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On the extension of certain maps with values in spheres

Biasi C., Libardi A. K., Pergher P. L., Spież S.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 2

177-182

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 Sn[sup]n-2 ⊂ S[sup]n be the standard inclusion, where S[sup]n is the n-sphere and n ≥ 3. We prove the following extension result: if h : V → S[sup]n-2 is a smooth map, then h extends to a smooth map g : E → S[sup]n transverse to S[sup]n-2 and with g[sup]-1(S[sup]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.

Note on sectioning of some vector bundles

Horanská Ľ.

Demonstratio Mathematica

2002

Vol. 35, nr 1

205-210