
Simply connected manifold
Posted:
Jul 2, 1996 4:34 PM


I would like some help in finding a proof that SL(2,C) is a simply connected manifold. (SL(2,C) being the 2x2 complex matrices of unit determinant) The case for SU(2) is easy, as it is diffeomorphic to the three sphere S^3, but the polynomial equation which describes SL(2,C) is considerably harder for me: as a subset of R^8, it is the set of points satisfying the following two equations: sum over i = {1,2,3,4} (x_i)^2  (y_i)^2 = 1 and sum over i of (x_i y_i) = 0, where the eight coordinates are the x_i's and the y_i's. (as a subset of C^4, it is sum over i (z_i)^2 = 1 ,where the z's are complex and the squared is _not_ modulus squared) I am at a loss to show that this is a simply connected set, although I have heard heuristic arguments that it is S^3 x R^3 (or diffeomorphic to it).
Also, if we let pi_1(G) be the first homotopy group of a lie group G, then is pi_1(GxH) = pi_1(G)xpi_1(H) where = means isomorphic, and G and H are lie groups?
Thanks for your help in advance,
Jake Mannix UCSC Undergrad Math and Physics

