On the Extension of Certain Maps with Values in Spheres

Carlos Biasi¹ ; Alice K. M. Libardi² ; Pedro L. Q. Pergher³ ; Stanis/law Spież⁴

¹ Departamento de Matemática ICMC-USP – Campus de São Carlos Caixa Postal 668 São Carlos, SP 13560-970, Brazil
² Departamento de Matemática IGCE-UNESP – Campus de Rio Claro Rio Claro, SP 13506-700, Brazil
³ Departamento de Matemática Universidade Federal de São Carlos Caixa Postal 676 São Carlos, SP 13565-905, Brazil
⁴ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8, P.O. Box 21 00-956 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) no. 2, pp. 177-182

Résumé

Let $E$ be an oriented, smooth and closed $m$-dimensional manifold with $m \ge 2$ and $V \subset E$ an oriented, connected, smooth and closed $(m-2)$-dimensional submanifold which is homologous to zero in $E$. Let $S^{n-2} \subset S^n$ be the standard inclusion, where $S^n$ is the $n$-sphere and $n \ge 3$. We prove the following extension result: if $h:V \to S^{n-2}$ is a smooth map, then $h$ extends to a smooth map $g:E \to S^n$ 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 \it ambiental bordism \rm question, which asks whether, given a smooth closed $n$-dimensional manifold $E$ and a smooth closed $m$-dimensional submanifold $V \subset E$, one can find a compact smooth $(m+1)$-dimensional submanifold $W \subset E$ such that the boundary of $W$ is $V$.

Détail

DOI : 10.4064/ba56-2-8

Keywords: oriented smooth closed m dimensional manifold subset oriented connected smooth closed m dimensional submanifold which homologous zero nbsp n subset standard inclusion where n sphere prove following extension result n smooth map extends smooth map transverse n n using result simpler proof theorem carlos biasi related ambiental bordism question which asks whether given smooth closed n dimensional manifold smooth closed m dimensional submanifold subset compact smooth dimensional submanifold subset boundary