$\mu$-constant monodromy groups and Torelli results for the quadrangle singularities and the bimodal series

Abstract

This paper is a sequel to [He11] and [GH17]. In [He11] a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_\mu^{mar}$ for marked singularities in one $\mu$-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichm\"uller space. It comes together with a $\mu$-constant monodromy group $G^{mar}\subset G_{\mathbb{Z}}$. Here $G_{\mathbb{Z}}$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_\mu^{mar}$ is connected. This is equivalent to $G^{mar}= G_{\mathbb{Z}}$. Also Torelli type conjectures were formulated. In [He11] and [GH17] $M_\mu^{mar}, G_{\mathbb{Z}}$ and $G^{mar}$ were determined and all conjectures were proved for the simple, the unimodal and the exceptional bimodal singularities. In this paper the quadrangle singularities and the bimodal series are treated. The Torelli type conjectures are true. But the conjecture $G^{mar}= G_{\mathbb{Z}}$ and $M_\mu^{mar}$ connected does not hold for certain subseries of the bimodal series.