Vanishing cycles of matrix singularities

TL;DR

This paper investigates the topology of singular Milnor fibres of matrix families, establishing vanishing cycles, proving a conjecture, and exploring monodromy and group relationships in matrix singularities.

Contribution

It defines vanishing cycles for matrix singularities, proves an extended Damon-Pike $=$ conjecture, and explores monodromy and group connections in this context.

Findings

Proved an extended Damon-Pike $=$ conjecture.

Established a Lyashko-Looijenga type theorem for simple matrix families.

Identified relationships between Shephard-Todd groups, simple odd functions, and Bruce-Tari matrix classification.

Abstract

The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of the Damon-Pike $\mu=\tau$ conjecture about the families of a special type, and make first steps towards understading of the monodromy of matrix singularities. We also prove a Lyashko-Looijenga type theorem for simple matrix families, and point out a surprising relationship between certain Shephard-Todd groups, simple odd functions and a sporadic part of the Bruce-Tari simple matrix classification.