分野別セミナー

【講演要旨】Consider a real analytic map-germ $f: (\mathbb{R}^M,0) \to (\mathbb{R}^K,0),$ the canonical projection
$\Pi_{I}:(\mathbb{R}^K,0)\to (\mathbb{R}^{I},0)$ for $1\leq I $<$K,$ and the composition map germ
$f_{I}:=\Pi_{I}\circ f : \mathbb{R}^M,0) \to (\mathbb{R}^I,0),$ both $f$ and $f_{I}$ under conditions that admits the Milnor tube fibrations. In this talk we will show some recents results connecting the topology of the boundaries of the Milnor fibers $\partial F_{f}$ and $\partial F_{I},$ of $f$ and $f_{I},$ respectively. We will prove that for each $1\leq I\leq K-1$ the Milnor boundary $\partial F_{I}$ is given by the double of the Milnor tube fiber $F_{I+1},$ and that if $K-I\geq 2$, then the pair $(\partial F_{I},\partial F_{f})$ is (according to our definition) a ``generalized $(K-I-1)$-open-book decomposition'' with binding $\partial F_{f}$ and page $F_{f} \backslash \partial F_{f}$ - the interior of the Milnor fibre $F_{f}.$
As applications, we will prove several (new) Euler characteristic formulae connecting the Milnor boundaries $\partial F_{f},$ $\partial F_{I},$ with the respective links $\mathcal{L}_{f}, \mathcal{L}_{I},$ for each $1\leq I $<$K,$ and a ``Lê-Greuel type formula" for the Milnor boundaries. If permits we will introduce the problem in the general case of compositions of $H=G\circ F.$
This is a joint work with A. Menegon (MIUM/Sweden), M. Ribeiro (UFES/Brazil), J. Seade (UNAM/Mexico) and
I. Santamaria (ICMC-USP/Brazil).

PDF: https://drive.google.com/file/d/1UyReB3t76dRGfoj36jok3idXVMA3P8VM/view?usp=sharing