It is fairly easy to see that the set of non invertible matrices is path connected. Are they simply connected? If not what is their fundamental group? What are their homotopy and homology groups. I'm looking for the answer to any of these questions. Any examples for particular (non 0 or 1) dimensions are welcome as well, as well as for real or complex coefficients. (Sorry about the phrasing, it is currently 3:30 am and I will edit the question in the morning)
The space of non-invertible matrices (with real or complex coefficients) is contractible. There is an explicit homotopy between the identity map and the constant map equal to the zero matrix, simply given by: $$H(A,t) = tA.$$ In particular it is simply connected, and all its higher homotopy and homology groups vanish.