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

Lee Rudolph

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arXiv · 0912.4974

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

Lee Rudolph

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Abstract

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.

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Lee Rudolph. 2009-12-25. Whitehead's Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number. https://arxiv.org/abs/0912.4974

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