Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions
2011 (English)In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 43, no 2, 251-266 p.Article in journal (Refereed) Published
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.
Place, publisher, year, edition, pages
2011. Vol. 43, no 2, 251-266 p.
IdentifiersURN: urn:nbn:se:uu:diva-151903DOI: 10.1112/blms/bdq097ISI: 000288567300003OAI: oai:DiVA.org:uu-151903DiVA: diva2:411631