Subject: another reply for the list
From: "J. Michael Boardman"
Date: Tue, 21 Nov 2006 10:13:16 -0500
Another reply:
This is all made quite explicit in Boardman-Johnson-Wilson,
Chapter 15 in "Handbook of Algebraic Topology" (North-Holland,
1995). (True, for the sake of simplicity (!) we treated only the
case E=F, but this condition can readily be removed.)
This paper contains no formal indeterminates; what were
formerly known as formal indeterminates are recognized as E-Chern
classes, as T.K. suggests. The relations indeed simply express
the naturality of cohomology operations for the multiplication
map on CP^\infty. There are three levels. The full strength
version, for all unstable operations, is (15.8). The medium
strength, for additive operations, is (14.5) and is far easier to
handle. The stable analogue is (13.6) in the previous paper,
Chapter 14, in the same book.
And finally, no, we did not cover Dyer-Lashof operations.
J. Michael Boardman, Department of Mathematics, JHU
>>
>> 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