Subject: Re: more on rational maps Date: Sun, 12 Dec 1999 22:58:47 -0500 From: "Clarence W. Wilkerson" This is a reply to the second posting by Jianzhong Pan: I was assumming that U(2n)/Sp(n) is a rational H-space, based on Ravenel's assertion about its cohomology. But 1) It seems to me that for G connected compact Lie with closed subgroup H, then for any positive integer k, the Lie k-th power map on G induces a well defined map on G/H, since it maps H into H. 2) The next question is what it does on cohomology. My memory is that Gugenheim-May settled, with the possible exception of p=2, the question of what H^*(G/H,F_p) is when H^*(BG,F_p) and H^*(BH,FP) are finitely generated polynomial algebras. It is additively the E_2 term of the Eilenberg-Moore SS for the fibration G/H -> BH -> BG . So one can work out the effect of the powermaps mentioned in 1) above. 3) We don't need that general a result for the case at hand, since H^*(BU(2n),Q) splits as a tensor product of H^*(BSp(n),Q) with another polynomial algebra on the remaining generators. Tor^1 of this second term gives the generators claimed. So I do believe that U(2n)/Sp(n) is a rational H-space. In any case, H^*(G/H) in this case injects back into H^*(G), so one can calculate the effect of the powermaps in H^*(G) . 4) I referred to the work of Zabrodsky. There is slightly later work of Richard Body, Douglas, and Sullivan, and Mimura-Toda-others on "universal" spaces with the property wrt to rational equivalences that I mentioned in my original posting. Thanks to John Opera for reminding me of this work. Clarence Wilkerson