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# Topology of Surfaces by Kinsey

I have used this book a couple of times as a required textbook for a course in introductory algebraic topology. Overall, a good book. Good exposition and plenty of exercises, with just a few complaints. Both the good and the bad inspired me in writing my own book, Topology Illustrated. My impressions, from back then, are below. Chapter 1, Introduction to topology: I wished for even more applied stuff here, especially applications and ideas outside mathematics. Chapter 2, Point-set topology in R^n: I was looking for a clearer connection to calculus here, beyond just the definition of continuity: continuity under algebraic operations, the Intermediate Value Theorem, sequences, etc. Chapter 3, Point-set topology is quite short, as it should be. Chapter 4, Surfaces: The definition of a "regular complex" is bad and I don't think it can be fixed. I understand that in such a book compromises are inevitable, but a simplified version of a CW-complex would be more appropriate. Meanwhile, dealing with the full proof of the Classification Theorem of Surfaces didn't fit the purpose I had for the book. Chapter 5, The euler characteristic: It is appropriately introduced as the very first topological invariant -- of graphs and surfaces -- and a lead into homology. Chapter 6, Homology: A first encounter with homology should be gentler in my opinion, holding the algebra back for a while. Meanwhile, the proof of the theorem about the homology of oriented surfaces lacks some details. Error: on page 139, the “cone” of the circle isn't the sphere but the “suspension” is. Chapter 7, Cellular functions: I wished for more examples of specific functions: inclusions, projections, quotients, etc. Chapter 8, Invariance of homology: that and the Simplicial Approximation Theorem require proofs that are quite challenging for such a course. Chapter 9, Homotopy: For such a topic, too few pictures here. Chapter 10, Miscellany: Good stuff here: the Jordan Curve Theorem, 3-manifolds, etc., but too little time... Chapter 11, Topology and calculus: More good stuff (even though the presentation is a bit sketchy): vector fields, ODEs, differentiable manifolds, etc.

Preface

Ch. 1 Introduction to topology

• An overview

I'd rather see more applied stuff here, in addition. And by applied I mean applications outside math.

Ch. 2 Point-set topology in $R^n$

I wish there was a good connection to calculus here, beyond just the definition of continuity: continuity under algebraic operations, the Intermediate value Theorem, sequences, etc.

Ch. 3 Point-set topology

Short, as it should be.

Ch. 4 Surfaces

The definition of "regular complex" is bad and I don't think it can be fixed. I understand that this is a compromise, but a simplified version of a CW-complex is easier to justify.

I am not sure that the full proof of the Classification Theorem of Surfaces is justified in such a book.

Ch. 5 The euler characteristic

Why isn't "euler" capitalized?

Ch. 6 Homology

A first encounter with homology should be gentler in my opinion.

The proofs of the theorem about the homology of oriented surfaces lacks some details.

ERROR: on page 139, the cone of the circle isn't the sphere. It's supposed to be the suspension.

Ch. 7 Cellular functions

I which for more examples of specific functions: identity function, inclusions, projections, quotients, etc.

Ch. 8 Invariance of homology

The proof quite challenging for such a course.

Ch. 9 Homotopy

Too few pictures here.

Ch. 10 Miscellany

Good stuff here. Too little time.

Ch. 11 Topology and calculus

Same here.

Appendix: Groups