It was written by Jean Gallier and Dianna Xu, and published in 2013 by Springer-Verlag as volume 9 of their Geometry and Computing series (doi:10.1007/978-3-642-34364-3, ISBN 978-3-642-34363-6).

An orientable surface of this type must be topologically equivalent (homeomorphic) to a sphere, torus, or more general handlebody, classified by its number of handles.

[2] This is a textbook aimed at the level of advanced undergraduates or beginning graduate students in mathematics,[2] perhaps after having already completed a first course in topology.

[4] However, by focusing on a single topic, the classification theorem, the book is able to prove the result rigorously while remaining at a lower overall level,[4][5] provide a greater amount of intuition and history,[4] and serve as "a motivating tour of the discipline’s fundamental techniques".

[1] Reviewer Clara Löh [de] complains that parts of the book are redundant, and in particular that the classification theorem can be proven either with the fundamental group or with homology (not needing both), that on the other hand several important tools from topology including the Jordan–Schoenflies theorem are not proven, and that several related classification results are omitted.