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.