# Text similar to chapter 9 of Topology from James Munkres

I'm self-studying chapter 9 of Topology from James Munkres.

I like to read different books about the same topic at the same time. Can someone recommend some text/book that is about the same subjects as found in chapter 9?

This chapter is about the fundamental group. It is from part 2 of the book, which is called algebraic topology. The sections are called homotopy of paths, the fundamental group, covering spaces, the fundamental group of the circle, retractions and fixed points, the fundamental theorem of algebra, the Borsuk-Ulam theorem, deformation retracts and homotopy type, the fundamental group of S* and fundamental groups of some surfaces.

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Massey's "A Basic Course in Algebraic Topology" or Hatcher's "Algebraic Topology".

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+1 Massey's book used to be the standard place to learn about the classification theorem of surfaces before Lee wrote his book. I think the low cost of Hatcher and it's ready availability has really launched it into the forefront of algebraic topology texts more then it's content-although it is a very good book overall. I just wish Hatcher had been more rigorous in his presentation.There are times he seems to yammer on incoherently over the pretty pictures and your eyes glaze over. AT is a hard enough subject to learn to begin with without too much handwaving. – Mathemagician1234 May 2 '14 at 22:16
@Mathemagician1234: Agree on all points, which explains the link I gave to a copy of Massey's book (in general, I do not do such things). – studiosus May 2 '14 at 23:14

John M.Lee's Introduction to Topological Manifolds is in many ways a more modern, more geometrically themed version of Munkres. I consider it the prototype for a comprehensive first course in topology at the upper level undergraduate/first year graduate level.I also think it will serve a student a lot better then Munkres in preparing them for a serious graduate course in algebraic topology while still teaching them all the basics, including the elements of category theory and diagram chasing. There are lots of terrific books on topology,but I think that one is probably closest to what you're looking for.

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My book Topology and Groupoids is the only text that fully treats these topics in $1$-dimensional homotopy theory from the modern viewpoint of groupoids. It includes results on orbit spaces not available elsewhere, and a useful gluing theorem for homotopy equivalences.

I also mention that this book is one of the few that defines a path of length $r \geqslant 0$ in a space $X$ to be a map $a: [0,r] \to X$. If paths of length $r,s$ are composable then their composite is of length $r+s$, which makes sense intuitively, and the composition is then associative and with strict identities. This saves a certain amount of bother. One defines paths $a,b$ to be equivalent if there are constant paths $u,v$ such that $u+a,v+b$ are defined, are of the same length, and are homotopic rel end points in the usual sense. This gives a definition of the fundamental groupoid $\pi_1 X$.

The book does not include the fundamental groups of surfaces, which is well treated in several books, and also in the book "Knots and surfaces" by Gilbert and Porter, (OUP) (1996).

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+1 for your excellent book and Gilbert/Porter, Ronald. I could have recommended both and thought to, but the lack of discussion of classical combinatorial topics in your book, such as the classification of compact surfaces,is a lacuna that Lee doesn't have. Also,the Gilbert/Porter book,unfortunately, is intended for very elementary students-even high school students-and consequently lacks rigor,which is the only reservation I have with it. These 2 quibbles made me opt for Lee's instead. That being said, yours combined with Gilbert/Porter is a nice alternative. – Mathemagician1234 May 2 '14 at 22:10
I wonder why so many algebraic topology books ignore the arguments for groupoids and more than one base point as presented for example in mathoverflow.net/questions/40945/…. I also find the arguments for some treatment of knots compelling, since the examples are so clear to students, and also to children and non mathematicians. See the web page www.popmath.org.uk for some knot sculptures and a knot exhibition. – Ronnie Brown May 4 '14 at 10:14

Algebraic Topology by F.H Croom will be a good choice for a beginner in algebraic topology

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Here is a cocktail of sources that are quite helpful

1) Chapter III from Gamelin and Greene "Intro to Topology."

http://www.amazon.com/Introduction-Topology-Second-Edition-Mathematics/dp/0486406806

2) An excellent chapter by John Lee on Simply Connected Spaces

http://www.math.washington.edu/~lee/Courses/441-2012/simplyconn.pdf?v2

3) An extremely nice set of videos by NJ Wildberger - several of which clearly present homotopy, the fundamental group, and covering spaces. There are plenty of examples and lots of pictures which add an intuitive understanding to the rigor: