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This is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself closely related to the annulus conjecture.

The same statement is then true for an homeomorphism $h$ of $\mathbb{S}^n$ : first, you can by composition with a (stable) homeomorphism of $\mathbb{S}^n$ assume that $h$ fixes the north pole $p$. Then by SHC $h$ restricted to $\mathbb{S}^n-p\simeq\mathbb{R}^n$ is a composition of homeomorphisms which are the identity on nonempty open sets. But an homeomorphism of $\mathbb{S}^n$ which is the identity on a non-empty open set is isotopic to the identity, by Alexander's trick, hence the conclusion.

PS : in fact, the detailed story is somewhat complicated. What was known relatively early (due to R. D. Anderson, G. M. Fisher around 1960) was that for any topological manifold $M$ (maybe non-compact, but paracompact), the group $H_c(M)$ of homeomorphisms generated by those compactly supported in domains of topological charts $\mathbb{R}^n\simeq U\subset M$ is the smallest nontrivial normal subgroup of $H(M)$ of all homeomorphisms, and that it is simple. In particular, $H_c(M)$ is arcwise connected. The proof is ingenious, but not very difficult.

The (much harder) methods and results of geometric topology in dimension $3$ (Bing, Moise,...) then allowed Fisher to prove that for a closed $3$-manifold $M$, $H_c(M)$ is open in $H(M)$ and thus coincides with the identity component $H_0(M)$. In particular $H(M)$ is locally arcwise connected, a not at all obvious fact -- later generalized by Cernavskii and Edwards-Kirby, who proved the local contractibilty (hence local arcwise connectedness) of $H(M)$ in any dimension.

Then M. Brown and H. Gluck named stable homeomorphisms, and studied stable structures on manifolds. A puzzling aspect of the notion of stable homeomorphism is that it is very "contagious" : if $h\in H(M)$ coincides with an element $f$ of $H_s(M)$ on a nonempty open set $U$, then $h$ is in $H_s(M)$, since $h^{-1}f$ is the identity on $U$. So this is seen locally everywhere, like orientation preservation (to which it was eventually identified).

After that came R. Kirby (and L. Siebenmann) in 1968, who proved (using the results of surgery by Wall et al), that SHC$_n$ (hence AC$_n$) is true in all dimensions $n> 4$.

But the remaining case SHC$_4$ was only solved by F. Quinn in 1982 (after work of A. Casson and M. Freedman), who proved AC$_4$, hence SHC$_4$ since the $n\leq 3$ cases were known. See the survey by Edwards.

This is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself closely related to the annulus conjecture.

The same statement is then true for an homeomorphism $h$ of $\mathbb{S}^n$ : first, you can by composition with a (stable) homeomorphism of $\mathbb{S}^n$ assume that $h$ fixes the north pole $p$. Then by SHC $h$ restricted to $\mathbb{S}^n-p\simeq\mathbb{R}^n$ is a composition of homeomorphisms which are the identity on nonempty open sets. But an homeomorphism of $\mathbb{S}^n$ which is the identity on a non-empty open set is isotopic to the identity, by Alexander's trick, hence the conclusion.

PS : in fact, the detailed story is somewhat complicated. What was known relatively early (due to R. D. Anderson, G. M. Fisher around 1960) was that for any topological manifold $M$ (maybe non-compact, but paracompact), the group $H_c(M)$ of homeomorphisms generated by those compactly supported in domains of topological charts $\mathbb{R}^n\simeq U\subset M$ is the smallest nontrivial normal subgroup of $H(M)$ of all homeomorphisms, and that it is simple. In particular, $H_c(M)$ is arcwise connected. The proof is ingenious, but not very difficult.

