Direct limits of regular Lie groups

TL;DR

This paper establishes conditions under which a regular Lie group formed as a union of smaller regular Lie groups can be viewed as their direct limit, with applications to diffeomorphism groups and test function groups.

Contribution

It proves that regular Lie groups with a direct limit chart are the direct limit of their subgroups, extending the understanding of infinite-dimensional Lie groups.

Findings

Regular Lie groups with a direct limit chart are the direct limit of their subgroups.

The group of compactly supported diffeomorphisms is a direct limit of groups supported in compact subsets.

Similar results hold for test function groups with values in a Lie group.

Abstract

Let G be a regular Lie group which is a directed union of regular Lie groups G_i (all modelled on possibly infinite-dimensional, locally convex spaces). We show that G is the direct limit of the G_i as a regular Lie group whenever G admits a so-called direct limit chart. Notably, this allows the regular Lie group Diff_c(M) of compactly supported smooth diffeomorphisms to be interpreted as a direct limit of the regular Lie groups Diff_K(M) of smooth diffeomorphisms supported in compact subsets K of M, even if the finite-dimensional smooth manifold M is merely paracompact (but not necessarily sigma-compact), which was not known before. Similar results are obtained for the test function groups C^k_c(M,F) with values in a Lie group F.

Equations

$$f:G\to H$$

$$C_K^k(M,F):=\{\gamma \in C_c^k(M,F):(\forall x\in M\setminus K)\gamma (x)=e\}$$

$${\mathrm{Diff}}_{K_1}(M)\subseteq {\mathrm{Diff}}_{K_2}(M)\subseteq \cdots$$

$${\mathrm{Diff}}_c(M)=\underset{⟶}{\lim }{\mathrm{Diff}}_{K_n}(M)\text{vspace}-1mm$$

$$C_c^{\infty }(M,F)=\underset{⟶}{\lim }{C}_{K_n}^{\infty }(M,F)\text{vspace}-1mm$$

$${\mathrm{Diff}}_c(M)=\underset{⟶}{\lim }{\mathrm{Diff}}_K(M)\text{vspace}-1mm$$

$$C_c^k(M,F)=\underset{⟶}{\lim }{C}_K^k(M,F)\text{vspace}-1mm$$

$$G:=\bigoplus_{j\in J}G_{j}:=\Big{\{}(g_{j})_{j\in J}\in\prod_{j\in J}G_{j}\colon\mbox{$g_{j}=e$ for all but finitely many $j$}\,\Big{\}}$$

$$\varphi :={\oplus }_{j\in J}{\varphi }_j:\underset{j\in J}{⨁}{V}_j\to \underset{j\in J}{⨁}{W}_j\subseteq \underset{j\in J}{⨁}{E}_j,(g_j{)}_{j\in J}\mapsto ({\varphi }_j(g_j){)}_{j\in J}$$

$$d^{(j)}f:U\times E^j\to F$$

$$(D_{y_j}\cdots D_{y_1}f)(x)=d^{(j)}f(x,y_1,\dots ,y_j)$$

$$G\times TG\to TG,\quad(g,v)\mapsto g.v:=T\lambda_{g}(v)\quad\mbox{for $g\in G$ and $v\in TG$,}$$

$$\delta (\gamma ):[0,1]\to \mathfrak{g},t\mapsto \gamma (t{)}^{-1}.{\gamma }^{\prime }(t),$$

$$\text{mbox}evol:C^k([0,1],\mathfrak{g})\to G,\gamma \mapsto \eta (1)$$

$$\psi\circ L(\eta_{i})=\psi_{i}\quad\mbox{for all $\,i\in I$.}$$

$$(\forall w\in W)(\exists j\in I)[0,1]w\subseteq W_j.$$

$${\gamma }_x:[0,1]\to G,t\mapsto {\varphi }^{-1}(t\varphi (x)).$$

$$(\forall x\in V)(\exists j\in I){\gamma }_x([0,1])\subseteq G_j\text{mbox}and{\gamma }_x{|}^{G_j}\in C^{\infty }([0,1],G_j).$$

$$f(x)$$

$$h^{\vee }:V\to C^{\infty }([0,1],W)\subseteq C^{\infty }([0,1],\mathfrak{g}),x\mapsto h(x,\cdot )$$

$$C^{\infty }([0,1],{\varphi }^{-1}):C^{\infty }([0,1],W)\to C^{\infty }([0,1],V),\gamma \mapsto {\varphi }^{-1}\circ \gamma$$

$$f{|}_V={\mathrm{evol}}_H\circ C^{\infty }([0,1],\psi )\circ \delta \circ C^{\infty }([0,1],{\varphi }^{-1})\circ h^{\vee }$$

