From: "Jianzhong Pan" <purduepjz@hotmail.com>
Subject: Re: Wilkerson comments
Date: Mon, 13 Dec 1999 09:21:46 CST




>From: dmd1@Lehigh.EDU (DON DAVIS)
>To: Distribution.List@lehigh.edu (toplist)
>Subject: Wilkerson comments
>Date: Fri, 10 Dec 1999 12:16:19 EST
>
>Date: Fri, 10 Dec 1999 10:00:05 -0500
>From: Clarence Wilkerson <wilker@math.purdue.edu>
>Subject: Re: response re Ravenel
>
>I think Ravenel's conclusion is correct but that the
>chain of thought presented is incomplete and not quite as
>straightforward as it might seem.
>According to Ravenel's assertion about the cohomology of U(2n)/Sp(n),
>it is an H-space after rationallization. This implies by work of
>Zabrodsky in the late 60's and early 70's that there are selfmaps
>analogous to the powermaps on an H-space).
>These selfmaps are rational equivalences that induce multiplication
>by  certain integers on the indecomposable quotient of the
>cohomology ring of  U(2n)/Sp(n).
>
>This means that U(2n)/Sp(n) has the interesting property that given
>a map f:X \to Y of  s.c finite type CW complexes which is a rational
>equivalence, and a map  h: U(2n)/Sp(n) \to Y, then there is a
>g: U(2n)/Sp(n) \to U(2n)/Sp(n) which is a rational equivalence
>so that the composition   g(h)  lifts to X. The proof is an execise
>in Postnikov style obstruction theory, using the powermaps to kill off
>the obstructions, which lie in H^*(U(2n)/Sp(n), \pi_*(X,Y) which are
>finite.
>
>Thus, one does not need to find  a map GEM \to BSp(n)  which is a
>rational equivalence.
>
>By the way, it is pretty easy to see that there is no map
>GEM \to BSp(n) which induces a non-zero map on mod p
>cohomology, using properties of the Steenrod operations,
>
>Clarence Wilkerson
>
>
>At 08:29 AM 12/10/99 -0500, you wrote:
> >From: "Jianzhong Pan" <purduepjz@hotmail.com>
> >Subject: remark on Ravenal's responce
> >Date: Fri, 10 Dec 1999 09:55:20 CST
> > >
> >Dear Prof.Ravenal:
> >Unfortunately,your proof that there is a rationally nontrivial
> >map from $X_n$ to $Y_n$ is not true.
> >The problem is that there is no map from $Y'_n$ to $Y_n$ which is
> >also a rational equivalence (this follows from results of phantom map
> >theory ).On the other hand the problem that"Is there a rational
>equivalence
> >from $\Omega X_n$ to $\Omega X'_n$?" is an open problem as posed in a
>paper
> >by McGibbon in "Handbook of Algebraic Topology" .
> >I guess maybe there is no rationally nontrivial map from
> >$X_n$ to $Y_n$ though I can not prove it.
> >Pan Jianzhong
> >
> >______________________________________________________
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>
It seems to me that the map wanted doesn't exist.
If $U(2n)/Sp(n)$ is indeed rational H-space , then Wilkerson's
argument gives the map wanted. This space is not rational H-space
according to my memory. I hope this is true.
On the other hand , a similar method in a paper
by Glover and another one in Trans.AMS,276(1983) (self maps
of flag manifolds) applied to nonrational H-space.
But it seems that it may not apply to the present
situation .
Hope this will be helpful
Pan Jianzhong

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