The (much harder) methods and results of geometric topology in dimension $3$ (Bing, Moise,...) then allowed Fisher to prove that for a closed $3$-manifold $M$, $H_c(M)$ is open in $H(M)$ and thus coincides with the identity component $H_0(M)$. In particular $H(M)$ is locally arcwise connected, a not at all obvious fact -- later generalized by Cernavskii and Edwards-Kirby, who proved the local contractibilty (hence local arcwise connectedness) of $H(M)$ in any dimension.

Then M. Brown and H. Gluck named stable homeomorphisms, and studied stable structures on manifolds. A puzzling aspect of the notion of stable homeomorphism is that it is very "contagious" : if $h\in H(M)$ coincides with an element $f$ of $H_s(M)$ on a nonempty open set $U$, then $h$ is in $H_s(M)$, since $h^{-1}f$ is the identity on $U$. So this is seen locally everywhere, like orientation preservation (to which it was eventually identified).

After that came R. Kirby (and L. Siebenmann) in 1968, who proved (using the results of surgery by Wall et al), that SHC$_n$ (hence AC$_n$) is true in all dimensions $n> 4$.

But the remaining case SHC$_4$ was only solved by F. Quinn in 1982 (after work of A. Casson and M. Freedman), who proved AC$_4$, hence SHC$_4$ since the $n\leq 3$ cases were known. See the survey by Edwards.

This is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself closely related to the annulus conjecture.

The SHC says that any orientation preserving homeomorphism of $\mathbb{R}^n$ is stable i.e. a (finite) product of homeomorphisms each of which is the identity on some non-empty open set.

The same statement is then true for an homeomorphism $h$ of $\mathbb{S}^n$ : first, you can by composition with a (stable) homeomorphism of $\mathbb{S}^n$ assume that $h$ fixes the north pole $p$. Then by SHC $h$ restricted to $\mathbb{S}^n-p\simeq\mathbb{R}^n$ is a composition of homeomorphisms which are the identity on nonempty open sets. But an homeomorphism of $\mathbb{S}^n$ which is the identity on a non-empty open set is isotopic to the identity, by Alexander's trick, hence the conclusion.

PS : in fact, the detailed story is somewhat complicated. What was known relatively early (due to R. D. Anderson, G. M. Fisher around 1960) was that for any topological manifold $M$ (maybe non-compact, but paracompact), the group $H_c(M)$ of homeomorphisms generated by those compactly supported in domains of topological charts $\mathbb{R}^n\simeq U\subset M$ is the smallest nontrivial normal subgroup of $H(M)$ of all homeomorphisms, and that it is simple. In particular, $H_c(M)$ is arcwise connected. The proof is ingenious, but not very difficult.

The (much harder) methods and results of geometric topology in dimension $3$ (Bing, Moise,...) then allowed Fisher to prove that for a closed $3$-manifold $M$, $H_c(M)$ is open in $H(M)$ and thus coincides with the identity component $H_0(M)$. In particular $H(M)$ is locally arcwise connected, a not at all obvious fact -- later generalized by Cernavskii and Edwards-Kirby, who proved the local contractibilty (hence local arcwise connectedness) of $H(M)$ in any dimension.

Fisher also considered the group of stable homeomorphisms $H_s(M)$ (without the name) of a connected $M$ (otherwise the notion is empty). He managed to prove that it coincides with the group of orientation preserving ones $H_+(M)$ for closed oriented $3$-manifolds $M$ admitting an orientation reversing homeomorphism. For $M=\mathbb{S}^3$, this implies that $H_+=H_0=H_c$ : any orientation preseving homeomorphism of $\mathbb{S}^3$ is isotopic to the identity (note that for all $n$, $H_s(\mathbb{S}^n)=H_c(\mathbb{S}^n)$). This is the $n=3$ case of your question.

Then R. Brown and H. Gluck named stable homeomorphisms, and studied stable structures on manifolds. A puzzling aspect of the notion of stable homeomorphism is that it is very "contagious" : if $h\in H(M)$ coincides with an element $f$ of $H_s(M)$ on a nonempty open set $U$, then $h$ is in $H_s(M)$, since $h^{-1}f$ is the identity on $U$. So this is seen locally everywhere, like orientation preservation (to which it was eventually identified).