$$\prod _{j\in \Phi }{\varphi }_j:\prod _{j\in \Phi }{V}_j\to \prod _{j\in \Phi }{W}_j$$

$$\prod _{j\in \Phi }{G}_j\to H,(g_j{)}_{j\in \Phi }\mapsto f{|}_{G_{j_1}}(g_{j_1})\cdot f{|}_{G_{j_2}}(g_{j_2})\cdot \dots \cdot f{|}_{G_{j_n}}(g_{j_n}),\text{vspace}-.4mm$$

$$\theta :U\to M\times M,v\mapsto ({\pi }_{TM}(v),\Sigma (v))$$

$$\Omega \subseteq \{X\in {\Gamma }_c(TM):X(M)\subseteq U\}$$

$$\varphi :\Omega \to {\mathrm{Diff}}_c(M),X\mapsto \Sigma \circ X$$

$${\Gamma }_c(TM)={\underset{⟶}{\lim }}_{K\in \mathcal{F}}{\Gamma }_K(TM)$$

$$C_c^k(M,\varphi ):C_c^k(M,V)\to C_c^k(M,W),\gamma \mapsto \varphi \circ \gamma$$

$$C_K^k(M,\varphi ):C_K^k(M,V)\to C_K^k(M,W),\gamma \mapsto \varphi \circ \gamma$$

Full text

Consider a Lie group $G$ (modelled on a locally convex space) which is a union $G=\bigcup_{i\in I}G_{i}$ of such Lie groups $G_{i}$, where $(I,\leq)$ is a directed set, $G_{i}\subseteq G_{j}$ for all $i\leq j$ in $I$ and all inclusion maps $\eta_{ji}\colon G_{i}\to G_{j}$ and $\eta_{i}\colon G_{i}\to G$ are smooth group homomorphisms. It is natural to ask whether $G$ is the direct limit of the Lie groups $G_{i}$ in the category of Lie groups and smooth group homomorphisms; or, equivalently, whether a group homomorphism

[TABLE]

to a Lie group $H$ is smooth if $f|_{G_{i}}\colon G_{i}\to H$ is smooth for all $i\in I$. This question and related ones (concerning direct limit properties of $G$ as a topological group, as a smooth manifold, or as topological space) are fairly well-understood if $I={\mathbb{N}}$, so that we are dealing with a union $G=\bigcup_{n\in{\mathbb{N}}}G_{n}$ of an ascending sequence $G_{1}\subseteq G_{2}\subseteq\cdots$ of Lie groups (see [6] and the references therein). For typical examples, consider a paracompact finite-dimensional smooth manifold $M$; then the group $\operatorname{Diff}_{c}(M)$ of all smooth diffeomorphisms $\psi\colon M\to M$ such that $\psi(x)=x$ off some compact set is a Lie group. Moreover, $\operatorname{Diff}_{K}(M):=\{\psi\in\operatorname{Diff}_{c}(M)\colon(\forall x\in M\setminus K)\;\psi(x)=x\}$ is a Lie subgroup of $\operatorname{Diff}_{c}(M)$ for each compact subset $K\subseteq M$ (cf. [17] and [12]). If $F$ is a Lie group with neutral element $e$ and $k\in{\mathbb{N}}_{0}\cup\{\infty\}$, then the “test tunction group” $C^{k}_{c}(M,F)$ is a Lie group, comprising all $C^{k}$-maps $\gamma\colon M\to F$ such that $\gamma(x)=e$ for $x\in M$ off some compact set $K$ [4, 8, 10, 11]. For fixed $K$,

[TABLE]

is a Lie subgroup of $C^{k}_{c}(M,F)$. If $M$ with $\mbox{dim}(M)>0$ is $\sigma$-compact, non-compact and $K_{1}\subseteq K_{2}\subseteq\cdots$ is an exhaustion of $M$ by compact sets, then

[TABLE]

is an ascending sequence of Lie groups and

[TABLE]

holds as a Lie group and as a topological group, but not as a smooth manifold (see [6]), nor as a topological space (see [20]). Likewise,

[TABLE]

holds as a Lie group and topological group but neither as a smooth manifold nor as a topological space if $M$ and the $K_{n}$ are as before and $F$ is a non-discrete finite-dimensional Lie group.

For ascending unions $G=\bigcup_{i\in I}G_{i}$ with uncountable index sets $I$, hardly anything is known concerning direct limit properties: neither general results, nor results concerning concrete examples. Notably, in the case of a paracompact finite-dimensional smooth manifold $M$ which fails to be $\sigma$-compact, it is an open problem whether

[TABLE]

holds as a Lie group, and whether

[TABLE]

holds as a Lie group for each finite-dimensional Lie group $F$ (see Problem 17.13 in the extended preprint version arxiv.math/0606078 of [6]).

