Subject: A question about Wu classes
Date: Mon, 2 Dec 2002 15:22:04 +0100
From: "Boccellari" <boccellari@inwind.it>
To: "Don Davis" <dmd1@lehigh.edu>

I am a PhD atudent in Mathematics at the Università degli Studi di Milano.
I have a simple question about Wu classes.

Following Browder  fix n >= 0 and define

V^{(n)} = Sq^{-1}(W^{-1})

where:

1) Sq^{-1} = 1 + c(Sq^1) + c(Sq^2) +...
and "c" is the  anti-automorphism of the mod 2 Steenrod algebra

2) W = 1 + w_1 + w_2 + ... +w_n the total
Stiefel-Whitney class in H^*(BO(n);Z_2) and
W^{-1}*W =  1 = W*W^{-1}.

Let v^{(n)} _0,...,v^{(n)} _i,...,v^{(n)} _n the omogeneous components of
V^{(n)} .

My question is the following:

a) is it known the expression of v^{(n)} _i (at least inductive on i or on
n) as a
polynomial  in  w_1,...,w_n ?

b) as each w_i can be considered as the i-th symmetric polynomial in
x_1,...,x_n is it known the expression of v^{(n)} _i (at least inductive on
i or on
n) as a polynomial  in  x_1,...,x_n ?

If something is known can you send me references?

I  found the following expression:

v^{(n)} _i =
\sum_{j = 0}^{\infty}( (x^{2^j-1}_n)*(v^{(n-1)} _{i-(2^j-1)} )

where v^{(m)} _k = 0 if k < 0 or k > m.

This expression leads me to some relations in the mod 2 Steenrod algebra and
to a property of number theory.

Thank you very much.

With best regards

Tommaso Boccellari