On the Extension of Certain Maps with Values in Spheres

Carlos Biasi ; Alice K. M. Libardi ; Pedro L. Q. Pergher ; Stanisław Spież

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008), p. 177-182 / Harvested from The Polish Digital Mathematics Library

Résumé

Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m-2)-dimensional submanifold which is homologous to zero in E. Let $S^{n-2} ⊂ Sⁿ$ be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if $h: V → S^{n-2}$ is a smooth map, then h extends to a smooth map g: E → Sⁿ transverse to $S^{n-2}$ and with $g^{-1}(S^{n-2}) = V$. Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental bordism question, which asks whether, given a smooth closed n-dimensional manifold E and a smooth closed m-dimensional submanifold V ⊂ E, one can find a compact smooth (m+1)-dimensional submanifold W ⊂ E such that the boundary of W is V.

Détail

Publié le : 2008-01-01

Zbl 1148.55012

EUDML-ID : urn:eudml:doc:286489