In this note, we explain that the situation improves if we restrict attention to the class of Lie groups which are regular (in the sense recalled in Section 1). Regularity is a key concept in infinite-dimensional Lie theory going back to Milnor [18] (in the case of sequentially complete modelling spaces); see [10], [12], and [19] for further information (cf. also [16]). Up to now, no examples of non-regular Lie groups modelled on sequentially complete (or Mackey complete) locally convex spaces have been found. Therefore, a focus on regular Lie groups hardly poses a restriction.

We consider a Lie group $G$ modelled on a locally convex spaces which is the union $G=\bigcup_{i\in I}G_{i}$ of a directed family $(G_{i})_{i\in I}$ of such Lie groups, as described above. We assume that $G$ “has a direct limit chart”, i.e., $G$ admits a chart around its neutral element $e$ which is built up from compatible charts of the Lie groups $G_{i}$ (see Section 1 for details). Then the following holds:

Theorem A. If the Lie group $G$ is an ascending union $G=\bigcup_{i\in I}G_{i}$ of Lie groups $G_{i}$ and $G$ admits a direct limit chart, then a group homomorphism $f\colon G\to H$ to a regular Lie group $H$ is smooth if and only if $f|_{G_{i}}\colon G_{i}\to H$ is smooth for each $i\in I$.

We record an immediate consequence:

Theorem B. If a regular Lie group $G$ is an ascending union $G=\bigcup_{i\in I}G_{i}$ of regular Lie groups $G_{i}$ and $G$ admits a direct limit chart, then $G={\displaystyle\lim_{\longrightarrow}}\,G_{i}$ in the category of regular Lie groups and smooth group homomorphisms. $\,\square$

Now consider a family $(G_{j})_{j\in J}$ of Lie groups $G_{j}$, where $G_{j}$ is modelled on the locally convex space $E_{j}$. Then the subgroup

[TABLE]

of $\prod_{j\in J}G_{j}$ can be made a Lie group in a natural way, modelled on the locally convex direct sum $\bigoplus_{j\in J}E_{j}$ (see [11], where the notation $\prod_{j\in J}^{*}G_{j}$ is used for $\bigoplus_{j\in J}G_{j}$). The Lie group $\bigoplus_{j\in J}G_{j}$ is called the weak direct product of the family $(G_{j})_{j\in J}$. If $\phi_{j}\colon V_{j}\to W_{j}\subseteq E_{j}$ is a chart of $G_{j}$ with $e\in V_{j}$ and $\phi(e)=0$, then $\bigoplus_{j\in J}V_{j}:=G\cap\prod_{j\in J}V_{j}$ is an open identity neighbourhood in $G$ and the map

[TABLE]

is a chart for $G$ (cf. [11]). Among other things, weak direct products are useful tools for the study of diffeomorphism groups and test functions groups (cf. [6], [8], [11], and [12]).

Theorem C (Direct limit properties of prime examples).

• (a)

If $G:=\bigoplus_{j\in J}G_{j}$ is the weak direct product of a family $(G_{j})_{j\in J}$ of Lie groups, then a group homomorphism $f\colon G\to H$ to a regular Lie group $H$ is smooth if and only if $f|_{G_{j}}\colon G_{j}\to H$ is smooth for each $j\in J$. Let ${\mathcal{F}}$ be the set of all finite subsets of $J$ $($directed via inclusion$)$. If $G$ and each $G_{j}$ is regular, then $G={\displaystyle\lim_{\longrightarrow}}_{\Phi\in{\mathcal{F}}}\prod_{j\in\Phi}G_{j}$ holds in the category of regular Lie groups and smooth group homomorphisms.

Now let $M$ be a paracompact, finite-dimensional smooth manifold. Let ${\mathcal{K}}$ be the set of compact subsets of $M$, directed via inclusion. Then we have:

• (b)

$\operatorname{Diff}_{c}(M)={\displaystyle\lim_{\longrightarrow}}_{K\in{\mathcal{K}}}\operatorname{Diff}_{K}(M)$ holds in the category of regular Lie groups and smooth group homomorphisms.

• (c)

If $F$ is a Lie group and $k\in{\mathbb{N}}_{0}\cup\{\infty\}$, then a group homomorphism $f\colon C^{k}_{c}(M,F)\to H$ to a regular Lie group $H$ is smooth if and only if $f|_{C^{k}_{c}(M,F)}\colon C^{k}_{K}(M,F)\to H$ is smooth for each $K\in{\mathcal{K}}$. If $C^{k}_{c}(M,F)$ and each $C^{k}_{K}(M,F)$ is regular, then $C^{k}_{c}(M,F)={\displaystyle\lim_{\longrightarrow}}\,C^{k}_{K}(M,F)$ in the category of regular Lie groups and smooth group homomorphisms.

