Tameness conditions and the Milnor fibrations for composite singularities

In this paper, we introduce a new regularity condition that characterizes the tameness of a composite singularity $H=G\circ F$ in a sharp way. Our approach provides a natural tool that links the topology of the Milnor tube fibrations through the Milnor fibers of the respective components of the map germs $F$, $G$ and $H = G\circ F$. We also study the invariance of tameness by $\mathcal{L}$-equivalence, $\mathcal{R}$-equivalence, and hence by $\mathcal{A}$-equivalence, and we give conditions for when two component map germs of the composite singularity $H=G\circ F$ being tame implies the third one is tame. As an application, we show how to relate the Euler characteristics of the Milnor fibers of $F,G$ and $H$ to each other.