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

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 these 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.

Immersions of 3-sphere into 4-space associated with Dynkin diagrams of typesAandD

“…In [2], Ekholm-Takase has given a formula for the Smale invariant of an immersion f : S 3 R 4 by using a singular Seifert surface for f . (iii) For any p ∈ V, the rank rk(dF p ) of the differential…”

“…In fact, the right hand side of (2.1) depends only on f , and does not depend on the choice of singular Seifert surface. (b) [6,2] The Smale invariant of f is given by…”

“…We recall the construction of immersions f n and g n . Consider the immersion of E(ξ 2 ) into R 4 [2]. Plumbing copies of this immersion along A n−1 or D n+2 , we obtain immersions P(A n−1 , 2) := P(A n−1 , 2, · · · , 2) and P(D n+2 , 2) := P(D n+2 , 2, · · · , 2) into R 4 .…”

“…The paper is organized as follows. In Section 2, we recall the formula for the Smale invariant in terms of singularities which is given by Ekholm-Takase [2]. In Section 3, we summarize some basic facts about Kirby calculus, which we use to construct singular Seifert surfaces for f n and g n .…”

Counting indices of critical points of rank two of polynomial selfmaps of R4

“…Papers [5,12,13] offer methods of computing the index I p (Df ) which require the germ f : (R 4 , p) → (R 4 , f (p)) to be written in a special form. However, if f is a polynomial then we usually are not able to find explicitely points in Σ 2 (Df ), so we cannot adopt these techniques in our case.…”