The integral monodromy of the cycle type singularities

TL;DR

This paper proves Orlik's conjecture for the integral monodromy of cycle type singularities, providing a detailed algebraic and combinatorial analysis that corrects previous errors and advances understanding of Milnor fiber homology.

Contribution

It confirms Orlik's conjecture for cycle type singularities and corrects earlier mistakes, offering new algebraic and combinatorial insights.

Findings

Proves Orlik's conjecture for cycle type singularities

Provides corrected and extended algebraic and combinatorial results

Enhances understanding of Milnor fiber homology and monodromy

Abstract

The middle homology of the Milnor fiber of a quasihomogeneous polynomial with an isolated singularity is a ${\mathbb Z}$-lattice and comes equipped with an automorphism of finite order, the integral monodromy. Orlik (1972) made a precise conjecture, which would determine this monodromy in terms of the weights of the polynomial. Here we prove this conjecture for the cycle type singularities. A paper of Cooper (1982) with the same aim contained two mistakes. Still it is very useful. We build on it and correct the mistakes. We give additional algebraic and combinatorial results.