Brown and Gluck proved that SHC$_n$ (SHC in dimension $n$) implies the annulus conjecture in dimension $n$ (AC$_n$), and that AC$_k$ in all dimensions $k\leq n$ imply SHC$_n$. But this was still stuck at SHC$_3$.

After that came R. Kirby (and L. Siebenmann) in 1968, who proved (using the results of surgery by Wall et al), that SHC$_n$ (hence AC$_n$) is true in all dimensions $n> 4$.

But the remaining case SHC$_4$ was only solved by F. Quinn in 1982 (after work of A. Casson and M. Freedman), who proved AC$_4$, hence SHC$_4$ since the $n\leq 3$ cases were known. See the survey by Edwards.

This is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself closely related to the annulus conjecture.

The SHC says that any orientation preserving homeomorphism of $\mathbb{R}^n$ is stable i.e. a (finite) product of homeomorphisms each of which is the identity on some non-empty open set.

The same statement is then true for an homeomorphism $h$ of $\mathbb{S}^n$ : first, you can by composition with a (stable) homeomorphism of $\mathbb{S}^n$ assume that $h$ fixes the north pole $p$. Then by SHC $h$ restricted to $\mathbb{S}^n-p\simeq\mathbb{R}^n$ is a composition of homeomorphisms which are the identity on nonempty open sets. But an homeomorphism of $\mathbb{S}^n$ which is the identity on a non-empty open set is isotopic to the identity, by Alexander's trick, hence the conclusion.

PS : in fact, the detailed story is somewhat complicated. What was known relatively early (due to R. D. Anderson, G. M. Fisher around 1960) was that for any topological manifold $M$ (maybe non-compact, but paracompact), the group $H_c(M)$ of homeomorphisms generated by those compactly supported in domains of topological charts $\mathbb{R}^n\simeq U\subset M$ is the smallest nontrivial normal subgroup of $H(M)$ of all homeomorphisms, and that it is simple. In particular, $H_c(M)$ is arcwise connected. The proof is ingenious, but not very difficult.

The (much harder) methods and results of geometric topology in dimension $3$ (Bing, Moise,...) then allowed Fisher to prove that for a closed $3$-manifold $M$, $H_c(M)$ is open in $H(M)$ and thus coincides with the identity component $H_0(M)$. In particular $H(M)$ is locally arcwise connected, a not at all obvious fact -- later generalized by Cernavskii and Edwards-Kirby, who proved the local contractibilty (hence local arcwise connectedness) of $H(M)$ in any dimension.

Fisher also considered the group of stable homeomorphisms $H_s(M)$ (without the name) of a connected $M$ (otherwise the notion is empty). He managed to prove that it coincides with the group of orientation preserving ones $H_+(M)$ for closed oriented $3$-manifolds $M$ admitting an orientation reversing homeomorphism. For $M=\mathbb{S}^3$, this implies that $H_+=H_0=H_c$ : any orientation preseving homeomorphism of $\mathbb{S}^3$ is isotopic to the identity (note that for all $n$, $H_s(\mathbb{S}^n)=H_c(\mathbb{S}^n)$). This is the $n=3$ case of your question.

Then M. Brown and H. Gluck named stable homeomorphisms, and studied stable structures on manifolds. A puzzling aspect of the notion of stable homeomorphism is that it is very "contagious" : if $h\in H(M)$ coincides with an element $f$ of $H_s(M)$ on a nonempty open set $U$, then $h$ is in $H_s(M)$, since $h^{-1}f$ is the identity on $U$. So this is seen locally everywhere, like orientation preservation (to which it was eventually identified).

