A friend of mine and I were trying to answer a question related to his research and he couldn't remember whether or not the special linear group over the complex numbers, SLn(C),was simply connected. (It IS,of course.)

This got me wondering:What are all the simply connected topological subgroups of the general linear group over C? Is there a simple characterization of all of them up to isomorphism? What about thier fundamental groups as topological spaces?Are THEY simply connected if the subgroup is? I would expect them to have fundamental groups as basepoints should be easy to choose via the identity matrix.

So is there such a characterization for the simply connected subgroups of GLn(C)?