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An expression for Hopf invariants in terms of secondary characteristic classes induced from the intersection pairing, also known as functional cup products (Steenrod 49) or homotopy periods (Sinha-Walter 13).
The original expression due to Whitehead 47 (see Bott-Tu 82, Prop. 17.22) is in terms of smooth functions to an n-sphere, which implies that wedge product/cup product of the pullback of the volume form with itself vanishes identically.
More generally the homotopy Whitehead formula applies to general cocycles in cohomotopy. Its existence was suggested in Haefliger 78, p. 17, worked out for the case of maps from the 3-sphere to the 2-sphere in Griffith-Morgan 81, Section 14.5 and stated generally but without proof in Sinha-Walter 13, Example 1.9. A transparent proof of the general expression via lifts in cohomotopy through Hopf fibrations is in FSS 19, relating the expression to the Hopf-Wess-Zumino term of the M5-brane.
J. H. C. Whitehead, An expression of Hopf’s invariant as an integral, Proc. Nat. Acad. Sci. USA 33 (1947), 117–123 (jstor:87688)
Norman Steenrod, Cohomology Invariants of Mappings, Annals of Mathematics Second Series, Vol. 50, No. 4 (Oct., 1949), pp. 954-988 (jstor:1969589)
Hassler Whitney, Section 31 in Geometric Integration Theory, 1957 (pup:3151)
André Haefliger, p. 3 of Whitehead products and differential forms, In: P.A. Schweitzer (ed.), Differential Topology, Foliations and Gelfand-Fuks Cohomology, Lecture Notes in Mathematics, vol 652. Springer 1978 (doi:10.1007/BFb0063500)
Raoul Bott, Loring Tu, Prop. 17.22 in Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/BFb0063500)
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)
Phillip Griffiths, John Morgan, Section 14.5 of Rational Homotopy Theory and Differential Forms, Progress in Mathematics Volume 16, Birkhauser (1981, 2013) (doi:10.1007/978-1-4614-8468-4)
Dev Sinha, Ben Walter, Lie coalgebras and rational homotopy theory II: Hopf invariants, Trans. Amer. Math. Soc. 365 (2013), 861-883 (arXiv:0809.5084, doi:10.1090/S0002-9947-2012-05654-6)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M5 WZ term level quantization, 2019 (arXiv:1906.07417)
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