Regular homotopy classes of links of simple singularities and immersions associated with their Dynkin diagrams

TL;DR

This paper classifies the regular homotopy classes of immersions related to simple singularities and their Dynkin diagrams, providing a comprehensive understanding of their topological invariants and homotopy properties.

Contribution

It determines the regular homotopy classes of links of simple singularities and relates them to Dynkin diagrams, extending previous work by computing complete invariants and Smale invariants.

Findings

The inclusion map of the link into the 5-sphere is regularly homotopic to the immersion associated with the Dynkin diagram.

Complete invariants of the immersions are computed using Wu and Saeki--Sz\

Smale invariants of Kinjo's immersions are explicitly determined.

Abstract

Our aim is to determine the regular homotopy classes of immersions related to Arnol'd's simple singularities. For every type of simple singularities, we determine the regular homotopy class of the inclusion map of the link into the 5-sphere. We further show that the inclusion map is regularly homotopic to the immersion associated with the corresponding Dynkin diagram, which was constructed by Kinjo. We prove these by computing the complete invariants of the immersions given by Wu and Saeki--Sz\H{u}cs--Takase. As an application, we also determine the Smale invariants of Kinjo's immersions.