The Lie group of bisections of a Lie groupoid

Abstract

In this article, we endow the group of bisections of a Lie groupoid with compact base with a natural locally convex Lie group structure. Moreover, we develop thoroughly the connection to the algebra of sections of the associated Lie algebroid and show for a large class of Lie groupoids that their groups of bisections are regular in the sense of Milnor.

Appendix: Locally convex manifolds and spaces of smooth maps

In this appendix we collect the necessary background on the theory of manifolds that are modelled on locally convex spaces and how spaces of smooth maps can be equipped with such a structure. Let us first recall some basic facts concerning differential calculus in locally convex spaces. We follow [3, 8].

Definition 7.1

Let $E, F$ be locally convex spaces, $U \subseteq E$ be an open subset, $f :U \rightarrow F$ a map and $r \in \mathbb {N}_{0} \cup \{\infty \}$. If it exists, we define for $(x,h) \in U \times E$ the directional derivative

$$\begin{aligned} df(x,h) := D_h f(x) := \lim _{t\rightarrow 0} t^{-1} (f(x+th) -f(x)). \end{aligned}$$

We say that $f$ is $C^r$ if the iterated directional derivatives

$$\begin{aligned} d^{(k)}f (x,y_1,\ldots , y_k) := (D_{y_k} D_{y_{k-1}} \cdots D_{y_1} f) (x) \end{aligned}$$

exist for all $k \in \mathbb {N}_0$ such that $k \le r$, $x \in U$ and $y_1,\ldots , y_k \in E$ and define continuous maps $d^{(k)} f :U \times E^k \rightarrow F$. If $f$ is $C^\infty$ it is also called smooth. We abbreviate $df := d^{(1)} f$.

From this definition of smooth map there is an associated concept of locally convex manifold, i.e. a Hausdorff space that is locally homeomorphic to open subsets of locally convex spaces with smooth chart changes. See [8, 20, 29] for more details.$\square$

Definition 7.2

(Differentials on non-open sets)

• (a) A subset $U$ of a locally convex space $E$ is called locally convex if every $x \in U$ has a convex neighbourhood $V$ in $U$.

• (b) Let $U\subseteq E$ be a locally convex subset with dense interior and $F$ a locally convex space. A continuous mapping $f :U \rightarrow F$ is called $C^r$ if $f|_{U^\circ } :U^\circ \rightarrow F$ is $C^r$ and each of the maps $d^{(k)} (f|_{U^\circ }) :U^\circ \times E^k \rightarrow F$ admits a continuous extension $d^{(k)}f :U \times E^k \rightarrow F$ (which is then necessarily unique). Analogously, we say that a continuous map $g :U \rightarrow M$ to a smooth manifold $M$ is of class $C^r$ if the tangent maps $T^{k} (f|_{U^\circ }) :U^\circ \times E^{2^k-1} \rightarrow T^kM$ exist and admit a continuous extension $T^{k}f :U \times E^{2^k-1} \rightarrow T^kM$. Note that we defined $C^k$-mappings on locally convex sets with dense interior in two ways for topological vector spaces (when viewed as manifolds). However, by [8], Lemma 1.14] both conditions yield the same class of mappings. If $U \subseteq \mathbb {R}$ and $g$ is $C^{1}$, we obtain a continuous map $g' :U \rightarrow TM, g'(x) := T_x g(1)$. We shall write $\frac{\partial }{\partial x}g(x) := g' (x)$.$\square$

Definition 7.3

Let $M$ be a smooth manifold. Then $M$ is called Banach (or Fréchet) manifold if all its modelling spaces are Banach (or Fréchet) spaces. The manifold $M$ is called locally metrisable if the underlying topological space is locally metrisable (equivalently if all modelling spaces of $M$ are metrisable). It is called metrisable if it is metrisable as a topological space (equivalently locally metrisable and paracompact).$\square$

Definition 7.4

Suppose $M$ is a smooth manifold. Then a local addition on $M$ is a smooth map , defined on an open neighbourhood $U$ of the submanifold $M\subseteq TM$ such that

• (a) $\pi \times {{\mathrm{\Sigma }}}:U\rightarrow M\times M$, $v\mapsto (\pi (v),{{\mathrm{\Sigma }}}(v))$ is a diffeomorphism onto an open neighbourhood of the diagonal $\Delta M\subseteq M\times M$ and

• (b) ${{\mathrm{\Sigma }}}(0_{m})=m$ for all $m\in M$.

We say that $M$ admits a local addition if there exist a local addition on $M$.$\square$

Lemma 7.5

(cf. [17], 10.11]) Suppose that is a local addition on $M$ and that $\tau :T(TM)\rightarrow T(TM)$ is the canonical flip on $T(TM)$. Then is a local addition on $TM$. In particular$,$ $TM$ admits a local addition if $M$ does so.