Criteria ensuring that $\bigoplus_{j\in J}G_{j}$ is regular are known (see [10]), and recalled in Section 3. Notably, each weak direct product of finite-dimensional Lie groups (or Banach-Lie groups) is regular. Criteria ensuring that $C^{k}_{c}(M,F)$ and its Lie subgroups $C^{k}_{K}(M,F)$ are regular are known as well (see [10]) and recalled in Section 3. Notably, $C^{k}_{c}(M,F)$ and $C^{k}_{K}(M,F)$ are regular whenever $F$ is a finite-dimensional Lie group or a Banach-Lie group.

Let $G$ be a Lie group with tangent space ${\mathfrak{g}}:=L(G):=T_{e}(G)$ at the neutral element. We say that $G$ has a smooth exponential function if there exists a (necessarily unique) smooth function $\exp_{G}\colon{\mathfrak{g}}\to G$ such that $\exp_{G}((t+s)v)=\exp_{G}(tv)\exp_{G}(sv)$ for all $v\in{\mathfrak{g}}$ and $s,t\in{\mathbb{R}}$, and \frac{d}{dt}\big{|}_{t=0}\exp_{G}(tv)=v for all $v\in{\mathfrak{g}}$. If $G$ has a smooth exponential function and $\exp_{G}$ is a local $C^{\infty}$-diffeomorphism at [math], then $G$ is called locally exponential. Every regular Lie group has a smooth exponential function. Many important examples of Lie groups are locally exponential (for example, all Lie groups modelled on Banach spaces), but there also are many important examples which fail to be locally exponential (like the diffeomorphism group $\operatorname{Diff}({\mathbb{S}})$ of the circle), cf. [18], [12], [19]. We record an easy observation:

Theorem D. Let $G$ be a locally exponential Lie group which is the ascending union $G=\bigcup_{i\in I}G_{i}$ of Lie groups $G_{i}$ with smooth exponential function. Assume that $L(G)={\displaystyle\lim_{\longrightarrow}}\,L(G_{i})$ as a locally convex space with limit maps $L(\eta_{i}):=T_{e}(\eta_{i})$, for the direct system $((L(G_{i})_{i\in I},(L(\eta_{ji})_{i\leq j})$ of locally convex spaces. Then $G={\displaystyle\lim_{\longrightarrow}}\,G_{i}$ in the category of Lie groups with smooth exponential function and smooth group homomorphisms.

Theorem D was the starting point for this work. It is a general strategy of infinite-dimensional Lie theory that one-parameter groups $t\mapsto\exp_{G}(tv)$ should be replaced with general smooth curves $\gamma\colon[0,1]\to G$ with $\gamma(0)=e$ in classical proofs, which are represented on the Lie algebra level not by a single element $v\in{\mathfrak{g}}$, but by an element in $C^{\infty}([0,1],{\mathfrak{g}})$ (the left logarithmic derivative of $\gamma$). Following this strategy, Theorems A and B can be considered as adaptations of Theorem D when $G$ fails to be locally exponential, so that smoothness of $f\circ\exp_{G}$ cannot guarantee smoothness of $f\colon G\to H$.

Acknowledgement. The research was supported by Deutsche Forschungsgemeinschaft (DFG), project GL 357/9-1.

1 Preliminaries and Notation

Write ${\mathbb{N}}:=\{1,2,\ldots,\}$ and ${\mathbb{N}}_{0}:={\mathbb{N}}\cup\{0\}$.

1.1 We shall work with $C^{k}$-maps between open subsets of locally convex topological vector spaces as introduced by Bastiani [2] for $k\in{\mathbb{N}}_{0}\cup\{\infty\}$ (a setting which is also known as Keller’s $C^{k}_{c}$-theory). More generally, consider locally convex spaces $E$ and $F$ and a subset $U\subseteq E$ which is locally convex in the sense that each point of $U$ has a convex neighbourhood in $U$. Following [12], we say that a map $f\colon U\to F$ is $C^{k}$ if $f$ is continuous and there exist continuous maps

[TABLE]

for all $j\in{\mathbb{N}}$ with $j\leq k$ such that the following iterated directional derivatives exist and are given by the right hand side:

