Difference index of vectorfields and the enhanced Milnor number

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THE MILNOR number p( .X) of a simple fibered link X = (S*” - ‘, K) is now usually defined as the rank of the middle homology of a fiber F for the link, but it was in fact first introduced by Milnor for links of singularities as a certain mapping degree; its equality with a Betti number of F was a theorem (Milnor [S]; he called it “multiplicity”). In [lo] it was observed that an invariant introduced in [12] for fibered links in S3 could be defined in any dimensions and its definition could be seen as a natural extension of Milnor’s definition of p(X). For this reason it was named the enhanced Milnor number. Suitably normalized it takes the form (U(X), n(X)), lying in Z 8 Z or 2@2/2 according as the ambient dimension is 3 or > 3. The invariant E.(Y)EZ or Z/2 is called the enhancement to the Milnor number. The enhanced Milnor number is defined as follows. Let X = (S*” - r, K) be a fibered link. We define a (2n - 2)-plane field in the stabilized tangent bundle TS2” - 1 0 W as follows: outside a tubular neighborhood N of K we use the tangent field to the fibration for X, along K we use (tangent field to K) @ R, and over the rest of N we interpolate as directly as possible between the above fields on aN and K. TS*” - 1 @ R has a trivialization coming from the embedding of S*” - l in R*“, so the above field defines a mapping A:S*” - ’ + G(2n - 2,2n) to the Grassman manifold of 2n - 2 planes in R*“. The homotopy class of this mapping in 7r2” _ 1 G(2n - 2,2n) is the enhanced Milnor number. As described in [lo], this homotopy group is isomorphic to Z @ Z for n = 2 and to Z @ Z/2 for n > 2, and the enhanced Milnor number has the form (( - l)“p(X), A(X)), where p(X) is the usual Milnor number. In fact the first summand is the image of 7~~” _ l(GC(n - 1, n)) in 7~~” _ 1 G(2n - 2,2n), so I vanishes if and only if the above field of real (2n - 2)-planes can be homotoped to a field of complex (n - 1 )-planes in @” 2 R*“. For simple links j.(X) is determined modulo 2 by the Seifert form L: