Reprinted with permission from CHOICE, © American Library Association.

Laurent Bessières (Université Joseph Fourier, Grenoble, France)
Gérard Besson (Université Joseph Fourier, Grenoble, France)
Michel Boileau (Université Paul Sabatier, Toulouse, France)
Sylvain Maillot (Université Montpellier II, France)
Joan Porti (Universitat Autònoma de Barcelona, Spain)

Geometrisation of 3-Manifolds

EMS Tracts in Mathematics Vol. 13
ISBN 978-3-03719-082-1
DOI 10.4171/082
September 2010, 247 pages, hardcover, 17 x 24 cm. 48.00 Euro

A mathematics library dedicated to serving undergraduates must nevertheless offer a wise selection of graduate-level material, both first-year graduate textbooks and books consolidating major recent triumphs and illuminating dominant trends. G. Perelman's settling of the celebrated Poincaré conjecture--and beyond that the full Thurston geometrisation conjecture--stands as the outstanding mathematical accomplishment of the new century. Since Perelman did not publish his work, several teams fleshed out his sketches in the form of journal articles, Internet notes, and one notable book, J. Morgan and G. Tian's Ricci Flow and the Poincaré Conjecture (2007). Thurston's conjecture inextricably binds issues of pure three-dimensional topology to Riemannian geometry, and Perelman's proof moreover introduces heavy analytical tools. While topologists reluctantly accept the invasion of their subject by exogenous techniques, they strive, as in this work by Bessiéres (Univ. Joseph Fourier, France) and colleagues, for more idiomatically topological methods. The authors' new proof replaces Perelman's "Ricci flow with surgery" with a simplification called "Ricci flow with bubbling-off," an evolving flow on an unchanging manifold where surgery occurs before singularities arise. Chapter 1, "The Geometrisation Conjecture," provides the best available survey of the whole business, and though it moves fast, the body of the book starts from square zero.

Summing Up: Recommended. Upper-division undergraduates through researchers/faculty

D. V. Feldman, University of New Hampshire