Vanishing polyhedron and collapsing map

TL;DR

This paper proves that the Milnor fiber of an isolated complex singularity can be viewed as a regular neighborhood of a polyhedron and describes its collapse onto the special fiber.

Contribution

It provides a detailed proof that Milnor fibers are neighborhoods of polyhedra and introduces a collapsing map onto the special fiber.

Findings

Milnor fiber is a regular neighborhood of a polyhedron

Existence of a collapsing map from the Milnor fiber to the special fiber

The collapsing map is a homeomorphism outside the polyhedron

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 $X_t \backslash P_t \to X_0 \backslash \{0\}$.