Subject: RE: two postings
From: "Neil Strickland"
Date: Mon, 20 Nov 2006 17:05:38 -0000
In answer to Takuji's question: the approach he mentions is more or less
what is done in Section 10 of the following paper:
@incollection {MR1718079,
AUTHOR = {Goerss, Paul G.},
TITLE = {Hopf rings, {D}ieudonn\'e modules, and {$E\sb *\Omega\sp
2S\sp
3$}},
BOOKTITLE = {Homotopy invariant algebraic structures (Baltimore, MD,
1998)},
SERIES = {Contemp. Math.},
VOLUME = {239},
PAGES = {115--174},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1999},
MRCLASS = {57T05 (16W30 55N20 55P35 55S05)},
MRNUMBER = {MR1718079 (2000j:57078)},
MRREVIEWER = {Neil P. Strickland},
}
Neil
__________________________________________________________________________
Subject: Response to inquiry from Kashiwabara
From: Michael Slack
Date: Mon, 20 Nov 2006 11:43:18 -0600
I don't have a reference for the explicit question you are asking about,
but I did once (some years ago) think about total Steenrod operations
(Milnor style) for BP cohomology. It's been a while since I thought about
this, but fortunately I wrote a paper which has the details. I never
really thought about the connection between the relations from my paper
and the Ravenel-Wilson relations, but perhaps it is not too hard to figure
out. Anyway, even if it isn't what you're looking for, you might find it
interesting to peruse:
Multiplicative operations in the Steenrod algebra for Brown-Peterson
cohomology. Fund. Math. 160, No. 1, pp 81-93.
Mike Slack
> Subject: Question on "adding formal indeterminate" (A message to be
posted to the Algebraic Topology Discussion List )
> From: Takuji Kashiwabara
> Date: Sun, 19 Nov 2006 23:37:01 +0100
>
> Hello,
> I wonder if there is any explicit written reference of the following
> (easy, and presumably well-known) fact :
>
> Consider the Ravenel-Wilson's main relations in
$R=E_*(\underline{F}_*)$.
> It takes the form of the equalities between the coefficient of some
formal
> power series in $R[[s,t]]$. The usual proof involves computing the
image of
> $\beta _i\otimes \beta _j$ 's in $E_*(CP{\infty } x CP{\infty })$ in two
ways
> and summing them up. However, if we consider
> $$R[[x_1^E,x_2^E]]\cong Hom ((E_*(CP{\infty } x CP{\infty }),R)$$
> the proof becomes almost trivial. (Then the relation is nothing but the
> equality between the two elements in R[[s,t]] corresponding to the two
> induced maps.)
>
> In other words, the "formal indeterminates" s, t are nothing but the
> orientation classes (or their induced maps).
>
> So, does anyone know of any written reference of this? Or something
similar?
> For example in the context of FGL's ( I am aware that this kind of idea
is used
> implicitly all over in FGL theory, but anything that looks like this
explicitly?
> ) or that of total Steenrod/Dyer-Lashof operations?
> Thank you in advance.
> Takuji Kashiwabara