On the Milnor fiber of a real map-germ. (Q700240)

The Milnor fibre is a tool for the analysis of singularities of (real or complex) analytic map germs \(f:(\mathbb{K}^n,0)\to (\mathbb{K}^k,0)\). It is defined as \(\overline{B(0,\varepsilon)}\cap f^{-1}(y)\), where \(\overline{B(0,\varepsilon)}\) is a small closed ball and \(y\) is a small regular value of \(f\). The Milnor number is a numerical invariant associated with complex Milnor fibres. The author proves a real analogue of a formula about the Milnor number. It provides a connection between the Euler characteristics of two sets associated with the real Milnor fibre and a factor ring of the ring of germs of analytic functions. The Milnor number mod 2 of the complexification of the real map germ is known to be a topological invariant. A new and more elementary proof of this fact is presented.