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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.
Wydawca
Rocznik
Tom
Strony
177--182
Opis fizyczny
Bibliogr. 9 poz.
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autor
autor
autor
- Departamento de Matemática ICMC-USP - Campus de São Carlos, Caixa Postal 668 São Carlos, SP 13560-970, Brazil, biasi@icmc.usp.br
Bibliografia
- [1] C. Biasi, On ambiental bordism, Pacific J. Math. 163 (1994), 73-80.
- [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.
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Bibliografia
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bwmeta1.element.baztech-article-BAT5-0029-0018