On the Milnor fibers of sandwiched singularities

TL;DR

This paper investigates the topology of Milnor fibers of sandwiched surface singularities, showing that different incidence matrices lead to non-diffeomorphic fibers, thus providing insights into the classification of Stein fillings.

Contribution

It extends previous results to show that Milnor fibers with different incidence matrices are not diffeomorphic, offering a lower bound on Stein fillings for sandwiched singularities.

Findings

Milnor fibers are distinguished by incidence matrices.

Different incidence matrices imply non-diffeomorphic Milnor fibers.

Provides a lower bound on the number of Stein fillings.

Abstract

The sandwiched surface singularities are those rational surface singularities which dominate birationally smooth surface singularities. de Jong and van Straten showed that one can reduce the study of the deformations of a sandwiched surface singularity to the study of deformations of a 1-dimensional object, a so-called decorated plane curve singularity. In particular, the Milnor fibers corresponding to their various smoothing components may be reconstructed up to diffeomorphisms from those deformations of associated decorated curves which have only ordinary singularities. Part of the topology of such a deformation is encoded in the incidence matrix between the irreducible components of the deformed curve and the points which decorate it, well-defined up to permutations of columns. Extending a previous theorem ofours, which treated the case of cyclic quotient singularities, we show that the Milnor fibers which correspond to deformations whose incidence matrices are different up to permutations of columns are not diffeomorphic in a strong sense. This gives a lower bound on the number of Stein fillings of the contact boundary of a sandwiched singularity.