Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions

TL;DR

This paper computes the Smale invariants for a family of 3-sphere immersions into 4-space derived from 2-sphere immersions with self-intersections, linking these invariants to cobordism classes and stable homotopy groups.

Contribution

It introduces a method to compute Smale invariants for a specific family of 3-sphere immersions using singularity resolution and relates these to cobordism and stable homotopy groups.

Findings

g_n immersions generate the stable 3-stem when n is divisible by 3

Smale invariants are computed via resolution of singularities

g_n represents a generator of the cobordism group under certain conditions

Abstract

A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self intersection points equal to -n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing the circle bundle immersions with their universal covering maps, we get for n>0 immersions g_n of the 3-sphere into 4-space. In this note, we compute the Smale invariants of g_n. The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends g_n. As an application, we determine the classes represented by g_n in the cobordism group of immersions which is naturally identified with the stable 3-stem. It follows in particular that g_n represents a generator of the stable 3-stem if and only if n is divisible by 3.