Subject: Re: correction to posted question
Date: Mon, 24 Jun 2002 22:45:01 -0500
From: Bill Richter <richter@math.northwestern.edu>

   Subject: stable homotopy group of Eilenberg-Maclane space
   Date: Mon, 24 Jun 2002 20:08:14 +0800
   From: "pjz" <pjz@mail.amss.ac.cn>

   The referee of one of my papers suggested that 2-primary part of
   2n-th stable homotopy group of Eilenberg-Maclane space K(G,n-1) is
   known where G is a finite abelian group.  Anyone know a reference?

I think the standard references are

@ARTICLE{B-M,
        AUTHOR = {W. D. Barcus  and J. P. Meyer},
        TITLE = {The suspension of a loop space},
        JOURNAL = {Amer. J. Math.},
        YEAR = {1958},
        VOLUME = {80},
        PAGES = {895--920}
}

@ARTICLE{Whitehead:homology-suspension(55),
        AUTHOR = {G. W. Whitehead},
        TITLE = {On the homology suspension},
        JOURNAL = {Ann. of Math.},
        YEAR = {1955},
        VOLUME = {2},
        PAGES = {254--268}
}

Let X = K(G,n).  As Mark Mahowald explained to me a long time ago,
what's important here is the Barcus-Meyers-Whitehead fibration

Omega X * Omega X -Hopf construction--> Sigma Omega X -evaluation--> X

which in the metastable range (which you are in) is a cofibration, and
also induced by the diagonal map

X ---> X smash X

Let's just assume that X is (n-1)-connected here.  You want to know

pi_{2n}^s( Omega X ) = pi_{2n+1}^s( Sigma Omega X)

and that's in a long exact sequence

pi_{2n+1}^s( X ) ---> pi_{2n+2}^s( X smash X ) --->

pi_{2n+1}^s( Sigma Omega X)

        ---> pi_{2n+1}^s( X ) ---> pi_{2n+1}^s( X smash X )

Hmm, I'm getting stuck, because we don't quite know pi_{2n+2}^s( X ).
You can build a tower of fibration cofibrations here, because

X = Omega K(G, n+1)

etc, and you oughta be able to work it out from that.  Eventually
stable homotopy become unstable homotopy, which we know, since they're
EM spaces.  I'll have to go think about it :)

The metastable stuff is stated below, but unfortunately it doesn't
contain a complete proof:

@InCollection{M-R2,
  author =       "M. Mahowald and W. Richter",
  title = "{$\Omega SU(n)$} does not split in 2 suspensions, for
{$n\ge3$}",
  booktitle =    " Alg. Top. Proc. Oaxtepec",
  publisher =    "Amer. Math. Soc",
  year =         1993,
  editor =       "M. Tangora",
  series =       "Cont. Math. Series",
  volume =       "146"
}