# Who discovered this definition of Stiefel-Whitney classes?

I would define Stiefel-Whitney classes as the pullbacks of generators of $H^*(BO, \mathbb{Z}/2)$ under a classifying map, and I gather this is a pretty common definition.

However, the book "Characteristic classes" by Milnor-Stasheff contains a different definition, as eigenvalues'' for the Steenrod squares acting on the fundamental class of a manifold in its normal bundle. I want to attribute this construction to somebody for a paper, but I'm not sure what the correct attribution is. Who is the discoverer of this definition?

• That is not a definition at all, as the generators" does not define elements of a ring. – Oscar Randal-Williams Dec 19 '16 at 21:52
• Sorry, I purposefully did not say "the generators" but maybe should have said "certain generators", which I realize is not actually a definition either, but is probably enough indication of what I'm referring to. – user84144 Dec 19 '16 at 22:13
• @OscarRandal-Williams I think he meant the homogeneous polynomial generators, which I believe in this case is in fact a characterization of the elements of $H^*(BO;\mathbb{Z}/2)$ we are talking about – Denis Nardin Dec 19 '16 at 22:32
• Well, you give me too much credit. What I actually have in my head is a specific choice of generator labelled $w_n$, corresponding to a specific symmetric polynomial under the embedding $H^*(BO(n)) \hookrightarrow H^*(BO(1)^n)$. I lazily decided not to be more precise, thinking that nobody would care, but that seems to have been a wrong presumption. – user84144 Dec 19 '16 at 22:47
• @user84144: Is this really a definition? The only way I can think of showing that map i) is injective, and ii) hits the symmetric polynomials, needs me to already have a definition of Stiefel-Whitney classes and have proved the sum formula for them. – Oscar Randal-Williams Dec 19 '16 at 23:42

## 3 Answers

For the relation $Sq(U) = \Phi(w)$, where $Sq$ is the total Steenrod squaring operation, $U$ is the Thom class, $\Phi$ is the Thom isomorphism and $w$ is the total Stiefel-Whitney class, I would cite Rene Thom's 1951 thesis, published as "Espaces fibres en spheres et carres de Steenrod" in Ann. Sci. Ecole Norm. Sup. (3) 69 (1952). In the introduction he explains that he found the results for manifolds embedded in Euclidean space, and that Henri Cartan then showed him how to obtain the general case. These results were first announced by Thom in two Comptes Rendus Acad. Sc. notes (t. 230, 1950, p. 427 and p. 507). Wen-Tsun Wu's formula $Sq(v)=w$ in the case of the tangent bundle of a closed manifold is not the same result. It was announced in the same volume of Comptes Rendus (p. 508 and p. 918). Its proof depends on the results of Thom.

The earliest reference for the connection between Stiefel-Whitney classes and Steenrod squares seems to be Wenjun Wu, Classes caractéristiques et i-carrés d'une variété, C.R. Acad. Sci. Paris 230, 508–511 (1950), followed a few years later by René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, 17–86 (1954).

Both works are discussed in this article:

• It seems to me that this article does not present the definition I referred to, but instead proves the "Wu formula", which can be used to give yet another definition (also related to Steenrod squares as you say). – user84144 Dec 19 '16 at 22:25

The AMS has spent a lot of money and effort over many years to develop a fine tool that easily allows for the investigation of such questions: it is called Math Reviews (MathSciNet). The oldest mention of SW classes in the reviews seems to be a review by Whitney of a 1947 paper by Wu. There is a fantastic review (in German) by Hirzebruch of Milnor and Stasheff's book, giving some history - e.g. the SW classes go back to the mid 1930's. I believe Milnor and Stasheff themselves discuss how to think about SW classes as obstructions, and I am pretty sure that this finding this interpretation was quite close to the discovery of the formula for the SW classes via Steenrod squares: this type of question is how the `reduced squares' were discovered in the first place.

Added later: ... see Thom, René Espaces fibrés en sphères et carrés de Steenrod. (French) Ann. Sci. Ecole Norm. Sup. (3) 69, (1952). 109–182.

In particular, section 3 of chapter 2 carefully develops the formula for the SW classes via Steenrod squares, and compares it to earlier constructions. Chapter 1 discusses what folks might now call the Thom isomorphism (though the Gysin sequence is already known).

Remark: the basic theory of fiber bundles and classifying spaces is just being figured out at this same time.