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.

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.