TY - JOUR
T1 - Embeddings of Sp × Sq × Sr in Sp+q+r+1
AU - Lucas, Laércio Aparecido
AU - Saeki, Osamu
PY - 2002/12
Y1 - 2002/12
N2 - Let f: Sp × Sq × Sr → Sp+q+r+1,2 ≤ p ≤ q ≤ r, be a smooth embedding. In this paper we show that the closure of one of the two components of Sp+q+r+1 - f(Sp × Sq × Sr), denoted by C1, is diffeomorphic to Sp × Sq × Dr+1 or Sp × Dq+1 × Sr or Dp+1 × Sq × Sr, provided that p + q ≠ r or p + q = r with r even. We also show that when p + q = r with r odd, there exist infinitely many embeddings which do not satisfy the above property. We also define standard embeddings of Sp × Sq × Sr into Sp+q+r+1 and, using the above result, we prove that if C1 has the homology of Sp × Sq, then f is standard, provided that q < r.
AB - Let f: Sp × Sq × Sr → Sp+q+r+1,2 ≤ p ≤ q ≤ r, be a smooth embedding. In this paper we show that the closure of one of the two components of Sp+q+r+1 - f(Sp × Sq × Sr), denoted by C1, is diffeomorphic to Sp × Sq × Dr+1 or Sp × Dq+1 × Sr or Dp+1 × Sq × Sr, provided that p + q ≠ r or p + q = r with r even. We also show that when p + q = r with r odd, there exist infinitely many embeddings which do not satisfy the above property. We also define standard embeddings of Sp × Sq × Sr into Sp+q+r+1 and, using the above result, we prove that if C1 has the homology of Sp × Sq, then f is standard, provided that q < r.
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U2 - 10.2140/pjm.2002.207.447
DO - 10.2140/pjm.2002.207.447
M3 - Article
AN - SCOPUS:0036975403
VL - 207
SP - 447
EP - 462
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
SN - 0030-8730
IS - 2
ER -