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Does anyone know of a good reference describing the action of the Steenrod algebra $\mathcal{A}_2$ on the cohomology algebra $$H^\ast(BO(k);\mathbb{F}_2)\cong\mathbb{F}_2[w_1,w_2,\ldots ,w_k]$$ of the classifiying space for $k$-dimensional vector bundles? This is a polynomial algebra on the universal Stiefel-Whitney classes. The action of the squares is determined by Wu's formula $$Sq^i(w_k) = \sum_r \binom{k+r-i-1}{r} w_{i-r}w_{k+r},$$ where $w_m=0$ for $m>k$.

Most of the references I've found seem to focus on the stable classifying space $BO$. I would like to see a detailed exposition for fixed $k$, in particular, relations on the $Sq^I(w_k)$ following from Wu's formula.

(This paper of Pengelley and Williams seems to contain useful information, but I have a feeling this is something more classical.)

Update: Remark 2.5 in the linked paper seems to be saying that the free unstable $\mathcal{A}_2$-module on $w_k$ injects into $H^*(BO(k);\mathbb{F}_2)$, or in other words that the $Sq^I(w_k)$ are linearly independent for $I$ admissible (the multi-index $I=(i_1,\ldots ,i_p)$ is admissible if $i_\ell\geq 2 i_{\ell+1}$ for $1\leq \ell \leq p-1$).

They refer to a paper of Lannes and Zarati, "Foncteurs dérivés de la déstabilisation", which seems to me to be overkill. I tried to give an elementary proof along the lines of the proof in Thom's paper that the $Sq^I(w_k)$ are linearly independent in $H^*(BO;\mathbb{F}_2)$ for $|I|\leq k$, but so far to no avail. Thom orders monomials in the $w_i$ lexicographically, then shows that the leading monomial in the expansion of $Sq^I(w_k)$ is $w_k\cdot w_I$. Does anyone know a slick proof of this claim (that the $Sq^I(w_k)$ are linearly independent in $H^*(BO(k);\mathbb{F}_2)$ for $I$ admissible)?

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Sauf erreur the inclusion $BO(k) \to BO$ induces on cohomology the map that sets $w_i$ to $0$ for $i>k$. This map commutes with the action of the Steenrod algebra so you already have a formula. –  Torsten Ekedahl Jul 31 '11 at 18:21
@Torsten: You're right. I was wondering if anyone has explored the consequences of this formula, for example found relations among the $Sq^I(w_k)$ for $I$ admissible. –  Mark Grant Jul 31 '11 at 19:35
You might find the action of Milnor basis elements more 'regular', as in the case of the action on the classifying space of the maximal 2-torus $BO(1)^k \longrightarrow BO(k)$. By 'relations among ...', might you be asking for a presentation of $H^∗BO(k)$ as a module over the Steenrod algebra? –  Robert Bruner Aug 3 '11 at 2:00
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2 Answers

up vote 2 down vote accepted

In the linked paper, David and I have a self-contained, elementary proof of Theorem 2.3, i.e. the freeness of the Steenrod action, from which follow Remarks 2.4 and 2.5. (It does not use Lannes-Zarati, we mentioned their result only for completeness.)

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Thank you sir. I will look more closely at your paper. –  Mark Grant Aug 25 '11 at 14:58
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This "Wu formula" is a direct consequence of the Cartan formula and the splitting principle. The actual calculation is usually left as an (implicit or explicit) exercise in books (e.g. page 197 in May's "A Concise Course in Algebraic Topology").

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this doesn't seem to answer the question. –  Dylan Wilson Aug 3 '11 at 5:43
Welcome to MO, Jesus! Dylan's right, I was accepting the Wu formula as given and asking for relations between the $Sq^I(w_k)$ (which I no longer think exist - see my updated question). –  Mark Grant Aug 3 '11 at 6:34
oops... as Mark noticed (and I am sure everyone else), I am new to MO. I did not note that there are spots for comments, and then spots for actual answers... and then I should have read carefully to realize what the actual question was 8) –  Jesus Gonzalez Aug 5 '11 at 23:31
@Jesus: Don't be discouraged - we're just glad to have you here :) –  Mark Grant Aug 8 '11 at 11:12
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