On the Extension of Certain Maps with Values in Spheres

Tom 56 / 2008

Carlos Biasi, Alice K. M. Libardi, Pedro L. Q. Pergher, Stanis/law Spież Bulletin Polish Acad. Sci. Math. 56 (2008), 177-182 MSC: Primary 55S36; Secondary 55S35. DOI: 10.4064/ba56-2-8


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 $V$.


  • Carlos BiasiDepartamento de Matemática
    ICMC-USP – Campus de São Carlos
    Caixa Postal 668
    São Carlos, SP 13560-970, Brazil
  • Alice K. M. LibardiDepartamento de Matemática
    IGCE-UNESP – Campus de Rio Claro
    Rio Claro, SP 13506-700, Brazil
  • Pedro L. Q. PergherDepartamento de Matemática
    Universidade Federal de São Carlos
    Caixa Postal 676
    São Carlos, SP 13565-905, Brazil
  • Stanis/law SpieżInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8, P.O. Box 21
    00-956 Warszawa, Poland

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