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Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$ setwise: that is, $f(Y)=Y$. Finally, we suppose that that $f$ is topologically isotopic to the identity map.

Does this imply, perhaps after some further smooth isotopy, that $f$ is the identity on $Y$: that is, fixes $Y$ pointwise?

If it helps, we can also assume that $X$ is simply connected.

One thing in dimension four that requires care is that the existence of antopological isotopy may not imply that there is a smooth isotopy. Danny Ruberman found the existence of such examples.

Another thing: if $\mathrm{Diff}^+(Y)$ is connected, then the above question has a positive answer. (For example $S^3$.)

Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$ setwise: that is, $f(Y)=Y$. Finally, we suppose that that $f$ is topologically isotopic to the identity map.

Does this imply, perhaps after some further smooth isotopy, that $f$ is the identity on $Y$: that is, fixes $Y$ pointwise?

If it helps, we can also assume that $X$ is simply connected.

One thing in dimension four that requires care is that the existence of a topological isotopy may not imply that there is a smooth isotopy. Danny Ruberman proved the existence of such examples.

Another thing: if $\mathrm{Diff}^+(Y)$ is connected, then the above question has a positive answer. (For example $S^3$.)

Let $X$ be an oriented, closed $4$-manifold and $Y$ be an oriented, closed $3$-manifold smoothly embedded in $X$. Suppose we have a diffeomorphism $f:X \to X$ which is fixing $Y$ as a set, i.e., $f(Y)=Y$. Moreover, we assume that $f$ is topologically isotopic to the identity map. Does it imply up to some further smooth isotopy that $f$ is identity on $Y$, i.e., fixing $Y$ pointwise? (Also for simplification, one can assume that $X$ is simply connected.)

One thing in dim-4 that one needs to be careful about about is, an existence of topological isotopy may not imply smooth isotopy (Danny Ruberman found the existence of such examples).

One thing to be noted that if $\mathrm{Diff}^+(Y)$ is connected, then the above question has a positive answer. (For example $S^3$.)

Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$ setwise: that is, $f(Y)=Y$. Finally, we suppose that that $f$ is topologically isotopic to the identity map.

Does this imply, perhaps after some further smooth isotopy, that $f$ is the identity on $Y$: that is, fixes $Y$ pointwise?

If it helps, we can also assume that $X$ is simply connected.

One thing in dimension four that requires care is that the existence of antopological isotopy may not imply that there is a smooth isotopy. Danny Ruberman found the existence of such examples.

Another thing: if $\mathrm{Diff}^+(Y)$ is connected, then the above question has a positive answer. (For example $S^3$.)

Let $X$ be an oriented, closed $4$-manifold and $Y$ be an oriented, closed $3$-manifold smoothly embedded in $X$. Now if we have a diffeomorphism $f:X \to X$ which is fixing $Y$ as a set, i.e., $f(Y)=Y$. And moreover, if we assume that $f$ is topologically isotopic to identity. Does it imply that upto some further smooth isotopy $f$ is identity on $Y$, i.e., fixing $Y$ pointwise? [Also for simplification, one can assume that $X$ is simply connected.]

One thing in dim-4 that one needs to be care about about is, existence of topological isotopy may not imply smooth isotopy [Danny Ruberman found existence of such examples first].

One thing to be noted that if $\mathrm{Diff}^+(Y)$ is connected, then above question has a positive answer. (For example $S^3$.)

Let $X$ be an oriented, closed $4$-manifold and $Y$ be an oriented, closed $3$-manifold smoothly embedded in $X$. Suppose we have a diffeomorphism $f:X \to X$ which is fixing $Y$ as a set, i.e., $f(Y)=Y$. Moreover, we assume that $f$ is topologically isotopic to the identity map. Does it imply up to some further smooth isotopy that $f$ is identity on $Y$, i.e., fixing $Y$ pointwise? (Also for simplification, one can assume that $X$ is simply connected.)

One thing in dim-4 that one needs to be careful about about is, an existence of topological isotopy may not imply smooth isotopy (Danny Ruberman found the existence of such examples).

One thing to be noted that if $\mathrm{Diff}^+(Y)$ is connected, then the above question has a positive answer. (For example $S^3$.)