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 $1\leq I<K.$ 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 $K-I\geq 2$, 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 $1\leq I<K,$ and a L\^e-Greuel type formula for the Milnor boundary.