Subject: another response for the list
From: Tyler Lawson
Date: Mon, 27 Feb 2006 19:50:20 -0500 (EST)
Note that an element A of SU(2) has a unique eigenvalue z
with Im(z) >= 0; if Im(z) > 0, the element A is then determined
it is the pushout of the diagram
{0,1} <- {0,1}x(SU(2)/T) -> [0,1]x(SU(2)/T)
where T is the diagonal torus. Taking products of this decomposition
allows you to understand the quotient.
Ultimately, association of eigenvalues gives a map
SU(2)^n/conj -> [0,1]^n.
The preimage of a point in the range is a space of isomorphism classes
of point arrangements in CP1; if the point has k coordinates z_i not
equal to 0 or 1, then the preimage is the space of isomorphism classes
of k-tuples of points.
If k=1, this preimage is a point.
If k=2, a point in the preimage is determined by an angle, so the
preimage is contractible.
If k=3, this is a space of configurations of point triples in S2
mod the action of SU(2); exercise, this space is homotopy equivalent to
S3.
So the net effect of this is:
- SU(2)/conj is contractible
- SU(2)2/conj is contractible
- SU(2)3/conj is homotopy equivalent to S6
More generally,
- SU(2)^n/conj has Euler characteristic 2^{n-2} for n >= 2
Using similar methods, one can show that U(3)2/conj has homology groups
abstractly isomorphic to those of (S1 x S1 x S8).
--
Tyler.