Re: SL(n,C)

in article <di5t85$bkt$1@xxxxxxxxxxxxxxxx>,
lee rudolph <lrudolph@xxxxxxxxx> wrote:

|Timothy Murphy <tim@xxxxxxxxxxxxxxxxxxxxxx> writes:
|
|>Rupert wrote:
|>
|>> Can anyone point me towards a proof that SL(n,C) is simply
|>> connected?
....
|It would also appear that, for n > 1, any proof of what Rupert wants
|should be pointed away from, as being bogus: for instance, in case n
|= 2, SL(2,C) is the complement in a 4-dimensional complex vectorspace
|with coordinates (a,b,c,d) of the non-singular complex hyperquadric
|ad-bc = 1, and any of a number of similar but distinct arguments
|shows that that complement has non-zero first homology and _a
|fortiori_ is not simply-connected.
|
|Lee Rudolph
|[P.S. In case any or all of my argument above is itself bogus, I beg
|indulgence; I am severely undercaffeinated this morning.]

you did have me going for a moment there, but that only shows how
shamefully little i know about basic practical homology calculations.
doesn't the theorem about the inclusion of a maximal compact subgroup
being a homotopy equivalence apply here, and isn't su(2) a maximal
compact subgroup here, and isn't su(2) a 3-sphere?

















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