Brown and Gluck proved that SHC$_n$ (SHC in dimension $n$) implies the annulus conjecture in dimension $n$ (AC$_n$), and that AC$_k$ in all dimensions $k\leq n$ imply SHC$_n$. But this was still stuck at SHC$_3$.

After that came R. Kirby (and L. Siebenmann) in 1968, who proved (using the results of surgery by Wall et al), that SHC$_n$ (hence AC$_n$) is true in all dimensions $n> 4$.

But the remaining case SHC$_4$ was only solved by F. Quinn in 1982 (after work of A. Casson and M. Freedman), who proved AC$_4$, hence SHC$_4$ since the $n\leq 3$ cases were known. See the survey by Edwards.

This is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself closely related to the annulus conjecture.

The SHC says that any orientation preserving homeomorphism of $\mathbb{R}^n$ is stable i.e. a (finite) product of homeomorphisms each of which is the identity on some non-empty open set.

The same statement is then true for an homeomorphism $h$ of $\mathbb{S}^n$ : first, you can by composition with a (stable) homeomorphism of $\mathbb{S}^n$ assume that $h$ fixes the north pole $p$. Then by SHC $h$ restricted to $\mathbb{S}^n-p\simeq\mathbb{R}^n$ is a composition of homeomorphisms which are the identity on nonempty open sets. But an homeomorphism of $\mathbb{S}^n$ which is the identity on a non-empty open set is isotopic to the identity, by Alexander's trick, hence the conclusion.

PS : in fact, the detailed story is somewhat complicated. What was known relatively early (due to R. D. Anderson, G. M. Fisher around 1960) was that for any topological manifold $M$ (maybe non-compact, but paracompact), the group $H_c(M)$ of homeomorphisms generated by those compactly supported in domains of topological charts $\mathbb{R}^n\simeq U\subset M$ is the smallest nontrivial normal subgroup of $H(M)$ of all homeomorphisms, and that it is simple. In particular, $H_c(M)$ is arcwise connected. The proof is ingenious, but not very difficult.

The (much harder) methods of geometric topology in dimensions $3$ (Bing, Moise,...) then allowed Fisher to prove that for a closed $3$-manifold $M$, $H_c(M)$ is open in $H(M)$ and thus coincides with the identity component $H_0(M)$. In particular $H(M)$ is locally arcwise connected, a not at all obvious fact -- later generalized by Cernavskii and Edwards-Kirby, who proved the local contractibilty (hence local arcwise connectedness) of $H(M)$ in any dimension.

Fisher also considered the group of stable homeomorphisms $H_s(M)$ (without the name) of a connected $M$ (otherwise the notion is empty). He managed to prove that it coincides with the group of orientation preserving ones $H_+(M)$ for closed oriented $3$-manifolds $M$ admitting an orientation reversing homeomorphism. For $M=\mathbb{S}^3$, this implies that $H_+=H_0=H_c$ : any orientation preseving homeomorphism of $\mathbb{S}^3$ is isotopic to the identity (note that for all $n$, $H_s(\mathbb{S}^n)=H_c(\mathbb{S}^n)$). This is the $n=3$ case of your question.

Then R. Brown and H. Gluck named stable homeomorphisms, and studied stable structures on manifolds. A puzzling aspect of the notion of stable homeomorphism is that it is very "contagious" : if $h\in H(M)$ coincides with an element $f$ of $H_s(M)$ on a nonempty open set $U$, then $h$ is in $H_s(M)$, since $h^{-1}f$ is the identity on $U$. So this is seen locally everywhere, like orientation preservation (to which it was eventually identified).

Brown and Gluck proved that SHC$_n$ (SHC in dimension $n$) implies the annulus conjecture in dimension $n$ (AC$_n$), and that AC$_k$ in all dimensions $k\leq n$ imply SHC$_n$. But this was still stuck at SHC$_3$.

After that came R. Kirby (and L. Siebenmann) in 1968, who proved (using results of surgery by Wall et al), that SHC$_n$ (hence AC$_n$) is true in all dimensions $n> 4$.

