Math Forum Discussions

Jacob Alexnder Mannix

Posts: 9
Registered: 12/12/04

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

Plain Text

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