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Embeddings of \(S^p\times S^q\times S^r\) in \(S^{p+q+r+1}\). (English) Zbl 1058.57022

The authors investigate codimension one smooth embeddings of the product of three spheres of dimensions \(p,q\) and \(r\) into the sphere of dimension \(p+q+r+1\).
The main results are as follows: Let \(2\leq p\leq q\leq r\) and let \(f:S^p\times S^q\times S^r\to S^{p+q+r+1}\) be a smooth embedding.
(1) The closure of one of the two components of \(S^{p+q+r+1}-f(S^p\times S^q\times S^r)\), denoted by \(C\), has the same homology as \(S^p\times S^q\) or \(S^p\times S^r\) or \(S^q\times S^r\).
(2) If the above space \(C\) is homotopy equivalent to \(S^p\times S^q\) or \(S^p\times S^r\) or \(S^q\times S^r\), then \(C\) is diffeomorphic to \(S^p\times S^q\times D^{r+1}\) or \(S^p\times D^{q+1}\times S^r\) or \(D^{p+1}\times S^q\times S^r\), respectively.
(3) The assumption of (2) is automatically fulfilled for the case when (a) \(p+q\not=r\), or (b) \(p+q=r\) with \(r\) even, or (c) \(p+q=r\) with \(r\) odd and \(C\) has the same homology as \(S^p\times S^q\).
For the other cases, the situation is different.
(4) If \(p+q=r\) with \(r\) odd, then there exist countably many, mutually distinct smooth embeddings which do not satisfy the assumption of (2).
The same problem as above for a smooth embedding \(f:S^p\times S^q\to S^{p+q+1}\) has been studied by A. Kosinski, C. T. C. Wall, and, e.g., by L. A. Lucas, O. M. Neto, O. Saeki [Kobe Math. J. 13, 145–165 (1996; Zbl 0876.57045)].

MSC:

57R40 Embeddings in differential topology
57R50 Differential topological aspects of diffeomorphisms
57R80 \(h\)- and \(s\)-cobordism

Citations:

Zbl 0876.57045
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