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# Topology of Surfaces by Kinsey

### From Mathematics Is A Science

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 $R^n$

- Open and closed sets in $R^n$ -- Relative neighborhoods -- Continuity -- Compact sets -- Connected sets -- Applications

*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

- Open sets and neighborhoods -- Continuity, connectedness, and compactness -- Separation axioms -- Product spaces -- Quotient spaces

*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

- Topological invariants -- Graphs and trees -- The euler characteristic and the sphere -- The euler characteristic and surfaces -- Map-coloring problems

*Why isn't "euler" capitalized?*

Ch. 6 Homology

- The algebra of chains -- Homology -- More computations -- Betti numbers and the euler characteristic

*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

- Cellular functions -- Homology and cellular functions -- Examples -- Covering spaces

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

Ch. 8 Invariance of homology

- Invariance of homology for surfaces -- The Simplicial Approximation Theorem

*The proof quite challenging for such a course.*

Ch. 9 Homotopy

- Homotopy and homology -- The fundamental group

*Too few pictures here. *

Ch. 10 Miscellany

- Applications -- The Jordan Curve Theorem -- 3-manifolds

*Good stuff here. Too little time.*

Ch. 11 Topology and calculus

- Vector fields and differential equations in $R^n$ -- Differentiable manifolds -- Vector fields on manifolds -- Integration on manifolds

*Same here.*

Appendix: Groups