On the Extension of Certain Maps with Values in Spheres

Carlos Biasi, et al. "On the Extension of Certain Maps with Values in Spheres." Bulletin of the Polish Academy of Sciences. Mathematics 56.2 (2008): 177-182. <http://eudml.org/doc/286489>.

@article{CarlosBiasi2008,
abstract = {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 $S^\{n-2\} ⊂ Sⁿ$ be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if $h: V → S^\{n-2\}$ is a smooth map, then h extends to a smooth map g: E → Sⁿ 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 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.},
author = {Carlos Biasi, Alice K. M. Libardi, Pedro L. Q. Pergher, Stanisław Spież},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {ambiental bordism; oriented cobordism group; Euler class; Alexander duality isomorphism; Lefschetz duality isomorphism; Thom class; Thom transversality theorem},
language = {eng},
number = {2},
pages = {177-182},
title = {On the Extension of Certain Maps with Values in Spheres},
url = {http://eudml.org/doc/286489},
volume = {56},
year = {2008},
}

TY - JOUR
AU - Carlos Biasi
AU - Alice K. M. Libardi
AU - Pedro L. Q. Pergher
AU - Stanisław Spież
TI - On the Extension of Certain Maps with Values in Spheres
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2008
VL - 56
IS - 2
SP - 177
EP - 182
AB - 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 $S^{n-2} ⊂ Sⁿ$ be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if $h: V → S^{n-2}$ is a smooth map, then h extends to a smooth map g: E → Sⁿ 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 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.
LA - eng
KW - ambiental bordism; oriented cobordism group; Euler class; Alexander duality isomorphism; Lefschetz duality isomorphism; Thom class; Thom transversality theorem
UR - http://eudml.org/doc/286489
ER -