But the remaining case SHC$_4$ was only solved by F. Quinn in 1982 (after work of A. Casson and M. Freedman), who proved AC$_4$, hence SHC$_4$ since the $n\leq 3$ cases were known. See the survey by Edwards.

This is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself closely related to the annulus conjecture.

The SHC says that any orientation preserving homeomorphism of $\mathbb{R}^n$ is stable i.e. a (finite) product of homeomorphisms each of which is the identity on some non-empty open set.

The same statement is then true for an homeomorphism $h$ of $\mathbb{S}^n$ : first, you can by composition with a (stable) homeomorphism of $\mathbb{S}^n$ assume that $h$ fixes the north pole $p$. Then by SHC $h$ restricted to $\mathbb{S}^n-p\simeq\mathbb{R}^n$ is a composition of homeomorphisms which are the identity on nonempty open sets. But an homeomorphism of $\mathbb{S}^n$ which is the identity on a non-empty open set is isotopic to the identity, by Alexander's trick, hence the conclusion.

PS : in fact, the detailed story is somewhat complicated. What was known relatively early (due to R. D. Anderson, G. M. Fisher around 1960) was that for any topological manifold $M$ (maybe non-compact, but paracompact), the group $H_c(M)$ of homeomorphisms generated by those compactly supported in domains of topological charts $\mathbb{R}^n\simeq U\subset M$ is the smallest nontrivial normal subgroup of $H(M)$ of all homeomorphisms, and that it is simple. In particular, $H_c(M)$ is arcwise connected. The proof is ingenious, but not very difficult.

The (much harder) methods and results of geometric topology in dimension $3$ (Bing, Moise,...) then allowed Fisher to prove that for a closed $3$-manifold $M$, $H_c(M)$ is open in $H(M)$ and thus coincides with the identity component $H_0(M)$. In particular $H(M)$ is locally arcwise connected, a not at all obvious fact -- later generalized by Cernavskii and Edwards-Kirby, who proved the local contractibilty (hence local arcwise connectedness) of $H(M)$ in any dimension.

Fisher also considered the group of stable homeomorphisms $H_s(M)$ (without the name) of a connected $M$ (otherwise the notion is empty). He managed to prove that it coincides with the group of orientation preserving ones $H_+(M)$ for closed oriented $3$-manifolds $M$ admitting an orientation reversing homeomorphism. For $M=\mathbb{S}^3$, this implies that $H_+=H_0=H_c$ : any orientation preseving homeomorphism of $\mathbb{S}^3$ is isotopic to the identity (note that for all $n$, $H_s(\mathbb{S}^n)=H_c(\mathbb{S}^n)$). This is the $n=3$ case of your question.

Then R. Brown and H. Gluck named stable homeomorphisms, and studied stable structures on manifolds. A puzzling aspect of the notion of stable homeomorphism is that it is very "contagious" : if $h\in H(M)$ coincides with an element $f$ of $H_s(M)$ on a nonempty open set $U$, then $h$ is in $H_s(M)$, since $h^{-1}f$ is the identity on $U$. So this is seen locally everywhere, like orientation preservation (to which it was eventually identified).

Brown and Gluck proved that SHC$_n$ (SHC in dimension $n$) implies the annulus conjecture in dimension $n$ (AC$_n$), and that AC$_k$ in all dimensions $k\leq n$ imply SHC$_n$. But this was still stuck at SHC$_3$.

After that came R. Kirby (and L. Siebenmann) in 1968, who proved (using the results of surgery by Wall et al), that SHC$_n$ (hence AC$_n$) is true in all dimensions $n> 4$.

But the remaining case SHC$_4$ was only solved by F. Quinn in 1982 (after work of A. Casson and M. Freedman), who proved AC$_4$, hence SHC$_4$ since the $n\leq 3$ cases were known. See the survey by Edwards.