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Topic: Simply connected manifold
Replies: 4   Last Post: Jul 9, 1996 10:34 AM

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 Jacob Alexnder Mannix Posts: 9 Registered: 12/12/04
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?

Jake Mannix

Date Subject Author
7/2/96 Jacob Alexnder Mannix
7/2/96 David L. Johnson
7/5/96 Mark McConnell
7/8/96 Jacob Alexnder Mannix
7/9/96 Mark McConnell