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

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.