[TABLE]

for all $x$ in the interior of $U$ and $y_{1},\ldots,y_{j}\in E$. As usual, $C^{\infty}$-maps are also called smooth. We refer to [3] and [12] for outlines of the corresponding concepts of manifolds and Lie groups modelled on arbitrary locally convex spaces (which need not satisfiy any completeness conditions). All Lie groups and manifolds we consider may be infinite-dimensional, unless the contrary is stated. Compare [18] for the case of sequentially complete modelling spaces, [13] for differential calculus on Fréchet spaces; see also [17]. If $U$ is an open subset of a locally convex space $E$, we identify its tangent bundle $TU$ with $U\times E$, as usual. If $M$ is a smooth manifold, we let $\pi_{TM}\colon TM\to M$ be the bundle projection. If $f\colon M\to U$ is a $C^{1}$-map, we write $df\colon TM\to E$ for the second component of $Tf\colon TM\to U\times E$; thus $Tf=(f\circ\pi_{TM},df)$.

1.2 Every Lie group $G$ acts smoothly on its tangent bundle from the left via

[TABLE]

where $\lambda_{g}\colon G\to G$, $x\mapsto gx$ is left translation by $g$. Write ${\mathfrak{g}}:=L(G):=T_{e}(G)$ for the tangent space of $G$ at the neutral element. If $\gamma\colon[0,1]\to G$ is a $C^{1}$-curve, we define its left logarithmic derivative as

[TABLE]

where $\gamma^{\prime}(t):=T\gamma(t,1)$. Endow $C([0,1],{\mathfrak{g}})$ with the compact-open topology. Given $k\in{\mathbb{N}}_{0}\cup\{\infty\}$, we equip $C^{k}([0,1],{\mathfrak{g}})$ with the so-called compact-open $C^{k}$-topology (the initial topology with respect to the maps $C^{k}([0,1],{\mathfrak{g}})\to C([0,1],{\mathfrak{g}})$, $\gamma\mapsto\gamma^{(j)}$ taking $\gamma$ to its $j$th derivative for all $j\in{\mathbb{N}}_{0}$ with $j\leq k$). If each $\gamma\in C^{k}([0,1],{\mathfrak{g}})$ arises as $\gamma=\delta(\eta)$ for a (necessarily unique) $C^{k+1}$-map $\eta\colon[0,1]\to G$ with $\eta(0)=e$ and the map

[TABLE]

is smooth, then $G$ is called $C^{k}$-regular. We mention that $C^{k}$-regularity implies $C^{\ell}$-regularity for all $\ell\in{\mathbb{N}}\cup\{\infty\}$ such that $\ell\geq k$. If $G$ is $C^{\infty}$-regular (the weakest regularity property), we simply say that $G$ is regular. See [18], [10], [12], and [19] for further information (cf. also [16]).

1.3 Consider a Lie group $G$ which is the union of a directed system $(G_{i})_{i\in I}$ of Lie groups, as described in the introduction, with inclusion maps $\eta_{i}\colon G_{i}\to G$ for $i\in I$ and $\eta_{ji}\colon G_{i}\to G_{j}$ for $i\leq j$ in $I$. Let $E$ be the modelling space of $G$ and $E_{i}$ be the modelling space of $G_{i}$, for $i\in I$. We say that a chart $\phi\colon V\to W\subseteq E$ of the smooth manifold $G$ is a direct limit chart if $e\in V$ and the following holds:

• (a)

$E=\bigcup_{i\in I}E_{i}$ and $E_{i}\subseteq E_{j}$ for $i\leq j$ in $I$; moreover the inclusion maps $\lambda_{i}\colon E_{i}\to E$ and $\lambda_{ji}\colon E_{i}\to E_{j}$ are continuous linear and $E$, with the maps $\lambda_{i}$, is the direct limit locally convex space $E={\displaystyle\lim_{\longrightarrow}}\,E_{i}$ of the direct system $((E_{i})_{i\in I},(\lambda_{ji})_{i\leq j})$ of locally convex spaces.

• (b)

There are open $e$-neighbourhoods $V_{i}\subseteq G_{i}$ and open [math]-neighbourhoods $W_{i}\subseteq E_{i}$ such that $\phi(V_{i})=W_{i}$ and $\phi_{i}:=\phi|_{V_{i}}\colon V_{i}\to W_{i}$ is a chart for $G_{i}$, for each $i\in I$, and moreover $V=\bigcup_{i\in I}V_{i}$ and $V_{i}\subseteq V_{j}$ (and hence also $W_{i}\subseteq W_{j})$ for all $i\leq j$ in $I$.

Then the $L(\eta_{i}):=T_{e}(\eta_{i})$ and $L(\eta_{ji}):=T_{e}(\eta_{ji})$ are injective and $L(G)$, with maps $L(\eta_{i})$, is the locally convex direct limit of $((L(G_{i}))_{i\in I},(L(\eta_{ji}))_{i\leq j})$. When convenient, we identify $L(G)$ with $E$ and $L(G_{i})$ with $E_{i}$ using the isomorphisms $d\phi|_{L(G)}$ and $d\phi_{i}|_{L(G_{i})}$.

