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

TL;DR

This paper explores Whitehead's integral formula for the Hopf invariant in relation to isolated critical points and the enhancement of the Milnor number, proposing geometric interpretations and raising open questions.

Contribution

It connects Whitehead's integral formula with the enhancement of the Milnor number, suggesting new geometric insights and raising questions about their relationship.

Findings

Whitehead's formula can be integrated along fibers to relate to the Hopf invariant.

The Hopf invariant lambda(K) is linked to the enhancement of the Milnor number.

Several conjectures and questions are proposed about the geometric significance of these invariants.

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.