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Suppose that $M$ is a finite-dimensional $C^{\infty}$-manifold, and let $\mathrm{Diff}\left(M\right)$ be the group of $C^{\infty}$-diffeomorphisms from $M$ to itself. When $M$ is compact, the usual compact-open (Whitney) topology turns $\mathrm{Diff}\left(M\right)$ into a Polish group (topological group whose topology is separable and completely-metrizable).

When $M$ is nonmerely $\sigma$-compact, the phrase "diffeomorphism groups" commonly refers to the group $\mathrm{Diff}_{c}\left(M\right)$ of compactly-supported diffeomorphisms from $M$ to itself. It is now the question of what topology is appropriate here. When taking the local-compact-open topology (weak Whitney, uniform convergence on each compact set), it does not control the behaviour at infinity. On the other hand, the strongeruniversal-compact-open topology of uniform convergence everywhere (strong Whitney, uniform convergence everywhere) is non-metrizable. Thus, the working-topology is "specified" by the following notion of convergence:

$\left(\ast\right)$ $f_{n}\to f$ if all of $f,f_{1},f_{2},...$ are supported on the same compact set and, on this compact set, $f_{n}$ with their derivatives converge uniformly to $f$.

For instance, this seems to be the approach of [1, page 2] in the introduction of the subject. This is also the description of the toplogy in [2].

My question then is the following:

If $M$ is $\sigma$-compact and non-compact, is there a topology on $\mathrm{Diff}_{c}\left(M\right)$ that turns it into a Polish group, such that the convergent sequences are as in $\left(\ast\right)$?

Here is why I think that this question is non-trivial. On one hand, it is true that there exist topologies that induce this notion of convergence as was shown in [3] (see Lemma 1.7, Corollary 2.3, Remark 2.6). However, those topologies seem not to be metrizable. A different cnadidate one may suggest is the direct limit of topologies, when the limit is taken over compact sub-manifold (as stated shortly in [1, page 2]). However, in [4, Theorem 6.1] it was proved that this direct limit topology does not result a group-topology (!).

References

[1] Banyaga, Augustin, The structure of classical diffeomorphism groups, Mathematics and its Applications (Dordrecht). 400. Dordrecht: Kluwer Academic Publishers. xi, 197 p. (1997). ZBL0874.58005.

[2] Vershik, A. M.; Gel’fand, I. M.; Graev, M. I., Representations of the group of diffeomorphisms, Russ. Math. Surv. 30, No. 6, 1-50 (1975). ZBL0337.58003.

[3] Michor, P., Manifolds of smooth maps, Cah. Topol. Géom. Différ. 19, 47-78 (1978). ZBL0382.58009.

[4] Tatsuuma, Nobuhiko; Shimomura, Hiroaki; Hirai, Takeshi, On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms, J. Math. Kyoto Univ. 38, No. 3, 551-578 (1998). ZBL0930.22002.

Suppose that $M$ is a finite-dimensional $C^{\infty}$-manifold, and let $\mathrm{Diff}\left(M\right)$ be the group of $C^{\infty}$-diffeomorphisms from $M$ to itself. When $M$ is compact, the usual compact-open (Whitney) topology turns $\mathrm{Diff}\left(M\right)$ into a Polish group (topological group whose topology is separable and completely-metrizable).

When $M$ is non-compact, the phrase "diffeomorphism groups" commonly refers to the group $\mathrm{Diff}_{c}\left(M\right)$ of compactly-supported diffeomorphisms from $M$ to itself. It is now the question of what topology is appropriate here. When taking the local-compact-open topology (weak Whitney, uniform convergence on each compact set), it does not control the behaviour at infinity. On the other hand, the stronger topology of uniform convergence everywhere (strong Whitney) is non-metrizable. Thus, the working-topology is "specified" by the following notion of convergence:

$\left(\ast\right)$ $f_{n}\to f$ if all of $f,f_{1},f_{2},...$ are supported on the same compact set and, on this compact set, $f_{n}$ with their derivatives converge uniformly to $f$.

For instance, this seems to be the approach of [1, page 2] in the introduction of the subject. This is also the description of the toplogy in [2].

My question then is the following:

If $M$ is non-compact, is there a topology on $\mathrm{Diff}_{c}\left(M\right)$ that turns it into a Polish group, such that the convergent sequences are as in $\left(\ast\right)$?

Here is why I think that this question is non-trivial. On one hand, it is true that there exist topologies that induce this notion of convergence as was shown in [3] (see Lemma 1.7, Corollary 2.3, Remark 2.6). However, those topologies seem not to be metrizable. A different cnadidate one may suggest is the direct limit of topologies, when the limit is taken over compact sub-manifold (as stated shortly in [1, page 2]). However, in [4, Theorem 6.1] it was proved that this direct limit topology does not result a group-topology (!).

References

[1] Banyaga, Augustin, The structure of classical diffeomorphism groups, Mathematics and its Applications (Dordrecht). 400. Dordrecht: Kluwer Academic Publishers. xi, 197 p. (1997). ZBL0874.58005.

[2] Vershik, A. M.; Gel’fand, I. M.; Graev, M. I., Representations of the group of diffeomorphisms, Russ. Math. Surv. 30, No. 6, 1-50 (1975). ZBL0337.58003.

[3] Michor, P., Manifolds of smooth maps, Cah. Topol. Géom. Différ. 19, 47-78 (1978). ZBL0382.58009.

[4] Tatsuuma, Nobuhiko; Shimomura, Hiroaki; Hirai, Takeshi, On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms, J. Math. Kyoto Univ. 38, No. 3, 551-578 (1998). ZBL0930.22002.