See [7] for generalities concerning direct limits of Lie groups and related topics. In the case of ascending sequences $G_{1}\subseteq G_{2}\subseteq\cdots$, direct limit charts were defined and exploited in [6]. The case of uncountable index sets was considered in [9].

2 Proof of Theorem A

To prove Theorem A, we write ${\mathfrak{g}}_{i}:=L(G_{i})$ for $i\in I$ and abbreviate ${\mathfrak{g}}:=L(G)$ and ${\mathfrak{h}}:=L(H)$. For $i\in I$, let $\eta_{i}\colon G_{i}\to G$ be the inclusion map and for $i\leq j$ in $I$, let $\eta_{ji}\colon G_{i}\to G_{j}$ be the inclusion map; all of these are group homomorphisms and smooth. Abbreviate $\psi_{i}:=L(f|_{G_{i}})\colon{\mathfrak{g}}_{i}\to{\mathfrak{h}}$. Since $f|_{G_{i}}=f|_{G_{j}}\circ\eta_{ji}$, we have $\psi_{i}=\psi_{j}\circ L(\eta_{ji})$ for all $i\leq j$ in $I$. As ${\mathfrak{g}}={\displaystyle\lim_{\longrightarrow}}\,{\mathfrak{g}}_{i}$ as a locally convex space, we deduce that there is a unique continuous linear map $\psi\colon{\mathfrak{g}}\to{\mathfrak{h}}$ such that

[TABLE]

Let $\phi\colon V\to W\subseteq{\mathfrak{g}}$ be a direct limit chart for $G$. Thus $V=\bigcup_{i\in I}V_{i}$ with open $e$-neighbourhoods $V_{i}\subseteq G_{i}$ and $W=\bigcup_{i\in I}W_{i}$ with open [math]-neighbourhoods $W_{i}\subseteq{\mathfrak{g}}_{i}$ such that $W_{i}\subseteq W_{j}$ and $V_{i}\subseteq V_{j}$ for all $i\leq j$ in $I$ (identifying ${\mathfrak{g}}_{i}$ with $L(\eta_{i})({\mathfrak{g}}_{i})\subseteq{\mathfrak{g}}$ as a vector space) and $\phi|_{V_{i}}\colon V_{i}\to W_{i}$ is a chart for $G_{i}$. We let $Q\subseteq W$ be an open [math]-neighbourhood which is balanced (i.e., $[{-1},1]Q\subseteq Q$). After replacing $W$ with $Q$, the set $V$ with $\phi^{-1}(Q)$, the set $W_{i}$ with $W_{i}\cap Q$ and $V_{i}$ with $\phi^{-1}(W_{i}\cap Q)$, we may assume that $W$ is balanced. Then

[TABLE]

In fact, let $w\in W$. For each $t\in[0,1]$, we have $tw\in W$ and find $i(t)\in I$ such that $tw\in W_{i(t)}$. As the map $[0,1]\to{\mathfrak{g}}_{i(t)}$, $s\mapsto sw$ is continuous and $W_{i(t)}\subseteq{\mathfrak{g}}_{i(t)}$ is open, $t$ has a neighbourhood $U(t)\subseteq[0,1]$ such that $U(t)w\subseteq W_{i(t)}$. By compactness of $[0,1]$, there exist $t_{1},\ldots,t_{n}\in[0,1]$ such that $[0,1]=U(t_{1})\cup\cdots\cup U(t_{n})$. Let $j\in I$ such that $i(t_{k})\leq j$ for all $k\in\{1,\ldots,n\}$. Then $U(t_{k})w\subseteq W_{i(t_{k})}\subseteq W_{j}$ for each $k$ and thus $[0,1]w\subseteq W_{j}$ (establishing (2)).

Given $x\in V$, we define a smooth curve $\gamma_{x}\colon[0,1]\to G$ with $\gamma_{x}(0)=e$ and $\gamma_{x}(1)=x$ via

[TABLE]

Now $[0,1]\phi(x)\subseteq W_{j}$ for some $j\in I$, by (2). Then $\gamma_{x}(t)=(\phi|_{V_{j}}^{W_{j}})^{-1}(t\phi(x))\in G_{j}$ for all $t\in[0,1]$, and this is a smooth $G_{j}$-valued function of $t$. Thus

[TABLE]

For $x$ and $j$ as before, we obtain

[TABLE]

