The 1930s were important years in the development of modern topology, pushed forward by the appearance of a few pivotal books, of which this is one. The focus is on combinatorial and algebraic topology, with as much point-set topology as needed for the main topics. One sees from the modern point of view that the authors are working in a category of spaces that includes locally finite simplicial complexes. (Their definition of manifold is more properly known today as a "triangulizable homology manifold".) Amazingly, they manage to accomplish a lot without the convenient tools of homological algebra, such as exact sequences and commutative diagrams, that were developed later. The main topics covered are: simplicial homology (coefficients in \(\mathbb{Z}\) or \(\mathbb{Z}_2\)), local homology, surface topology, the fundamental group and covering spaces, three-manifolds, Poincaré duality, and the Lefschetz fixed point theorem.

Few prerequisites are necessary. A final section reviews the lemmas and theorems from group theory that are needed in the text. As stated in the introduction to the important book by Alexandroff and Hopf (which appeared a year after Seifert and Threlfall): "Its lively and instructive presentation makes this book particularly suitable as an introduction or as a textbook."

Reviews

"The exposition proceeds by easy stages with examples and illustrations at every turn."

-- Bulletin of the AMS

"The great strength of this book is the geometric insight it has given to generations of readers."

-- Mathematical Reviews

Table of Contents

• Anschauungsmaterial
• Simplizialer Komplex
• Homologiegruppen
• Simpliziale Approximation
• Eigenschaften im Punkte
• Flächentopologie
• Fundamentalgruppe
• Überlagerungskomplexe
• Dreidimensionale Mannigfaltigkeiten
• \(n\)-dimensionale Mannigfaltigkeiten
• Stetige Abbildungen
• Hilfssätze aus der Gruppentheorie