Title: Open book on the boundaries of real singularities

Speaker: Raimundo Nonato Araújo dos Santos (ICMC/USP-Brazil)

Abstract: Given a tame real analytic map germ $f:(\mathbb{R}^{N},0)\to (\mathbb{R}^{M},0), N>M\geq 2,$ with isolated critical value it is known that it admits a Milnor's tube fibration with projection $f_{|}:B^{N}_{\epsilon}(0)\cap f^{-1}(B_{\eta}^{K}(0)\setminus \{0\})\to B_{\eta} ^{K}(0)\setminus \{0\},$ for all small enough $0<\eta \ll \epsilon.$ In general, not many information is known about the topology of the Milnor fiber $F_{f},$ nor the boundary $\partial F_{f}=F_{f}\cap S^{N-1}_{\epsilon}.$ 

One may consider, however, the canonical projection $\Pi:  (\mathbb{R}^{M},0)\to  (\mathbb{R}^{K},0), M>K\geq 1,$ and the germ of composition $g=\Pi \circ f: (\mathbb{R}^{N},0)\to (\mathbb{R}^{K},0).$ In such a case, it is not hard to show that $g$ also admits a tube fibration with fiber $F_{g}$ and boundary $\partial F_{g}=F_{g}\cap S^{N-1}_{\epsilon}$ such that $\partial F_{f} \hookrightarrow \partial F_{g}.$ 

Then, one may ask: Is there any interesting (geometrical/topological) property on might expect from such embedding $\partial F_{f} \hookrightarrow \partial F_{g}?$

In this talk, we intend to show that under subtle condition such embedding turns the pair $(\partial F_{g}, \partial F_{f})$ a {\it generalized open-book structure}, since $M-K\geq 2.$ 

If time permits, further information will be provided connecting the topologies of the Milnor fibers, the Milnor boundaries and the links.

This is  part of an ongoing project with the collaborators: Aurelio Menegon Neto (Mid Sweden University/Sweden), Maico F. Ribeiro (Federal University of Espirito Santo-Brazil), Ivan Santamaria (Colombia), J. Seade (UNAM/Mexico), O. Saeki (Kyushu University/Japan).