and three-dimensional spaces, and cosmic repulsion. He also analyzes the data from WMAP and describes the controversy that has arisen over its interpretation.

This is not a textbook, and it has no exercises. There are several very good expository chapters but the overall organization is somewhat eccentric. Combined with another book or two (for example, Jeffrey Weeks’ *The Shape of Space* and perhaps a more standard topology reference) this would make for a terrific “topics” or independent research course. Anyone even remotely interested in the subject would find this a fascinating book.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.