Tameness conditions and the Milnor fibrations for composite singularities

TL;DR

This paper introduces a new regularity condition to characterize the tameness of composite singularities, linking the topology of Milnor fibrations of component maps and their compositions.

Contribution

It provides a natural tool to analyze the tameness of composite singularities and explores invariance under various equivalences, relating Milnor fiber Euler characteristics.

Findings

New regularity condition characterizes tameness of composite singularities.

Invariance of tameness under $\

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Abstract

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.