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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 20th 2016
• (edited Jan 20th 2016)

I have been making trivial edits (adding references, basic statements, cross-links ) to Hopf invariant and a bunch of related entries, such as Kervaire invariant, Hopf invariant one problem, Arf-Kervaire invariant problem, normed division algebra.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 26th 2019

Discussion via differential forms/rational homotopy theory

• J. H. C. Whitehead, An expression of Hopf ’s invariant as an integral, Proc. Nat. Acad. Sci. U. S. A.33 (1947), 117–123 (jstor:87688)

• Dev Sinha, Ben Walter, Lie coalgebras and rational homotopy theory II: Hopf invariants (arXiv:0809.5084)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 26th 2019

and this one:

• André Haefliger, p. 3 of Whitehead products and differential forms, In: Schweitzer P.A. (eds) Differential Topology, Foliations and Gelfand-Fuks Cohomology. Lecture Notes in Mathematics, vol 652. Springer, Berlin, Heidelberg (doi:10.1007/BFb0063500)
• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeApr 26th 2019

If at the beginning the requirement is $n \gt 1$ in $\phi \;\colon\; S^{2n-1} \longrightarrow S^n$, then I guess the real Hopf fibration shouldn’t be included.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeApr 26th 2019

Mind you, Adams and Atiyah allow that case, so presumably change to $n \gt 0$.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeApr 29th 2019

• Dale Husemöller, chapter 15 of Fibre Bundles, Graduate Texts in Mathematics 20, Springer New York (1966)
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 21st 2019

added publication data for this here:

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 30th 2019

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJun 3rd 2019
• (edited Jun 3rd 2019)

added the claim (here) that the Hopf invariant of a map $\phi$ may be read off as the unique free coefficient of the Sullivan model of $\phi$.

This follows straightforwardly, and I’d like to cite this from a canonical RHT source, if possible. But I don’t see it in the textbooks (FHT, …). If anyone knows opus, page and verse for a canonical citation of this fact, please let me know.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJun 4th 2019

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 5th 2019

and

• Lee Rudolph, Whitehead’s Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number, Pure and Applied Mathematics Quarterly Volume 6, Number 2, 2010 (arXiv:0912.4974)
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 12th 2019