TITLE: Research in topology for a master student
QUESTION [8 upvotes]: 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

REPLY [3 votes]: 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.