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This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge, or too informal to afford students a comprehensive insight into the subject. Its dedicated, student-centred approach details a near-complete proof of this theorem, widely admired for its efficacy and formal beauty. The authors present the technical tools needed to deploy the method effectively as well as demonstrating their use in a clearly structured, worked example.
Ideal for students whose mastery of algebraic topology may be a work-in-progress, the text introduces key notions such as fundamental groups, homology groups, and the Euler-Poincaré characteristic. These prerequisites are the subject of detailed appendices that enable focused, discrete learning where it is required, without interrupting the carefully planned structure of the core exposition. Gently guiding readers through the principles, theory, and applications of the classification theorem, the authors aim to foster genuine confidence in its use and in so doing encourage readers to move on to a deeper exploration of the versatile and valuable techniques available in algebraic topology.
Ideal for students whose mastery of algebraic topology may be a work-in-progress, the text introduces key notions such as fundamental groups, homology groups, and the Euler-Poincaré characteristic. These prerequisites are the subject of detailed appendices that enable focused, discrete learning where it is required, without interrupting the carefully planned structure of the core exposition. Gently guiding readers through the principles, theory, and applications of the classification theorem, the authors aim to foster genuine confidence in its use and in so doing encourage readers to move on to a deeper exploration of the versatile and valuable techniques available in algebraic topology.
From the reviews:
"This highly focused book by Gallier (Univ. of Pennsylvania) and Xu (Bryn Mawr College) does both, delivering rigor to undergraduates by developing minimal doses of homotopy and homology theory, and without even presuming familiarity with group theory. ... The present careful treatment of a major result that draws from several branches of mathematics makes the book an excellent resource for a capstone course. Summing Up: Highly recommended. Upper-division undergraduates and above." -- D. V. Feldman, Choice, Vol. 51 (1), September, 2013
"The book is geared toward an audience including first-year graduate students but also strongly motivated upper-level undergraduates ... . the book under review offers a beautiful introduction to one of the oldest and most fascinating theorems in algebraic topology ... . the historical remarks concerning the classification theorem for compact surfaces are highly enlightening to anyone interested in topology, and the many related biographies and photographs are just as entertaining. ... No doubt, this is an introductory text of remarkable didactic value." -- Werner Kleinert, zbMATH, Vol. 1270, 2013
"This highly focused book by Gallier (Univ. of Pennsylvania) and Xu (Bryn Mawr College) does both, delivering rigor to undergraduates by developing minimal doses of homotopy and homology theory, and without even presuming familiarity with group theory. ... The present careful treatment of a major result that draws from several branches of mathematics makes the book an excellent resource for a capstone course. Summing Up: Highly recommended. Upper-division undergraduates and above." -- D. V. Feldman, Choice, Vol. 51 (1), September, 2013
"The book is geared toward an audience including first-year graduate students but also strongly motivated upper-level undergraduates ... . the book under review offers a beautiful introduction to one of the oldest and most fascinating theorems in algebraic topology ... . the historical remarks concerning the classification theorem for compact surfaces are highly enlightening to anyone interested in topology, and the many related biographies and photographs are just as entertaining. ... No doubt, this is an introductory text of remarkable didactic value." -- Werner Kleinert, zbMATH, Vol. 1270, 2013