Proof

Let $0_{M}:M\rightarrow TM$ denote the zero-section of $\pi _{M}:TM\rightarrow M$.

The diffeomorphism $\tau :T(TM)\rightarrow T(TM)$ is locally given by $(m,x,y,z)\mapsto (m,y,x,z)$ and makes the diagrams

commute [17], 1.19]. Then ${{\mathrm{\Sigma }}}\circ 0_{M}=\mathrm{id}_{M}$ implies that $T {{\mathrm{\Sigma }}}$ is defined on the open neighbourhood $TU$ of $T0_{M}(TM)$ in $T(TM)$ and satisfies $T {{\mathrm{\Sigma }}}\circ T0_{M} =\mathrm{id}_{TM}$. This implies that $T {{\mathrm{\Sigma }}}\circ \tau$ is defined on the open neighbourhood $\tau (T U)$ of $0_{TM}(TM)$. It satisfies $T {{\mathrm{\Sigma }}}\circ \tau \circ 0_{TM}=\mathrm{id}_{TM}$ and thus Definition 7.4(b) by construction. Moreover, if $\pi _{M}\times {{\mathrm{\Sigma }}}$ is a diffeomorphism from $U$ onto , then $T(\pi _{M}\times {{\mathrm{\Sigma }}})=(T \pi _{M}\times T{{\mathrm{\Sigma }}})$ is a diffeomorphism from $TU$ onto . Thus $(\pi _{TM}\times T {{\mathrm{\Sigma }}}\circ \tau )$ is a diffeomorphism from $\tau (TU)$ onto $TV$. This establishes Definition 7.4(a). $\square$

Definition 7.6

Let $M,N$ be smooth manifolds. Then we endow the smooth maps $C^{\infty }(M,N)$ with the initial topology with respect to

$$\begin{aligned} C^{\infty }(M,N)\hookrightarrow \prod _{k\in \mathbb {N}_{0}}C^0(T^{k}M,T^{k}N)_{c.o.},\quad f\mapsto (T^{k}f)_{k\in \mathbb {N}_{0} }, \end{aligned}$$

where $C^0(T^{k}M,T^{k}N)_{c.o.}$ denotes the space of continuous functions endowed with the compact-open topology.$\square$

From [29], Proposition 7.3 and Theorem 5.14] we recall the following result.

Theorem 7.7

Let $E\rightarrow M$ be a vector bundle over the compact manifold $M$ such that the fibres are locally convex spaces. Then the space of sections $\Gamma (M\xleftarrow {}E)$ is a closed subspace of $C^{\infty }(M,E)$ and a locally convex space with respect to point-wise addition and scalar multiplication. If the fibres of $E\rightarrow M$ are metrisable$,$ then so is $\Gamma (M\xleftarrow {} E)$ and if the fibres are Fréchet spaces$,$ then so is $\Gamma (M\xleftarrow {} E).$

Our main tool will be the following excerpt from [29], Theorem 7.6]. Note that if we assume that $N$ is modelled on a Fréchet space, then the space $C^\infty (M,N)$ will also be modelled on such a space. Then the differential calculus used in this paper and the convenient setting coincide and Theorem 7.8 can be deduced from [13], Chapter 42]. Moreover, Theorem 7.9 is then a consequence of [13], Theorem 42.17]. As we have in general not assumed any completeness conditions, the theorems stated in this appendix are slightly more general than the versions available in the literature.

Theorem 7.8

Let $M$ be a compact manifold and $N$ be a locally convex and locally metrisable manifold that admits a local addition $.$ Set $V:=(\pi \times {{\mathrm{\Sigma }}})(U),$ which is an open neighbourhood of the diagonal $\Delta N$ in $N\times N.$ For each $f\in C^{\infty }(M,N)$ we set

$$\begin{aligned} O_{f}:=\{g\in C^{\infty }(M,N)\mid (f(x),g(x))\in V \}. \end{aligned}$$

Then the following assertions hold.

• (a) The set $O_{f}$ contains $f,$ is open in $C^{\infty }(M,N)$ and the formula $(f(x),g(x))=(f(x),{{\mathrm{\Sigma }}}(\varphi _{f}(g)(m)))$ determines a homeomorphism

  $$\begin{aligned} \varphi _{f}:O_{f}\rightarrow \{h\in C^{\infty }(M,TN)\mid \pi (h(x))=f(x)\}\cong \Gamma (f^{*}(TN)) \end{aligned}$$

  from $O_{f}$ onto the open subset $\{h\in C^{\infty }(M,TN)\mid \pi (h(x))=f(x)\}\cap C^{\infty }(M,U)$ of $\Gamma (f^{*}(TN)).$

• (b) The family $(\varphi _{f}:O_{f}\rightarrow \varphi _{f}(O_{f}))_{f\in C^{\infty }(M,N)}$ is an atlas$,$ turning $C^{\infty }(M,N)$ into a smooth locally convex and locally metrisable manifold$.$

• (c) The manifold structure on $C^{\infty }(M,N)$ from (b) is independent of the choice of the local addition ${{\mathrm{\Sigma }}}$.

