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Nicholas Kuhn
Nicholas Kuhn
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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`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.

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.

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.

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Nicholas Kuhn
Nicholas Kuhn
  • 11.1k
  • 31
  • 59

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). Searching for `Stiefel-Whitney classes' brings up a 1947 review The oldest mention of an article by Chern, in which SW classes are mentioned as if everyone knows what they are already: see below. The notion of a classifying space is also just being figured out at this time, e.g.in the first books about fiber bundles appear around 1950 with proofsreviews seems to be a review by Whitney of basic theorems like pullbacks are homotopy invariant. The folks figuring out this stuff are the generation who taught Milnor, Serre, and later Dold, Adams, ..a 1947 paper by Wu.

MR0023053 There is a fantastic review (9,297iin German) Reviewed Chern, Shiing-shen On the characteristic ringby Hirzebruch of a differentiable manifold. Acad. Sinica Science Record 2Milnor and Stasheff's book, giving some history (1947). 1–5. 56.0X

Let H=H(n,N) be the Grassmann manifold of n-planes through O in Euclidean space En+N. An imbedding of a differentiable manifold M in En+N defines a mapping f of M into H e. This induces a homomorphism f∗ of the cohomology ring R(H) of H into that of Mg. The ring f∗(R(H)) is called the characteristic ring of M; its elements, characteristicSW classes. The latter include go back to the Stiefel-Whitney classes Wkmid 1930's. The ring R(H) is studied with 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 helpdiscovery of Schubert chains. Theorem: the classes Wk generate f∗(R(H)). The formula for ring products leads to nonimbedding theorems. For instance, if n is even and is notthe SW classes via Steenrod squares: this type of question is how the form 2(2k−1), k≥1, then projective n-space cannot be imbedded`reduced squares' were discovered in En+2. Proofs are only sketched. Reviewed by Hthe first place. Whitney

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). Searching for `Stiefel-Whitney classes' brings up a 1947 review of an article by Chern, in which SW classes are mentioned as if everyone knows what they are already: see below. The notion of a classifying space is also just being figured out at this time, e.g. the first books about fiber bundles appear around 1950 with proofs of basic theorems like pullbacks are homotopy invariant. The folks figuring out this stuff are the generation who taught Milnor, Serre, and later Dold, Adams, ...

MR0023053 (9,297i) Reviewed Chern, Shiing-shen On the characteristic ring of a differentiable manifold. Acad. Sinica Science Record 2, (1947). 1–5. 56.0X

Let H=H(n,N) be the Grassmann manifold of n-planes through O in Euclidean space En+N. An imbedding of a differentiable manifold M in En+N defines a mapping f of M into H. This induces a homomorphism f∗ of the cohomology ring R(H) of H into that of M. The ring f∗(R(H)) is called the characteristic ring of M; its elements, characteristic classes. The latter include the Stiefel-Whitney classes Wk. The ring R(H) is studied with the help of Schubert chains. Theorem: the classes Wk generate f∗(R(H)). The formula for ring products leads to nonimbedding theorems. For instance, if n is even and is not of the form 2(2k−1), k≥1, then projective n-space cannot be imbedded in En+2. Proofs are only sketched. Reviewed by H. Whitney

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.

Source Link
Nicholas Kuhn
Nicholas Kuhn
  • 11.1k
  • 31
  • 59

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). Searching for `Stiefel-Whitney classes' brings up a 1947 review of an article by Chern, in which SW classes are mentioned as if everyone knows what they are already: see below. The notion of a classifying space is also just being figured out at this time, e.g. the first books about fiber bundles appear around 1950 with proofs of basic theorems like pullbacks are homotopy invariant. The folks figuring out this stuff are the generation who taught Milnor, Serre, and later Dold, Adams, ...

MR0023053 (9,297i) Reviewed Chern, Shiing-shen On the characteristic ring of a differentiable manifold. Acad. Sinica Science Record 2, (1947). 1–5. 56.0X

Let H=H(n,N) be the Grassmann manifold of n-planes through O in Euclidean space En+N. An imbedding of a differentiable manifold M in En+N defines a mapping f of M into H. This induces a homomorphism f∗ of the cohomology ring R(H) of H into that of M. The ring f∗(R(H)) is called the characteristic ring of M; its elements, characteristic classes. The latter include the Stiefel-Whitney classes Wk. The ring R(H) is studied with the help of Schubert chains. Theorem: the classes Wk generate f∗(R(H)). The formula for ring products leads to nonimbedding theorems. For instance, if n is even and is not of the form 2(2k−1), k≥1, then projective n-space cannot be imbedded in En+2. Proofs are only sketched. Reviewed by H. Whitney