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.
Categories: math.DG