I am reading Munkres and know exactly how to find the fundamental groups of surfaces, using pi_1 and reducing it down to simpler problems. However, I'm completely lost when looking at my final exam it says to find the fundamental groups of matrices! How do you go about doing that! There are no explainations in teh book, or the net (that I can find). The two main ones I'm having trouble with are here, if you could quickly help that would be great. Find the fundamental groups of 1) The space of 2x2 matrices over C (complex) with 0 determinant with the topology given by the embedding in the vector space of all 2x2 matrices over C. 2) The space of upper triangular matrices of the size 2x2 over C with determinant 1 considered as a closed subspace in the vector space of all matrices size 2x2 over C. I know the answer to 1, because someone told me it, but I don't know how they came to that conclusion. The answer to one is the integers, Z. Thank you!