On the topology of the Milnor Boundary for real analytic singularities

Abstract

We study the topology of the boundaries of the Milnor fibers of real analytics map-germs $f:({\mathbb{R}}^M,0)\to ({\mathbb{R}}^K,0)$ and $f_I:={\Pi }_I\circ f:({\mathbb{R}}^M,0)\to ({\mathbb{R}}^I,0)$ that admit Milnor's tube fibrations, where ${\Pi }_I:({\mathbb{R}}^K,0)\to ({\mathbb{R}}^I,0)$ is the canonical projection for $1\le I<K.$ For each $I$ we prove that the Milnor boundary $\partial F_I$ is given by the double of the Milnor tube fiber $F_{I+1}.$ We prove that if $K-I\ge 2$, then the pair $(\partial F_I,\partial F_f)$ is a generalized $(K-I-1)$-open-book decomposition with binding $\partial F_f$ and page $F_f\setminus \partial F_f$ - the interior of the Milnor fibre $F_f$ (see the definition below). This allows us to prove several new Euler characteristic formulae connecting the Milnor boundaries $\partial F_f,$ $\partial F_I,$ with the respectives links ${\mathcal{L}}_f,{\mathcal{L}}_I,$ for each $1\le I<K,$ and a Lê-Greuel type formula for the Milnor boundary.