Suppose that $M$ is a finite-dimensional $C^{\infty}$-manifold, and let $\mathrm{Diff}\left(M\right)$ be the group of $C^{\infty}$-diffeomorphisms from $M$ to itself. When $M$ is compact, the usual compact-open (Whitney) topology turns $\mathrm{Diff}\left(M\right)$ into a Polish group (topological group whose topology is separable and completely-metrizable).

When $M$ is merely $\sigma$-compact, the phrase "diffeomorphism groups" commonly refers to the group $\mathrm{Diff}_{c}\left(M\right)$ of compactly-supported diffeomorphisms from $M$ to itself. It is now the question of what topology is appropriate here. When taking the local-compact-open topology (weak Whitney, uniform convergence on each compact set), it does not control the behaviour at infinity. On the other hand, the universal-compact-open topology (strong Whitney, uniform convergence everywhere) is non-metrizable. Thus, the working-topology is "specified" by the following notion of convergence:

$\left(\ast\right)$ $f_{n}\to f$ if all of $f,f_{1},f_{2},...$ are supported on the same compact set and, on this compact set, $f_{n}$ with their derivatives converge uniformly to $f$.

For instance, this seems to be the approach of [1, page 2] in the introduction of the subject. This is also the description of the toplogy in [2].

My question then is the following:

If $M$ is $\sigma$-compact and non-compact, is there a topology on $\mathrm{Diff}_{c}\left(M\right)$ that turns it into a Polish group, such that the convergent sequences are as in $\left(\ast\right)$?

Here is why I think that this question is non-trivial. On one hand, it is true that there exist topologies that induce this notion of convergence as was shown in [3] (see Lemma 1.7, Corollary 2.3, Remark 2.6). However, those topologies seem not to be metrizable. A different cnadidate one may suggest is the direct limit of topologies, when the limit is taken over compact sub-manifold (as stated shortly in [1, page 2]). However, in [4, Theorem 6.1] it was proved that this direct limit topology does not result a group-topology (!).

References

[1] Banyaga, Augustin, The structure of classical diffeomorphism groups, Mathematics and its Applications (Dordrecht). 400. Dordrecht: Kluwer Academic Publishers. xi, 197 p. (1997). ZBL0874.58005.

[2] Vershik, A. M.; Gel’fand, I. M.; Graev, M. I., Representations of the group of diffeomorphisms, Russ. Math. Surv. 30, No. 6, 1-50 (1975). ZBL0337.58003.

[3] Michor, P., Manifolds of smooth maps, Cah. Topol. Géom. Différ. 19, 47-78 (1978). ZBL0382.58009.

[4] Tatsuuma, Nobuhiko; Shimomura, Hiroaki; Hirai, Takeshi, On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms, J. Math. Kyoto Univ. 38, No. 3, 551-578 (1998). ZBL0930.22002.

Source Link
HUO
HUO
  • 426
  • 3
  • 10

Topologies on diffeomorphisms groups

Suppose that $M$ is a finite-dimensional $C^{\infty}$-manifold, and let $\mathrm{Diff}\left(M\right)$ be the group of $C^{\infty}$-diffeomorphisms from $M$ to itself. When $M$ is compact, the usual compact-open (Whitney) topology turns $\mathrm{Diff}\left(M\right)$ into a Polish group (topological group whose topology is separable and completely-metrizable).

When $M$ is non-compact, the phrase "diffeomorphism groups" commonly refers to the group $\mathrm{Diff}_{c}\left(M\right)$ of compactly-supported diffeomorphisms from $M$ to itself. It is now the question of what topology is appropriate here. When taking the local-compact-open topology (weak Whitney, uniform convergence on each compact set), it does not control the behaviour at infinity. On the other hand, the stronger topology of uniform convergence everywhere (strong Whitney) is non-metrizable. Thus, the working-topology is "specified" by the following notion of convergence:

$\left(\ast\right)$ $f_{n}\to f$ if all of $f,f_{1},f_{2},...$ are supported on the same compact set and, on this compact set, $f_{n}$ with their derivatives converge uniformly to $f$.

For instance, this seems to be the approach of [1, page 2] in the introduction of the subject. This is also the description of the toplogy in [2].

My question then is the following:

If $M$ is non-compact, is there a topology on $\mathrm{Diff}_{c}\left(M\right)$ that turns it into a Polish group, such that the convergent sequences are as in $\left(\ast\right)$?

Here is why I think that this question is non-trivial. On one hand, it is true that there exist topologies that induce this notion of convergence as was shown in [3] (see Lemma 1.7, Corollary 2.3, Remark 2.6). However, those topologies seem not to be metrizable. A different cnadidate one may suggest is the direct limit of topologies, when the limit is taken over compact sub-manifold (as stated shortly in [1, page 2]). However, in [4, Theorem 6.1] it was proved that this direct limit topology does not result a group-topology (!).

References

[1] Banyaga, Augustin, The structure of classical diffeomorphism groups, Mathematics and its Applications (Dordrecht). 400. Dordrecht: Kluwer Academic Publishers. xi, 197 p. (1997). ZBL0874.58005.

[2] Vershik, A. M.; Gel’fand, I. M.; Graev, M. I., Representations of the group of diffeomorphisms, Russ. Math. Surv. 30, No. 6, 1-50 (1975). ZBL0337.58003.

[3] Michor, P., Manifolds of smooth maps, Cah. Topol. Géom. Différ. 19, 47-78 (1978). ZBL0382.58009.

[4] Tatsuuma, Nobuhiko; Shimomura, Hiroaki; Hirai, Takeshi, On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms, J. Math. Kyoto Univ. 38, No. 3, 551-578 (1998). ZBL0930.22002.