The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Jacob Alexnder Mannix

Posts: 9
Registered: 12/12/04
Simply connected manifold
Posted: Jul 2, 1996 4:34 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.