• (d) If $L$ is another locally convex and locally metrisable manifold$,$ then a map $f:L\times M\rightarrow N$ is smooth if and only if $\widehat{f}:L\rightarrow C^{\infty }(M,N)$ is smooth. In other words$,$

  $$\begin{aligned} C^{\infty }(L\times M,N)\rightarrow C^{\infty }(L,C^{\infty }(M,N)), \quad f\mapsto \widehat{f} \end{aligned}$$

  is a bijection $($which is even natural$).$

• (e) Let $M'$ be compact and $N'$ be locally metrisable such that $N'$ admits a local addition. If $\mu :M'\rightarrow M,$ $\nu :N\rightarrow N'$ are smooth$,$ then

  $$\begin{aligned} \nu _{*} \mu ^{*}:C^{\infty }(M,N)\rightarrow C^{\infty }(M',N'),\quad \gamma \mapsto \nu \circ \gamma \circ \mu \end{aligned}$$

  is smooth.

• (f) If $M'$ is another compact manifold$,$ then the composition map

  $$\begin{aligned} \circ :C^{\infty }(M',N)\times C^{\infty }(M,M')\rightarrow C^{\infty }(M,N),\quad (\gamma ,\eta )\mapsto \gamma \circ \eta \end{aligned}$$

  is smooth.

Theorem 7.9

Let $M$ be a compact manifold and $N$ be a locally convex and locally metrisable manifold that admits a local addition. There is an isomorphism of vector bundles

given by

$$\begin{aligned} \Phi _{M,N}:T C^{\infty }(M,N)\rightarrow C^{\infty }(M,TN),\quad [t\mapsto \eta (t)]\mapsto \left( m\mapsto [t\mapsto \eta ^\wedge (t,m)]\right) . \end{aligned}$$

Here we have identified tangent vectors in $C^{\infty }(M,N)$ with equivalence classes $[\eta ]$ of smooth curves $\eta :]-\varepsilon ,\varepsilon [\rightarrow C^{\infty }(M,N)$ for some $\varepsilon >0$. The isomorphism $\varphi _{M,N}$ is natural with respect to the morphisms from (e), i.e. the diagrams

commute. In particular$,$ $T_{f}C^{\infty }(M,N)$ is naturally isomorphic $($as a topological vector space$)$ to $\Gamma (f^{*}TN)$ and with respect to this isomorphism we have

$$\begin{aligned}&T_{f}(\mu ^{*}):\Gamma (f^{*}TN)\rightarrow {\Gamma ((f \circ \mu )^{*}TN)},\quad \sigma \mapsto \sigma \circ \mu \\&T_{f}(\nu _{*}):\Gamma (f^{*}TN)\rightarrow {\Gamma ((\nu \circ f)^{*}TN')},\quad \sigma \mapsto T \nu \circ \sigma . \end{aligned}$$

Proof

First note that $TN$ is also locally convex and locally metrisable and from Lemma 7.5 we infer that it also admits a local addition. Let ${{\mathrm{\Sigma }}}:TN \supseteq \Omega \rightarrow N$ be the local addition on $N$ and $\tau :T^2N \rightarrow T^2N$ be the canonical flip (cf. Lemma 7.5). Then $T{{\mathrm{\Sigma }}}\circ \tau$ is a local addition on ${ TN}$. Furthermore, $M$ is compact and thus Theorem 7.8 implies that $C^\infty (M,N)$, $TC^\infty (M,N)$ and $C^\infty (M,{ TN})$ are locally convex manifolds. We can now argue as in [17], 10.12] to see that the charts $(\varphi _{0\circ f})_{f\in C^\infty (M,N)}$ cover $C^\infty (M,TN)$. In fact, the charts $(\varphi _{0\circ f})_{f\in C^\infty (M,N)}$ are bundle trivialisations for $(\pi _{{ TN}})_* :C^\infty (M,TN) \rightarrow C^\infty (M,N)$ (see [17], 10.12 2. Claim]). The map $\Phi _{M,N}$ will be an isomorphism of vector bundles if we can show that it coincides fibre-wise with the isomorphism of vector bundles constructed in the proof of [17], Theorem 10.13]. Note that the proof of [17], Theorem 10.13] deals only with the case of a finite-dimensional target $N$. However, the local addition constructed in Lemma 7.5 allows us to copy the proof of [17], Theorem 10.13] almost verbatim.⁵ To prove that $\Phi _{M,N}$ is indeed of the claimed form, fix $f \in C^\infty (M,N)$. We will evaluate $\varphi _{0\circ f} \circ \Phi _{M,N}$ on the equivalence class $[t\mapsto c(t)]$ of a smooth curve $c :]-\varepsilon , \varepsilon [ \rightarrow C^\infty (M,N)$ with $c(0)=f$:

