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?
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Run, don't walk, to Milnor's "Topology From The Differentiable Viewpoint."
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 self-contained, 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Spanier "Algebraic Topology": probably not a good first book, a little outdated, but much less annoying than Hatcher's. There is a lot to learn from it!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": "reader-friendly" 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 self-contained introductory exposition with exercises, suitable for beginners in Algebraic Topology (knowledge of general point-set 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 mini-problems 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 self-contained 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": self-contained, in particualar requires no prior knowledge of Algebraic Topology;
Milnor's "Topology from Differentiable Viewpoint": self-contained, 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": self-contained, 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 self-explanatory;
Kreck's "Differential algebraic topology - from stratifolds to exotic spheres": advanced text.
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 K-theory. 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.