Introduction to Homotopy Theory (Universitext) by Martin Arkowitz (PDF)

Description

This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows: Basic Homotopy; H-spaces and co-H-spaces; fibrations and cofibrations; exact sequences of homotopy sets, actions, and coactions; homotopy pushouts and pullbacks; classical theorems, including those of Serre, Hurewicz, Blakers-Massey, and Whitehead; homotopy Sets; homotopy and homology decompositions of spaces and maps; and obstruction theory. The underlying theme of the entire book is the Eckmann-Hilton duality theory. It is assumed that the reader has had some exposure to the rudiments of homology theory and fundamental group theory. These topics are discussed in the appendices. The book can be used as a text for the second semester of an advanced ungraduate or graduate algebraic topology course.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I bought this book while I was taking algebraic topology. At the time, I was having a hard time understanding what this duality in homotopy was all about. Now I can understand the essence of lots of duality in all of algebraic topology, thanks to the author.

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⭐I thought this was a well written introduction to the fundamentals of homotopy theory. I found the section on fibrations/cofibrations particularly helpful.

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⭐After suggesting that this book be used as the text in the second semester of a graduate topology course, the blurb on the back cover continues “The book could also be used by anyone with a little background in topology who wishes to learn some homotopy theory.” That seemed to fit me, since although being a topological dilettante with a career outside mathematics, I’d already read a couple of books on algebraic topology. I figured that even if I could make it partway into Chapter 2 (there are nine chapters) I could benefit from it.Alas, no such luck. The author opts for long, dense paragraphs of prose dominated by symbols. By p. 8 there is a chain of 15 definitions presented in one relentless paragraph, without any graphical support whatsoever. This style continues throughout Chapter 1. Homotopy is an inherently visual topic. While this book isn’t exactly devoid of diagrams, it’s definitely under-endowed in that regard. Nor does the prose style make many concessions to those who might benefit from having the big picture explained in English occasionally.This is clearly a book for grad students or others for whom mathematics is a full-time occupation, who like a predominantly algebraic (non-diagrammatic) approach, and who have the time and motivation to unpack each sentence one by one. I’m comforted by the fact that I bought this at discount in a Yellow Sale, but disappointed that it will probably be a long time before I pull it off the shelf again.

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