On the extension of certain maps with values in spheres

• Autorzy: Biasi C. , Libardi A. K. , Pergher P. L. , Spież S.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

ambiental bordism; oriented cobordism group; Euler class; Alexander duality isomorphism; Lefschetz duality isomorphism; Thorn class; Thorn transversality theorem

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2008
• Tom: Vol. 56, no 2
• Strony: 177--182
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Biasi C.

autor

Libardi A. K.

autor

Pergher P. L.

autor

Spież S.

• Departamento de Matemática ICMC-USP - Campus de São Carlos, Caixa Postal 668 São Carlos, SP 13560-970, Brazil,

Bibliografia

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• [2] G. E. Bredon, Topology and Geometry, Grad. Texts in Math. 139, Springer, New York, 1993.
• [3] M. Hirsch, On embedding of bounding manifolds in euclidean space, Ann. of Math. 74 (1961), 494-497.
• [4] W. S. Massey, A Basic Course in Algebraic Topology, Springer, New York, 1991.
• [5] J. W. Milnor, Topology from the Differentiable Viewpoint, Univ. Press of Virginia, Charlottesville, 1965.
• [6] N. Sato, Cobordism of semiboundary links, Topology Appl. 18 (1984), 225-234.
• [7] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
• [8] N. E. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, NJ, 1951.
• [9] R. Thorn, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17-86.