Topological invariants of real Milnor fibres
Let \(F\) be an analytic mapping defined on an open neighbourhood \(D\) of \(0 \in {\mathbb R}^n\), \[ F:(D,0) \rightarrow ({\mathbb R}^m,0),\quad n>m, \] such that the origin is an isolated singular point of \(F^{-1}(0)\). For any regular value \(z \in {\mathbb R}^m\) which is close enough to the origin and a sufficiently small ball \(B_r=B(0,r)\), the fibre \(W_z=F^{-1}(z) \cap B_r\) is called the Milnor fibre. In complex case the topological type of the Milnor fiber does not depend on \(z\). In the real case this is no longer true. The author considers an analytic function \(g\) defined on \(B_r\) such that its restriction to the generic Milnor fibre is Morse. He studies the properties of the global index \[ \lambda (g, W_z) = \sum (-1)^{\lambda(p)}, \] where \(\lambda(p)\) is the Morse index of the critical point \(p\) of \(g_ {| W_z}\) and the sum extends over all such critical points. The main result of the paper is the proof that under some general assumptions there exist a matrix \(\Theta\) with analytic entries and analytic functions \(\theta_i\) such that \[ \lambda (g, W_z)= \text{signature}(\Theta(z))= \sum \text{sgn} : \theta_i (z). \]
scientific article; zbMATH DE number 1922437