I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which one should I start with? and can you recommend books on either one?

5$\begingroup$ I'm not sure what is the rationale for closing here; pretty much the same question, asked about algebraic geometry, is staying there pretty well: mathoverflow.net/questions/2446/…, so do you guys think algebraic geometry is"research level" mathematics while algebraic / differential topology somehow isn't? $\endgroup$ – Fan Zheng Apr 18 '15 at 20:28

$\begingroup$ @FanZheng: this is a fairly different question. It's asking for a value judgement on learning a topic in a particular order. Generally speaking these kinds of "study advice" questions aren't treated particularly kindly here on MO. If asked in a fairly refined way, they do get tolerated, though. $\endgroup$ – Ryan Budney Apr 20 '15 at 22:31

1$\begingroup$ To answer the question, there's no reason to start with either. Why not just learn both at the same time? $\endgroup$ – Ryan Budney Apr 20 '15 at 22:37
Run, don't walk, to Milnor's "Topology From The Differentiable Viewpoint."

1$\begingroup$ BTW, it happens to have exactly one reference to homology theory :) $\endgroup$ – M.G. Apr 18 '15 at 15:02

$\begingroup$ @July It does! And I think I agree with you that differential topology often requires some algebraic topology, but Milnor's little book is such a pleasure, it should be enjoyed as soon as possible. $\endgroup$ – David Steinberg Apr 18 '15 at 15:10


$\begingroup$ I got the impression that homology and cohomology theory are important yet there is little to no reference to them in the book, why is that? $\endgroup$ – user70684 Apr 18 '15 at 15:16

$\begingroup$ Because the material in that book can do without them and so the focus of the book is as its title suggests. As I said, however, you cannot have a nice mature picture of Differential Topology if you leave aspects of Algebraic Topology out of it. $\endgroup$ – M.G. Apr 18 '15 at 15:39
I would say, it depends on how much Differential Topology you are interested in. Generally speaking, Differential Topology makes use of Algebraic Topology at various places, but there are also books like Hirsch' that introduce Differential Topology without (almost) any references to Algebraic Topology. Having said that, topological theory built on differential forms needs background/experience in Algebraic Topology (and some Homological Algebra). In other words, for a proper study of Differential Topology, Algebraic Topology is a prerequisite.
Addendum (book recommendations):
1) For a general introduction to Geometry and Topology:
 Bredon "Topology and Geometry": I can wholeheartedly recommend it! First part covers all the necessary and important general topology, then moves on to Differentiable Manifolds, after which it goes to Algebraic Topology (fundamental groups, (co)homology,homotopy theory). I think it is a good first book because it is selfcontained, has exercises and gives a taste of different basic parts of modern geometry and topology without leaving the impression that they are isolated from each other.
2) For Algebraic Topology:
Hatcher "Algebraic Topology": personally, I find his book overrated and quite annoying. It made me hate algebraic topology in my undergraduate years! YMMV.
Spanier "Algebraic Topology": probably not a good first book, a little outdated, but much less annoying than Hatcher's. There is a lot to learn from it!
Switzer "Algebraic Topology": great second book for Algebraic Topology, covering various topics of modern topology.
Rotman "Introduction to Algebraic Topology": good introduction to very basic Algebraic Topology.
tom Dieck "Algebraic Topology": good introduction to Algebraic Topology, but it contains several typos, so it's probably not so great for beginners. On the bright side, that keeps you on your toes to make sure you are paying attention :P It covers a good amount of topics too.
Greenberg and Harper "Algebraic Topology... a first course": "readerfriendly" introduction to basic Algebraic Topology, but some prior experience with the language of category might be helpful.
Davis and Kirk "Lecture notes on algebraic topology": since they are lecture notes, the material is "compressed" without filling in between (which is nice in my opinion) and covers various important topics of Algebraic Topology. The notes can be downloaded from their homepage .
May "A concise course in algebraic topology": in my opinion, not suitable for readers without prior experience with Algebraic Topology (unless they are very gifted students).
3) For Algebraic Topology with homotopical focus:
Selick "Introduction To Homotopy Theory": it is suited for readers who already have some experience with the basic concepts of Algebraic Topology, Category theory and Homological Algebra. Although the first part deals with all these prerequisites, the material would get dense for readers without prior exposure to them.
Gray "Homotopy Theory  An Introducton to Algebraic Topology": nice selfcontained introductory exposition with exercises, suitable for beginners in Algebraic Topology (knowledge of general pointset topology is assumed)
Fomenko, Fuchs "Homotopic Topology"  it is one of those books where you have to work your way through it (i.e. lots of stuff left to the reader as exercises).
Aguilar, Gitler, Prieto "Algebraic Topology from a Homotopical Viewpoint": only requires solid understanding of basic general topology. Prior experience with Algebraic Topology and Category Theory might be helpful, but not necessary.
Strom "Modern Classical Homotopy Theory": little gem that only assumes solid understanding of basic general topology. All the theory is presented as miniproblems to work through. What better way to learn something than to work it out on your own. Thus some experience with Algebraic Topology might be helpful, but not strictly necessary. And it is selfcontained in the sense that it takes care of the necessary category theory.
Whitehead "Elements of homotopy theory": requires a first course in algebraic topology.
Warner "Topics in Topology and Homotopy Theory": not suitable for beginners, it is more of an encyclopedia (mere 900+ pages).
4) For Differential Topology:
Hirsch' "Differential Topology": selfcontained, in particualar requires no prior knowledge of Algebraic Topology;
Milnor's "Topology from Differentiable Viewpoint": selfcontained, in particular requires no prior knowledge of Algebraic Topology;
Milnor's "Morse Theory": the classic book on Morse theory;
Guillemin and Pollack's "Differential Topology": selfcontained, the last chapter introduces some cohomology theory, thus it does not omit this important tool.
5) For Differential Algebraic Topology:
Milnor and Stasheff's "Characteristic Classes": A first course in smooth manifolds might be helpful, but not necessary.
Bott and Tu's "Differential Forms in Algebraic Topology": another classic text, the title is selfexplanatory;
Kreck's "Differential algebraic topology  from stratifolds to exotic spheres": advanced text.

$\begingroup$ please go ahead and list books from the homotopical viewpoint, I don't think having more recommendations hurts anyone. $\endgroup$ – bananastack Apr 18 '15 at 15:50

$\begingroup$ Done. I have also improved the structure and added more comments (=my own opinions) about the various texts. I hope this helps. $\endgroup$ – M.G. Apr 18 '15 at 17:13

$\begingroup$ I downvoted this otherwise excellent list, but I must admit it was an emotional reaction to the negative comments about Hatcher's book. Now the software won't allow me to reverse my downvote, unless the answer is edited. Perhaps this will motivate you to remove these subjective comments? ;) $\endgroup$ – Mark Grant Apr 21 '15 at 9:30

1$\begingroup$ So, in order to avoid the wrath of Hatcher's book fanboys, I should have written "great book, recommended to any beginner in AT" instead, which would be less subjective?... I know many people who feel the same way about this book, but they are simply much less vocal. Since his book is recommended pretty much in every other similar MO topic by the same people over and over again, I think it's healthy to have a negative opinion once in a while. But you are right, it motivates me to substantiate my criticism towards this textbook better. $\endgroup$ – M.G. Apr 21 '15 at 12:35
If you read Bott and Tu, you can learn both some differential and some algebraic topology at the same time! Unless you're unusually uncompromising, though, this will be heavy going for a beginner. Reading Milnor's book first, as David Steinberg suggests, seems a great idea (and won't take too long either).
One of the most common references for basic algebraic topology is Hatcher's book that you can download for free from his website. Another good books for somewhat different topics are Milnor and Stasheff's Characteristic classes and maybe Atiyah's Ktheory. If you're still interested do not forget to read Adams' "Stable homotopy and generalized cohomology" and "Infinite loop spaces" books. They are very nice, even if they are starting to show their age.