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

Keywords:

embedding; sphere; diffeomorphism

Zbl 0876.57045
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