The Classification Theorem: Informal Presentation.- Surfaces.- Simplices, Complexes, and Triangulations.- The Fundamental Group, Orientability.- Homology Groups.- The Classification Theorem for Compact Surfaces.- Viewing the Real Projective Plane in R3.- Proof of Proposition 5.1.- Topological Preliminaries.- History of the Classification Theorem.- Every Surface Can be Triangulated.- Notes.
From the reviews:“Undoubtedly, one of the most beautiful pieces of
the mathematical achievements is the classification of compact
surfaces.... Each chapter provides a good list of references
connnecting its topic to the literature. At the end, some
appendices complete the work, the history of the classification
problem being specially interesting.” (Marco Castrillon Lopez,
European Mathematical Society, euro-math-soc.eu, January, 2018)
“The book would make an excellent graduate or advanced
undergraduate project. … audience would be instructors or
researchers looking to reconnect to some topology fundamentals.
Practitioners in topology-adjacent fields might find it valuable as
a concise reference. … a textbook or supplemental reference for a
second topology course with some careful planning … . Academic
libraries should strongly consider this book as it offers a broad
look at material common to a standard undergraduate mathematics
curriculum.” (Bill Wood, MAA Reviews, March, 2014)“This book aims
to give students … access to a true classic in algebraic topology:
the classification of compact surfaces (up to homeomorphism). … The
book includes many historical comments throughout and contains lots
of pictures––both of the mathematicians involved in the
classification theorem and of examples illustrating the
mathematical content. Many examples are given that help one
understand the basic tools … .” (Clara Lӧh, Mathematical Reviews,
December, 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)
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