Vanishing polyhedron and collapsing map

Abstract

In this paper we give a detailed proof that the Milnor fiber $X_t$ of an analytic complex isolated singularity function defined on a reduced $n$-equidimensional analytic complex space $X$ is a regular neighborhood of a polyhedron $P_t\subset X_t$ of real dimension $n-1$. Moreover, we describe the degeneration of $X_t$ onto the special fiber $X_0$, by giving a continuous collapsing map ${\Psi }_t:X_t\to X_0$ which sends $P_t$ to $\{0\}$ and which restricts to a homeomorphism $X_t\backslash P_t\to X_0\backslash \{0\}$.