Mathematics – Geometric Topology
Scientific paper
2009-03-02
Mathematics
Geometric Topology
Several troubles about signs, including the formulation of the main theorem, have been fixed
Scientific paper
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.
Ekholm Tobias
Takase Masamichi
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