$$\begin{aligned} \varphi _{0\circ f} \circ \Phi _{M,N} ([t\mapsto c(t)])= & {} \varphi _{0\circ f} (m\mapsto [t\mapsto c^\wedge (t,m)])\nonumber \\ \!= & {} \! \left( m \mapsto (\pi _{T^2N}, T{{\mathrm{\Sigma }}}\circ \tau )^{-1} (0\circ f (m), [t\mapsto c^\wedge (t,m)])\right) \qquad \end{aligned}$$

(33)

By construction we obtain an element in $\Gamma ((0\circ f)^*T^2N) = \Gamma ((0\circ f)^* T^2N|N)$ where $T^2N|N$ is the restriction of the bundle $T^2N$ to the zero-section of $TN$. Consider the vertical lift $V_{TN} :TN \oplus TN \rightarrow V(TN)$ given locally by $V((x,a),(x,b)) := (x,a,0,b)$. Recall that $\tau$ and $V_{TN}$ are vector bundle isomorphisms. Now we argue as in [17], 10.12] to obtain a canonical isomorphism

$$\begin{aligned} I_f := (f^* (V_{TN})^{-1} \circ f^* \tau )_* :\Gamma ((0\circ f)^* T^2N|N) \rightarrow \Gamma (f^*TN) \oplus \Gamma (f^*TN). \end{aligned}$$

(Notice that there is some abuse in notation for $f^*\tau$, explained in detail in [17], 10.12]). We will now prove that $I_f$ is the inverse of $\varphi _{0\circ f} \circ \Phi _{M,N} \circ T\varphi _f^{-1}$. A computation in canonical coordinates for $T^2N$ yields

$$\begin{aligned} T\varphi _f ([t \mapsto c(t)])]= & {} (m \mapsto [t \mapsto (\pi _{TN}, {{\mathrm{\Sigma }}})^{-1} (f (m), c^\wedge (t,m))])) \nonumber \\= & {} (m \mapsto V_{TN}^{-1} \circ T(\pi _{TN},{{\mathrm{\Sigma }}})^{-1} (0\circ f,[t\mapsto c^\wedge (t,\cdot )]))\nonumber \\&\in \Gamma (f^*TN) \oplus \Gamma (f^*TN). \end{aligned}$$

(34)

Here we have used the identifications $C_f^\infty (M,TN\oplus TN) \cong \Gamma (f^*(TN\oplus TN)) \cong \Gamma (f^*TN) \oplus \Gamma (f^*TN)$. Since $\tau$ is an involution on $T^2N$ we can compute as follows

$$\begin{aligned}&I_f \circ \varphi _{0\circ f} \circ \Phi _{M,N} ([t\mapsto c(t)])\nonumber \\&\quad \mathop {=}\limits ^{(33)}\left( m \mapsto V_{TN}^{-1} \circ \tau \circ (\pi _{T^2N}, T{{\mathrm{\Sigma }}}\circ \tau )^{-1} (0\circ f (m), [t\mapsto c^\wedge (t,m)])\right) \nonumber \\&\quad =\left( m \mapsto V_{TN}^{-1} \circ (\pi _{T^2N} \circ \tau , T{{\mathrm{\Sigma }}}\circ \tau \circ \tau )^{-1} (0\circ f (m), [t\mapsto c^\wedge (t,m)])\right) \nonumber \\&\quad =(m \mapsto V_{TN}^{-1} \circ T(\pi _{TN} , {{\mathrm{\Sigma }}})^{-1} (0\circ f,[t\mapsto c^\wedge (t,\cdot )])). \end{aligned}$$

(35)

Hence the right-hand side of (35) coincides with the right-hand side of (34). Summing up the map $I_f$ is the inverse of $\Phi _{M,N}|_{T_f C^\infty (M,N)}^{C^\infty _{0\circ f} (M,TN)}$. We conclude that $\Phi _{M,N}^{-1}$ is the isomorphism of vector bundles described in [17], Theorem 10.13]. The statements concerning the tangent maps of the smooth maps discussed in Theorem 7.8(e) then follow from [17], Corollary 10.14]. $\square$