# First book on algebraic topology

Is there a good book on introduction to Algebraic Topology which is self-contained and does not assume any background in topology, only standard calculus and linear algebra courses?

• Here's a non-recommendation for a first book on algebraic topology: Edwin Spanier's Algebraic Topology. I know, I tried to read Spanier first! Cheers! – Robert Lewis May 20 '15 at 16:23
• @RobertLewis: +1. As much as I loved algebraic topology, I hated that book. – anomaly May 20 '15 at 16:33
• @anomaly: yeah, what a bear! but it's all there. I think it's better as a reference for people who already know most of that stuff, although I still have fond memories of the sections on the Lefschetz fixed point theorem! – Robert Lewis May 20 '15 at 16:44
• If you can read german check "Algebraische Topologie - Eine Einführung" by R. Stocker, H. Zieschang, it's really good. – PtF May 20 '15 at 16:59
• Since I haven't studied the book in details, I'll leave this just as a comment, but A First Course in Algebraic Topology, by Czes Kosniowski covers the minimum needed in point-set topology in the first few chapters. Maybe it is worth looking. – Ivo Terek May 20 '15 at 17:09

Warning : the following books are not algebraic topology books in the classical sens, but anyway it's impossible to get in AT without any background in topology : these two books will give you the necessary background and go after in the beginning of algebraic topology.

"Introduction to Topological Manifolds" by Lee is very readable and starts from scratch. It covers classic topology (4 chapters), one chapter on CW complex and classic topics in algebraic topology (fundamental group, covering space and groups) with emphasis on surfaces (with classification of compacts surface) with finally last chapters about homology and cohomology.

Another reference is the book of Munkres, Topology, which covers with lot (and lot) of details general topology, and also fundamental group and covering space.

For both, no background is needed (There is an appendix in Lee about group theory, and a chapter consacred to Set theory in Munkres !). For more advanced topics in algebraic topology (homology theory for example, or differentiable topology) I think calculus in $$\mathbb R^n$$, general topology and abstract algebra are highly recommended.

• I agree too, but first the OP ask a book of introduction to algebraic topology. I don't really think Spanier is really suitable for an introduction. Secondly, he asked no background in general topology, and obviously it's needed for doing algebraic topology, so why not a book which covers both ? In Lee there is only 4 chapters but almost all is used after. I think it would be ridiculous to recommand a normal book on algebraic topology. If he wants learn more, he can still ask again. – user171326 May 20 '15 at 16:51
• Clearly I did not read carefully (eg at all) what was asked; I now agree with your recommendations. I've deleted my first comment. (Might I suggest a statement, though, that these aren't really algebraic topology books?) – user98602 May 20 '15 at 16:53
• I edited my post ! – user171326 May 20 '15 at 16:56
• I liked munkres too. Very readable. – krishnab May 20 '15 at 19:58

The canonical reference is probably Hatcher's "Algebraic Topology," which in addition to being a very well-written text also has the advantage of being available online for free in its entirety. It stays in the category of CW-complexes for the most part, and there's a self-contained appendix describing enough of its topology to get you through the book. The only drawback I'd mentioned of it is that the more advanced topics (e.g., obstruction theory) covered in the appendices to its individual chapters often assume a bit more background or general sophistication than the rest of the book. It's not really a problem, and it's definitely not a reason to decide against the book, but you will probably notice a sharp difficulty spike if you go through those sections in order.

On the other hand, there's unfortunately no way of getting around the prerequisites in point-set topology. You'll find it easier after the first half of Hatcher, when things get more algebraic, but you should at least be comfortable with basic concepts like connectedness, compactness, metric spaces, various extensions lifting properties, etc. For that, I think Munkres' Topology is the canonical reference, but I can't recommend it to anyone; it's pedantic, tedious, and dull. I can't remember exactly what Lee's Introduction to Topological Manifolds, which N.H. mentioned above, covers in basic topology; still, it's a good book, and I'd recommend reading it regardless of whether you plan on continuing in algebraic topology.

• See this mathoverflow discussion mathoverflow.net/questions/40945/… for some of the issues behind my own book "Topology and Groupoids", (T&G), issues ignored by almost all other texts. T&G gives good motivation, assumes only some knowledge of analysis, has lots of figures, and exercises, does not do homology, introduces category theory, and includes topics such as orbit groupoids not even ventured into by other texts. It allows paths as maps $[0,r] \to X$, not just $r=1$. – Ronnie Brown May 21 '15 at 9:51
• I ought to give the url pages.bangor.ac.uk/~mas010/topgpds.html, which has a link to a £5 e-version. For more on groupoids, see the free download tac.mta.ca/tac/reprints/articles/7/tr7abs.html, ad how to fit this with homology theory see pages.bangor.ac.uk/~mas010/nonab-a-t.html, with a free pdf, and which discusses history, intuitions, and motivation. – Ronnie Brown May 21 '15 at 10:14
• @RonnieBrown : I'm curious, what is the main interest of allowed path as map $[0,r] \to X$ ? – user171326 May 21 '15 at 15:37
• @RonnieBrown: Neat, thanks for the link! I'm not familiar with the text (I've used Hatcher when presenting introductory algebraic topology in the past), but I like the approach. – anomaly May 21 '15 at 15:52
• @N.H. With this approach, the paths with composition form a category, with associativity, etc. Also it is more intuitive: if you think of a path as a journey, we are used to journeys of different length, and their composition. This avoids some proofs students find tricky, and technical, and which do not really convey much. Of course you have at some stage to discuss reparametrization, and the definition of equivalence of paths. – Ronnie Brown May 21 '15 at 16:54

The question about the first topology textbook has been asked and answered several times but what about specifically undergraduate textbook? Munkres has a good reputation but it's too dry in my opinion for a good undergraduate textbook. Kosniowski, mentioned above, is a very good choice with illustrations on almost every page. The only problem I have with it (and Munkres, and Basic Topology by Armstrong) is that its first topic in algebraic topology is the fundamental group. I prefer homology. The advantages are: (1) you can start with no point-set topology, (2) homology studies topological features of all dimensions instead of just one (cuts, tunnels, voids, etc.), and (3) you can connect homology back to calculus. For a short introduction to homology (with some point-set topology) I recommend Topology of Surfaces by Kinsey and for a longer one my own Topology Illustrated.