Is every homeomorphism $h$ of $S^n\times S^1$ isotopic to a homeomorphism $g$ of $S^n\times S^1$ such that $g(S^n\times x)=S^n\times x$ for some $x\in S^1$? I would be satisfied with an answer for $n>3$.
Homeomorphisms of S^n\times S^1
olga
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