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J. H. C. Whitehead gave an elegant integral formula for the Hopf invariant H(p) of a smooth map p from the 3-sphere to the 2-sphere. Given an open book structure b on the 3-sphere (or, essentially equivalently, an isolated critical point of a map F from 4-space to the plane), Whitehead's formula can be "integrated along the fibers" to express H(p) as the integral of a certain 1-form over the circle. In case p is geometrically related to b (or F) -- for instance, if p is the map (one component of the fiberwise generalized Gauss map of F) whose Hopf invariant lambda(K) is the "enhancement of the Milnor number" of the fibered link K in the 3-sphere associated to F (or b), previously studied by the author and others -- it might be hoped that this 1-form has geometric significance. This note makes that hope somewhat more concrete, in the form of several speculations and questions.

Whitehead’s Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number