How to know if you are "tough enough" to study Algebraic Topology I am graduating with a BA this summer and I am very interested in topology. I admit it, I never went that deep into topology and all I know is about point-set topology (metric spaces etc.) but from personal readings, I love the style of proofs in topology and I have a general idea about how things go there, so I think I can go through Munkres' which will virtually pave the way for me to start studying algebraic topology. I have a solid grasp of abstract algebra, differential geometry and basically everything a typical undergrad should have covered and I think that I am ready to take on something challenging.
However, I heard from many grad students that algebraic topology is very difficult in comparison to other areas and that I should have known much more before now to start with serious stuff. So I am bit scared of taking on something which might later turn out to be out of my reach. 
To frame my question well: is algebraic topology that difficult compared to other areas of research in which one might pursue their graduate studies? How much should you know and how mathematically powerful should you be in order to successfully obtain a PhD in algebraic topology in a reasonable time?
 A: Algebraic topology, by it's very nature,is not an easy subject because it's really an uneven mixture of algebra and topology unlike any other subject you've seen before.However,how difficult it can be  to me depends on how you present algebraic topology and the chosen level of abstraction. 
If you want to use a high-tech and fully general approach, where everything is presented via diagram chases and category theory a la Peter May or Tammo Tom Dieck's texts-then yes,the subject can be brutal even for a well prepared student. In this case, the subject really resembles an algebra course a lot more then what you're used to in a topology or geometry course.For this kind of course,a very strong background in algebra with category theory is more important then any topology or geometry background. 
However, if your professor prefers a more classical,geometric approach like Allen Hatcher's book or an approach via differential forms on manifolds such as in Bott/Tu's classic text or Bredon's book, then the ability to visualize and connect to constructions in locally Euclidean spaces becomes much more important.In this case,your background in differential geometry is going to come in quite handy.  
Again, it'll really depend on your professor's background and what approach he/she prefers.A book I'd suggest you look at as a warmup to a serious algebraic topology course is John M. Lee's An Introduction To Topological Manifolds. It covers all the prerequisites needed for a serious algebraic topology course-the basics of point set topology, homotopy theory,commutative diagrams and category theory and the classification of compact surfaces-in a completely modern and very geometric way. I suspect you'll find a serious algebraic topology course a lot easier if you work your way through this book first.  
