Konstantis, Panagiotis . A counting invariant for maps into spheres and for zero loci of sections of vector bundles. Abh. Math. Semin. Univ. Hamburg. HEIDELBERG: SPRINGER HEIDELBERG. ISSN 1865-8784
Full text not available from this repository.Abstract
The set of unrestricted homotopy classes [M, S-n] where M is a closed and connected spin (n + 1)-manifold is called the n-th cohomotopy group pi(n)(M) of M. Using homotopy theory it is known that pi(n)(M) = H-n(M;Z) circle plus Z(2). We will provide a geometrical description of the Z(2) part in pi(n)(M) analogous to Pontryagin's computation of the stable homotopy group pi(n+1)(S-n). This Z(2) number can be computed by counting embedded circles in M with a certain framing of their normal bundle. This is a similar result to the mod 2 degree theorem for maps M -> Sn+1. Finally we will observe that the zero locus of a section in an oriented rank n vector bundle E -> M defines an element in pi(n)(M) and it turns out that the Z(2) part is an invariant of the isomorphism class of E. At the end we show that if the Euler class of E vanishes this Z(2) invariant is the final obstruction to the existence of a nowhere vanishing section.
| Item Type: | Journal Article | ||||||||
| Creators: |
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| URN: | urn:nbn:de:hbz:38-310690 | ||||||||
| DOI: | 10.1007/s12188-020-00228-6 | ||||||||
| Journal or Publication Title: | Abh. Math. Semin. Univ. Hamburg | ||||||||
| Publisher: | SPRINGER HEIDELBERG | ||||||||
| Place of Publication: | HEIDELBERG | ||||||||
| ISSN: | 1865-8784 | ||||||||
| Language: | English | ||||||||
| Faculty: | Unspecified | ||||||||
| Divisions: | Unspecified | ||||||||
| Subjects: | no entry | ||||||||
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| URI: | http://kups.ub.uni-koeln.de/id/eprint/31069 |
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