From: "Jianzhong Pan" <purduepjz@hotmail.com>
Subject: Re: more from Wilkerson
Date: Tue, 14 Dec 1999 09:43:26 CST
>
>Subject: Re: more on rational maps
>Date: Sun, 12 Dec 1999 22:58:47 -0500
>From: "Clarence W. Wilkerson" <wilker@ip209-183-67-150.ts.indy.net>
>
>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
>
I am sorry I was wrong.
The fact that $U(2n)/Sp(n) $ is a rational H-space has also be proved
in a text book "Topology of Lie groups" by Mimura and Toda as
Theorem 3.6.7.
On the other hand McGibbon proved that for nilpotent rational H-space
which is also a finite CW , then for each prime p and sufficient large
integer n the grading homomorphism $x \mapto (q^i)x$
is realizable where $i=deg(x)$ and $q=p^n$ (c.f.McGibbon,On the localization
genus of a space,1994 Barcelona conference proceeding. ).
This applies to $U(2n)/Sp(n) $ . Then Wilkerson's argument in
the previous post applies to give the desired map.
Pan Jianzhong

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