Research in topology for a master student I hope here is the best place to ask this, I will begin my master degree very soon, I've already attended the regular undergraduate courses included Real Analysis, Analysis on manifolds, Abstract Algebra, Field Theory, point-set topology, Algebraic Topology, etc...
I like very much algebraic topology and I found it really beautiful, I would like to know which areas of algebraic topology are the most interesting to begin to work with and which books I can study with my background in order to get the prerequisites to begin to study this subject. 
I want as soon as possible has a "taste" of a current research field in algebraic topology, and I know that an algebraic topologist can give me a "shortest way" while I attend the regular courses of my master degree.
Thank you
 A: If you want to learn about algebraic topology, you can begin by very classical readings. When I was a Ph-D student, I first read Milnor Stasheff's book on "Characteristic classes", here you will learn a lot of differential and algebraic topology. There are so many good books to read, J.-F. Adams "Infinite loop spaces" or his blue book on "stable homotopy and generalised homologies", J. Milnor on Morse theory. 
I highly recommand Andrew Ranicki's homepage where you will find a lot of cool stuff about algebraic and geometric surgery, PL-topology, exotic spheres. 
Jacob Lurie also has some very good notes of his courses on his homepage.
Dan Freed is giving a course on the cobordism and his notes are very nice, and you will find plenty of references here.
You can also look at H. Miller notes "Notes on cobordism" and "Vector fields on spheres" (just google it). And J. P. May has also a list of very good books on his homepage.
And overall, read classical papers by Adams, Pontryagin, Quillen, Serre, Sullivan, Thom...John Francis has a list of classical papers for the Kan seminar on his homepage.
I am sure my list is too long and I have forgotten plenty of good references (homepages, notes of courses and books).
A: For your first research problem, I recommend that you find an adviser in your department.
If there is no algebraic topologist in your department, find some other adviser, and
ask to suggest an interesting problem.
It is very unlikely that, as a master student, you will be able to find and solve
a reasonable research
problem yourself, without a help from an experienced adviser.
In this site, people can give you only reading recommendations, and this is probably not enough
to begin your own research.
But of course, there were rare exceptions in history when self-taught mathematicians
did good research.
Here is an outstanding problem in algebraic topology on this site:
fedja (mathoverflow.net/users/1131), Two commuting mappings in the disk, Two commuting mappings in the disk (version: 2009-11-25)
To understand the statement of the problem, little knowledge is required.
What does one need to learn to solve this problem, nobody knows:-)
I wish you luck.
