Fundmental groups of matrix groups and covering relation between them

Question

I am self studying algebraic topology, the fundamental groups and covering spaces and homologous groups.

I have read the topological properties of classical matrix groups like connectedness, path connectedness and compactness.

Now I want to study the algebraic topological properties of these groups. In particular what are the fundmental groups of patnconnected components of the matrix groups?

Does there exists any relation among the matrix groups via covering spaces?

If anyone provide some referances regarding these questions that will be very much helpful.

Thanks in advance.