Note that $h\colon V\times[0,1]\to W\subseteq{\mathfrak{g}}$, $(x,t)\mapsto t\phi(x)$ is a smooth map, whence also

[TABLE]

is smooth (see [1, Theorem B]). Now $C^{\infty}([0,1],V)$ is an open identity neighbourhood in $C^{\infty}([0,1],G)$ and the map

[TABLE]

is a $C^{\infty}$-diffeomorphism by the construction of the Lie group structure on $C^{\infty}([0,1],G)$ (see, e.g., [12]; cf. [4]). Now $\delta\colon C^{\infty}([0,1],G)\to C^{\infty}([0,1],{\mathfrak{g}})$, $\gamma\mapsto\delta(\gamma)$ is smooth (see [10, Lemma 2.1]) and $C^{\infty}([0,1],{\mathfrak{g}})\to C^{\infty}([0,1],{\mathfrak{h}})$, $\gamma\mapsto\psi\circ\gamma$ is smooth (and continuous linear). Since $\phi^{-1}\circ h^{\vee}(x)=\gamma_{x}$ for each $x\in V$, using (3) we see that

[TABLE]

is smooth. Like every homomorphism between Lie groups which is smooth on an open identity neighbourhood, $f$ is smooth. $\square$

3 Proof of Theorem C

Proof of (a). The map $\phi$ from (1) is a direct limit chart for $\bigoplus_{j\in J}G_{j}=\bigcup_{\Phi\in{\mathcal{F}}}\prod_{j\in\Phi}G_{j}$, as $\bigoplus_{j\in J}V_{j}=\bigcup_{\Phi\in{\mathcal{F}}}\prod_{j\in\Phi}V_{j}$, $\bigoplus_{j\in J}W_{j}=\bigcup_{\Phi\in{\mathcal{F}}}\prod_{j\in\Phi}W_{j}$ and $\phi$ restricts to the chart

[TABLE]

of $\prod_{j\in\Phi}G_{j}$ around $e$, for each $\Phi\in{\mathcal{F}}$. Also note that the restriction of $f$ to $\prod_{j\in\Phi}G_{j}$ with $\Phi=\{j_{1},\ldots,j_{n}\}$ is the map

[TABLE]

which is smooth if and only if $f_{j}$ is smooth for all $j\in\Phi$. Therefore all assertions follow from Theorems A and B.

(b) Write $\pi_{TM}\colon TM\to M$ for the bundle projection and let $\Gamma_{c}(TM)$ be the locally convex space of all compactly supported smooth vector fields on $M$. Given a compact set $K\subseteq M$, write $\Gamma_{K}(TM)$ for the Fréchet space of all smooth vector fields $X\colon M\to TM$ which are supported in $K$. Let $\Sigma\colon U\to M$ be a local addition for $M$, i.e., a smooth map on an open neighbourhood $U\subseteq TM$ of $0_{M}:=\{0_{p}\in T_{p}M\colon p\in M\}$ such that $\Sigma(0_{p})=p$ for all $p\in M$ and moreover the map

[TABLE]

has open image and is a $C^{\infty}$-diffeomorphism onto its image.111It is well-known that such local additions always exist; one can take the Riemannian exponential map for a Riemannian metric on $M$ and restrict it to a suitable open set $U$. We can (and shall) assume, moreover, that $T\Sigma|_{U\cap T_{p}M}(0,\cdot)=\operatorname{id}_{T_{p}M}$ for all $p\in M$. There is an open subset

[TABLE]

containing $0_{M}$ such that $\Sigma\circ X\in\operatorname{Diff}_{c}(M)$ for all $X\in\Omega$ and the map

[TABLE]

has open image $\Omega^{\prime}$ and is a $C^{\infty}$-diffeomorphism onto its image. Then $\Omega_{K}:=\Omega\cap\Gamma_{K}(TM)$ is an open subset of $\Gamma_{K}(TM)$ and the restriction $\phi_{K}$ to $\Omega_{K}$ has open image $\Omega_{K}^{\prime}=\Omega^{\prime}\cap\operatorname{Diff}_{K}(M)$ and is a $C^{\infty}$-diffeomorphism onto the latter. Identifying $\Gamma_{c}(TM)$ with $T_{\operatorname{id}_{M}}(\operatorname{Diff}_{c}(M))$ by means of $T\phi(0,\cdot)$ and $\Gamma_{K}(TM)$ with $T_{\operatorname{id}_{M}}(\operatorname{Diff}_{K}(M))$ by means of $T\phi_{K}(0,\cdot)$, the inclusion map $\operatorname{Diff}_{K}(M)\to\operatorname{Diff}_{c}(M)$ has the inclusion map $\Gamma_{K}(TM)\to\Gamma_{c}(TM)$ as its tangent map at $\operatorname{id}_{M}$. Since

[TABLE]

as a locally convex space, we see that $\phi^{-1}$ is a direct limit chart for $\operatorname{Diff}_{c}(M)$. As $\operatorname{Diff}_{c}(M)$ and the Lie groups $\operatorname{Diff}_{K}(M)$ are regular (see, e.g., [11]), we can apply Theorems A and B.

