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Is every orientation preserving homeomorphism $h$ of $S^n\times S^1$ with identity action in $\pi_1$ pseudo 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 $n>3$.

Is every homeomorphism $h$ of $S^n\times S^1$ with identity action in $\pi_1$ pseudo isotopic to a homeomorphism $g$ of $S^n\times S^1$ such that $g(S^n\times x)=S^n\times x$ for each $x\in S^1$? I would be satisfied with an answer for $n>3$.

Is every homeomorphism $h$ of $S^n\times S^1$ pseudo 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 $n>3$.

Is every orientation preserving homeomorphism $h$ of $S^n\times S^1$ with identity action in $\pi_1$ pseudo 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 $n>3$.

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 $n>3$.

Is every homeomorphism $h$ of $S^n\times S^1$ pseudo 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 $n>3$.