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

Tobias Ekholm; Masamichi Takase

doi:10.1112/blms/bdq097

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arXiv · 0903.0238

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

Tobias Ekholm·Masamichi Takase

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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.

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BibTeX RIS

Tobias Ekholm, Masamichi Takase. 2010-09-28. Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions. https://doi.org/10.1112/blms%2Fbdq097

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