# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## On the Extension of Certain Maps with Values in Spheres

### Tom 56 / 2008

Bulletin Polish Acad. Sci. Math. 56 (2008), 177-182 MSC: Primary 55S36; Secondary 55S35. DOI: 10.4064/ba56-2-8

#### Streszczenie

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 $V$.

#### Autorzy

• Carlos BiasiDepartamento de Matemática
ICMC-USP – Campus de São Carlos
Caixa Postal 668
São Carlos, SP 13560-970, Brazil
e-mail
• Alice K. M. LibardiDepartamento de Matemática
IGCE-UNESP – Campus de Rio Claro
Rio Claro, SP 13506-700, Brazil
e-mail
• Pedro L. Q. PergherDepartamento de Matemática
Caixa Postal 676
São Carlos, SP 13565-905, Brazil
e-mail
• Stanis/law SpieżInstitute of Mathematics