nLab Whitehead integral formula

Idea

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 (in which case the 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 Griffiths & 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.

Statement

The base case

Consider a smooth function $f \,\colon\, S^3 \xrightarrow{\;} S^2$. Write $n \in \mathbb{N}$ for its Hopf invariant, hence for the element in the homotopy group of spheres $\pi_3(S^2) \,\simeq\, \mathbb{Z}$ that is represented by (the underlying continuous function of) $f$.

Write $F_2 \coloneqq dvol_{S^2} \in \Omega^2_{dR}(S^2)$ for the unit volume form of the 2-sphere, or any other 2-form with unit integral:

$$\textstyle{\int_{S^2}} \, F_2 \;=\; 1 \,.$$

Observe that the pullback of $F_2$ along $f$ admits a de Rham coboundary $A_1 \in \Omega^1_{\mathrm{dR}}(S^3)$, since the de Rham cohomology of the 3-sphere vanishes in degree 2, $H^2_{dR}(S^3) \,\simeq\, 0$:

$$\mathrm{d}\, A_1 \;=\; f^\ast F_2 \,.$$

Proposition

For any such choice of $A_1$, the integral

$$\textstyle{\int_{S^3}} \, A_1 \wedge f^\ast F_2 \;=\; n$$

computes the Hopf invariant.

This is the original statement of Whitehead 1947. In the above modernized form it is stated, e.g., in Bott & Tu 1982, p 228 and in Griffiths & Morgan 2013 p 134.

Related concepts

• functional cup product

References

• 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, 50 4 (1949) 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 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:10.1007/978-1-4757-3951-0]

• Lee Rudolph: Whitehead’s Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number, Pure and Applied Mathematics Quarterly 6 2 (2010) [arXiv:0912.4974]

• Phillip Griffiths, John Morgan, Section 14.5 of: Rational Homotopy Theory and Differential Forms, Progress in Mathematics 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]

• Felix Wierstra, Hopf Invariants in Real and Rational Homotopy Theory (2017) [diva:146246, pdf]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, pp. 18 of: Twisted Cohomotopy implies M5 WZ term level quantization, Comm. Math. Phys. 384 (2021) 403-432 [arXiv:1906.07417, doi:10.1007/s00220-021-03951-0]