(c) Let $E$ be the modelling space of $F$ and $\phi\colon V\to W\subseteq E$ be a chart for $F$ such that $e\in V$, $\phi(e)=0$ and $d\phi|_{L(F)}=\operatorname{id}_{L(F)}$. Then $C^{k}_{c}(M,V)$ is an open identity-neighbourhood in $C^{k}_{c}(M,F)$; moreover, $C^{k}_{c}(M,W)$ is an open [math]-neighbourhood in the locally convex space $C^{k}_{c}(M,E)$ and

[TABLE]

is a chart for $C^{k}_{c}(M,F)$. For each compact subset $K\subseteq M$, this chart restricts to the chart

[TABLE]

of $C^{k}_{K}(M,F)$. We now identify $T_{e}(C^{k}_{c}(M,F))$ with $C^{k}_{c}(M,E)$ by means of the restriction of $dC^{k}_{c}(M,\phi)$ to an isomorphism between the latter. Likewise, using $dC^{k}_{K}(M,\phi)$ we identify $T_{e}(C^{k}_{K}(M,F))$ with $C^{k}_{K}(M,E)$. Then the tangent map of the inclusion map $C^{k}_{K}(M,F)\to C^{k}_{c}(M,F)$ is the inclusion map $C^{k}_{K}(M,E)\to C^{k}_{c}(M,E)$. Since

[TABLE]

as a locally convex space, we see that $C^{k}_{c}(M,\phi)$ is a direct limit chart for $C^{k}_{c}(M,F)$. Thus Theorem A applies and if $C^{k}_{c}(M,F)$ and each of the Lie groups $C^{k}_{K}(M,F)$ is assumed regular, then also Theorem B applies. $\,\square$

Remark. The regularity requirements in Theorem C (a) and (c) are satisfied in the following situations:

(a) If $k\in{\mathbb{N}}_{0}$ and $G_{j}$ is $C^{k}$-regular for each $j\in J$, then $\bigoplus_{j\in J}G_{j}$ is a $C^{k+1}$-regular Lie group (see [10, Corollary 13.6]) and hence regular.

(b) If $F$ is a regular Lie group, then $C^{k}_{K}(M,F)$ is regular (this can be shown like [10, Proposition 12.1]).

(c) If $F$ is $C^{k}$-regular for some $k\in{\mathbb{N}}_{0}$, then $C^{k}_{c}(M,F)$ is $C^{k+1}$-regular (see [10, Proposition 12.3]) and hence regular.

Classes of Lie groups which are $C^{k}$-regular for finite $k$ can be found in [10] and [11].222All Lie groups which are measurably regular are $C^{0}$-regular. Notably, every Banach-Lie group is $C^{0}$-regular (and hence each finite-dimensional Lie group). Moreover, all direct limits of ascending sequences of finite-dimensional Lie groups are $C^{0}$-regular, and also all of the Lie groups $\operatorname{Diff}_{c}(M)$ and $\operatorname{Diff}_{K}(M)$.

4 Proof of Theorem D

Let $H$ be a Lie group with smooth exponential function and $f\colon G\to H$ be a homomorphism of groups such that $f|_{G_{i}}\colon G_{i}\to H$ is smooth for each $i\in I$. Then $L(f|_{G_{i}})=L(f|_{G_{j}}\circ\eta_{ji})=L(f|_{G_{j}})\circ L(\eta_{ji})$ for all $i\leq j$ in $I$. By the universal property of the direct limit, there is a unique continuous linear map $\psi\colon L(G)\to L(H)$ such that $\psi\circ L(\eta_{i})=L(f|_{G_{i}})$ for all $i\in I$. For each $i\in I$,

[TABLE]

and thus $\exp_{H}(\psi(v))=f(\exp_{G}(v))$ for all $v\in L(\eta_{i})(L(G_{i}))$. As $L(G)=\bigcup_{i\in I}L(\eta_{i})(L(G_{i}))$, we deduce that $f\circ\exp_{G}=\exp_{H}\circ\psi$, which is a smooth map. As $\exp_{G}$ is a local $C^{\infty}$-diffeomorphism at [math], we deduce that $f$ is smooth on some open identity neighbourhood in $G$. Since $f$ is a homomorphism, smoothness of $f$ follows. $\,\square$.