nLab
Whitehead integral formula

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Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

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see also algebraic topology

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Cohomology

cohomology

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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, 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.

References

Last revised on June 19, 2019 at 03:41:43. See the history of this page for a list of all contributions to it.