uu.seUppsala University Publications

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Immersions and their self intersectionsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 1998 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Acta Universitatis Upsaliensis , 1998. , 119 p.
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 12
##### Keyword [en]

Mathematics, Regular homotopy, generic immersion, finite order invariant, Vassiliev
invariant, self intersection, diffeotopy, framing, spin structure, pin
structure, embedding, Seifert surface, linking, 24
##### Keyword [sv]

MATEMATIK
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-1215ISBN: 91-506-1303-0OAI: oai:DiVA.org:uu-1215DiVA: diva2:160772
##### Public defence

1998-10-02, Room 247, Building 2, Polacksbacken, Uppsala University, Uppsala, 13:15
#####

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Available from: 1998-09-11 Created: 1998-09-11Bibliographically approved

Regular homotopy classes of immersions of the *k*-dimensional sphere *S*^{k} into (k + *n*)-dimensional Euclidean space R^{k+n} are known to form finitely generated Abelian groups. It is known how to calculate the regular homotopy class of a generic immersion *S*^{k} → R^{2k} in terms of its self intersection. A formula for the regular homotopy class of a generic immersion *S*^{k} → R^{2k-r}, *r* = 1, 2 and *k* ≥ 4, is found. It involves only terms depending on the geometry of the self intersection of the generic immersion. It is shown that, for immersions *S*^{3} → R^{5}, such a formula does not exist. More precisely, regular homotopy classes of immersions *S*^{3} → R^{5} constitute an infinite cyclic group. The classes containing embeddings form a subgroup. It is proved that this subgroup has index 24 and the obstruction for a generic immersion to be regularly homotopic to an embedding is expressed in terms of geometric invariants of its self intersection. It is shown that, up to regular homotopy through generic immersions, a generic immersion in the metastable range of a sufficiently high-connected manifold into Euclidean space depends only on the geometry of its self intersection. In the cases when the self intersection has dimension 0, 1, or 2, numerical invariants of self intersections are found and provedto give a complete classification of generic immersions. All finite order invariants of generic immersions *S*^{k} → *R*^{2k-r}, *r* = 1, 2 and k ≥ 2*r* + 2 are found. It is showed that they are not sufficient to separate immersions which cannot be deformed into each other by regular homotopy through generic immersions. Two independent first order invariants *J* and St of generic immersions *S*^{3}; → R^{5}; are constructed. It is proved that any first order invariant is a linear combination of these.

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