# Realizing Stiefel-Whitney classes via vector bundles

Let $$X$$ be a CW complex. If $$E$$ is a vector bundle over $$X$$, then it's well-known that the Stiefel-Whitney classes $$w_j(E) \in H^j(X,\mathbb F_2)$$ of $$E$$ are determined from the classes $$w_{2^k}(E)$$ (for $$2^k \leq j$$) via the Wu formula, using the cup product and the action of the Steenrod algebra.

Question 1: Does the Wu formula imply any further relations? This is a purely algebraic question which I make more precise in (a) and (b) below.

That is, let $$H$$ be a nonnegatively-graded $$\mathbb F_2$$-algebra with an unstable action of the Steenrod algebra satisfying the Cartan formula and $$Sq^{|x|}(x) = x^2$$ for all homogenenous $$x \in H$$. Let $$W$$ be the set of sequences $$(w_j \in H^j)_{j \in \mathbb N}$$ with $$w_0 = 1$$ and satisfying the Wu formla.

(a) For any sequence $$(v_{2^k} \in H^{2^k})_{k \in \mathbb N}$$, does there exist $$w \in W$$ (necessarily unique) with $$w_{2^k} = v_{2^k}$$ for all $$k \in \mathbb N$$?

Presumably (i) the Whitney sum formula and (ii) the universal formula for the Stiefel-Whitney classes of a tensor product of vector bundles are compatible with the Wu formula, so that $$W$$ is a commutative ring using (i) for addition and (ii) for multiplication.

(b) Is $$W$$ a polynomial algebra on whichever generators from (a) do exist?

Question 2: What restrictions beyond the Wu formula are there restricting the Stiefel-Whitney classes of a vector bundle $$E$$ on a CW complex $$X$$? This is a genuinely topological question.

Over here Mark Grant describes one such restriction, but ideally I'd like a more systematic discussion.

If it simplifies matters to assume that $$X$$ is finite, or even a compact manifold, then that's fine.

• I once asked a related question: mathoverflow.net/questions/239482/… Jul 3, 2020 at 6:53
• Isn't Question 2 answered by the integral cohomology of $BO(n)$? Jul 3, 2020 at 15:55
• @JohnGreenwood I'm not sure what you mean? I agree that the integral cohomology of $BO(n)$ will pull back to "integral characteristic classes" which probably contain some information that the Stiefel-Whitney classes don't. Do you have something more specific in in mind? Jul 3, 2020 at 18:34
• Answers to mathoverflow.net/q/257617/41291 might (or might not) contain relevant information Jan 15, 2022 at 18:02

Here is a start at answering Question 1: there are indeed further relations between the $$w_{2^k}$$, or at least conditions on the $$w_{2^k}$$.

For example, consider the case where $$H$$ has multiplication which is null except for what is implied by the multiplication being unital. (For example, $$H$$ may be the cohomology of a suspension space.)

In this case, the Wu formula reduces to

$$Sq^i(w_j) = \binom{j-1}{i} w_{i+j}$$

So if the $$w_{2^k}$$'s are given, we are forced to define $$w_{2^k + j'} = Sq^{j'} w_{2^k}$$ for $$0 \leq j' \leq 2^k-1$$, which gives us the definition of each $$w_j$$. So now in the case where $$j = 2^k+1$$ and $$1 \leq i \leq 2^k - 1$$, the Wu formula stipulates that $$Sq^i Sq^1 w_{2^k} = 0$$. This is always the case for $$i = 1$$, but for all other $$i$$, the relation $$Sq^i Sq^1 = 0$$ does not hold in the Steenrod algebra, so I believe there are examples of $$H$$'s and $$w_{2^k} \in H^{2^k}$$ where this equation does not hold. So this is an example of some kind of further condition which $$w_{2^k}$$ may be required by the Wu formula to satisfy.

• This answer reflects a fundamental confusion on my part: The Wu formula is a formula satisfied by the Stiefel-Whitney classes of the tangent bundle of a smooth manifold. Presumably it is not satisifed by an arbitrary vector bundle. So the case considered here, where the multiplication is trivial, is not terribly relevant. Jan 15, 2022 at 1:00
• The Wu formula does hold for arbitrary vector bundles. I am having trouble locating Wu's original paper at the moment, but see for example May's Concise p.197. The Wu formula describes the action of the Steenrod algebra on the mod 2 cohomology of BO (generated as a Z/2-algebra by the universal Stiefel-Whitney classes) Jan 15, 2022 at 9:26
• What does hold only for closed manifolds, and also goes under the name "Wu's formula", at least in Milnor-Stasheff, is the formula Sq(total Wu class) = total Stiefel-Whitney class. This restriction to closed manifolds is to be expected, since the Wu classes use the (mod 2) Poincare duality structure in their definition. However, the "Wu's formula" linked to in the question is the one concerning the Steenrod algebra action on H*(BO;Z/2). Jan 15, 2022 at 9:57
• @AleksandarMilivojevic Ah, thanks, that's helpful! Jan 15, 2022 at 17:20