Algebraic Topology

by Allen Hatcher

In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers.

Genres: Mathematics, Textbooks, Science, Algebra, Nonfiction, Reference, Geometry

556 pages, Paperback

First published November 15, 2001

About the author

Allen Hatcher is an American research mathematician and author currently at Cornell University. He specializes in Topology.

Reviews

[jeremy wang]

i love math!!!!!!!!!!!!!!!!!!!

[Tinwerume · Rating 4 out of 5]

Its reputation for being extremely unrigorous is deserved. On its own it's definitely inadequate for learning algebraic topology. But it does a good job of providing high level explanations and motivations for self study.

Update: It really suffers from the lack of rigor when it goes into more detailed constructions, e.g. in the chapter on cohomology.

[Anthony · Rating 4 out of 5]

Great introduction to algebraic topology. For those who have never taken a course or read a book on topology, I think Hatcher's book is a decent starting point. However, ( IMO) you should have a working familiarity with Euclidean Geometry, College Algebra, Logic or Discrete Math, and Set Theory before attempting this book. While some people might complain about the book's lack of rigor, I think this is irrelevant. What is important is that the book manages to convey the essentials of topology. If you are an intuitive learner with a rudimentary background in topology , you'll find this book to be more accessible than a book which focuses on rigor. However, if you are an academic, then you might find this book unsatisying.

Why study topology? Topology is like geometry on steroids. It has practical applications in the physical sciences, computer graphics, and statistics. Furthermore, there is an inherent beauty in topology which cannot be easily described in words. Topology expands our means of working with geometric figures. Unlike Euclidean Geometry, the geometry which you encounter in a high school math class, Topology is primarily concerned with transformations of connected geometric objects through bending, stretching, and twisting. Algebraic topology translates topological figures into algebraic images.

[Phyllis · Rating 5 out of 5]

I have this love-and-hate relationship with Hatcher.

[Kai · Rating 5 out of 5]

As an amateur going into topology next fall this was an extremely thought provoking book. A good complement to learning about point set topology and epsilon delta formalism. The idea of categorizing the shape of different spaces by the behavior of algebraic objects formed within them, (homotopy/homology groups), is almost Wittgensteinian in its beauty.

[Wayne]

Really good book! I really read this as a reference for the homology and cohomology sections for uni.

Will return soon for a more in-depth read through.

Shoutout sepehr for this hand-me-down copy

[Chris · Rating 3 out of 5]

As a beginner, prepped by Armstrong's Topology, I found this book, although readable, difficult and at times frustrating and exhausting. I spent seven months working the problems in chapters 0-3, with a handful of supplementary sections. Some say this book is not rigorous, but I disagree, and would say rather that I struggled because it isn't systematic enough for me: the geometric agenda gets in the way of revealing the fullness of the algebraic structure. Examples and exercises are often tricky and subtle applications from which it is hard to extract general patterns: examples often rely on geometric intuition and some exercises require more inspired homological algebra than provided in the text. Anyway, I'm just a beginner and this book is clearly correct and comprehensive, so I will just recommend that if you use this book for self-study you have another text as well and that you register for Math.SE.

[Harris · Rating 5 out of 5]

One of the least formal math books I've read, which was refreshing. The writing is always clear and the material is incredibly interesting. Will likely give this a more thorough read sometime in the future, as I took an odd route through it as dictated by my class on the same material.

[M · Rating 5 out of 5]

This book is not nonrigorous so much as it is post-rigorous. To quote Tao ,

The “post-rigorous” stage [is the stage] in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory.

Algebraic topology is as good a place to make this transition as any.

I found this book challenging upon first read-through when I was starting graduate school, because so many claims took me so long to process. I was transitioning from rigor to post-rigor. Struggling through proving the statements that were provided with proof outlines is exactly what helped me develop the intuition for algebraic topology that has served me well. And, as a now-topologist, this book contains just the right amount of explanation to be re-read pleasantly without getting bogged down in the details; I certainly can't say this about many other early graduate school texts.

In other words, I recommend new learners embrace the pain.

[Jeff · Rating 4 out of 5]

This is an excellent introduction to algebraic topology. However there are still some concepts that I struggled with while reading this book. For example, I didn't find the singular homology section super enlightening. Although that subject is dense in my opinion, it didn't help that Hatcher insisted on using lowercase delta for several different meanings within the same section. Also, I wasn't super impressed with Chapter 0, and I didn't feel like it did a sufficient job of bringing the reader up to speed on CW complexes, which is crucial to the later material

[Ivan Karamazov · Rating 4 out of 5]

One of the better introductory books out there on algebraic topology. The language is fluid and clean, the text is filled with useful analogies and the exercises are abundant and of good quality.

Quite non-rigorous in its treatment of material, but the author probably intended it to be so. This motivates and allows readers to assimilate concepts without getting lost in rigor, which also means you might want to use another more rigorous book together with this one.

[Jackson Morris · Rating 4 out of 5]

Hatcher is a bit hand wavy. I used this text for my first Algebriac Topology course in the Fall of 2020. It was nice sometimes: I like how it defined the cup product very directly, for example. But a lot of the proofs assumed a lot and were not as deep as I would’ve liked, or were poorly worded. The tone of the book is nice though.

[Sean Cottrell · Rating 5 out of 5]

Read the chapters on Fundamental Group, covering space theory, and simplicial, singular, & cellular homology for my topology class. Not bad if accompanied by a good lecture. Really thorough. Not super rigorous in a lot of proofs. If I were to try to teach myself directly out of this book without help I’d jump off a bridge. 5 stars.

[Chris · Rating 5 out of 5]

when someone spends 50 years thinking about diffeomorphisms, and is an excellent expositor, you get a master text.


Give a look to chapter 0 even if you're a non-mathematician. It will change what you think A+B−A−B means. And whether mathematics is about numbers or knitting.

[Nikos · Rating 5 out of 5]

Best book on the subject

[Lucas · Rating 5 out of 5]

best math textbook I've read

[Tim · Rating 5 out of 5]

Excellent introduction to the fundamentals of algebraic topology. It has tons of good exercises for every section. Also the secret fifth chapter on spectral sequences is an excellent intro to that subject!

Lack of rigor is a criticism that this text receives often which is at times warranted, although a student with decent background in algebra and point set topology should not have too much difficulty in filling the gaps.

[Bonnie]

Hatcher, I see your “what if we tried to make a formal dual of homology,” and I raise you “what if we took all the definitions from homology and just put ‘co-‘ in front of them.”

[Ella · Rating 4 out of 5]

I hate this book. It's fantastic.

[Phil · Rating 3 out of 5]

The best thing about this book is that its free on the internet! Hatcher's book is a good introduction to algebraic topology. Its full of examples and tons of extra material beyond the basics, which can actually make it difficult to find what you need. This book is NOT concise: if you have a lot of time on your hands and don't mind learning things that you probably won't ever use, this is a good book. But it can be hard to find what you need, and often the original examples that he gives aren't worth the effort, relative to how much they actually illuminate the material. I haven't tried many of the exercises.

[NoName66 · Rating 4 out of 5]

I have to say, the book is crazy, please gives specific definitions before you start to talk about theorems and examples. The good things of this book are the valuable exercises, examples, however, you need to fill in some details for the proof, which is good for some reasons (for me too), but if you don't want to learn algebraic topology in-depth, then it is not a friendly book, or you can study this book with some topology background

I finished 3 topics.

[Stephen]

ebook,non-fiction,mathematics,topology

[Lulu]

Algebraic topology