Manifolds of mappings on cartesian products

Given smooth manifolds $M_1,\ldots, M_n$ (which may have a boundary or corners), a smooth manifold $N$ modeled on locally convex spaces and $\alpha\in({\mathbb N}_0\cup\{\infty\})^n$, we consider the set $C^\alpha(M_1\times\cdots\times M_n,N)$ of all mappings $f\colon M_1\times\cdots\times M_n\to N$ which are $C^\alpha$ in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $\leq \alpha_j$ in the $j$th variable for $j\in\{1,\ldots, n\}$, in local charts. We show that $C^\alpha(M_1\times\cdots\times M_n,N)$ admits a canonical smooth manifold structure whenever each $M_j$ is compact and $N$ admits a local addition. The case of non-compact domains is also considered.


Introduction and statement of the results
As known from classical work by Eells [9], the set C ℓ (M, N ) of all C ℓ -maps f : M → N can be given a smooth Banach manifold structure for each ℓ ∈ N 0 , compact smooth manifold M and σ-compact finite-dimensional smooth manifold N . More generally, C ℓ (M, N ) is a smooth manifold for each ℓ ∈ N 0 ∪ {∞}, locally compact, paracompact smooth manifold M with rough boundary in the sense of [15] (this includes finite-dimensional manifolds with boundary, and manifolds with corners as in [7,8,21]) and each smooth manifold N modeled on locally convex spaces such that N admits a local addition (a concept recalled in Definition 5.6); see [16,21,22,25,4,14] for discussions in different levels of generality, and [20] for manifolds of smooth maps in the convenient setting of analysis. For compact M , the modeling space of C ℓ (M, N ) around f ∈ C ℓ (M, N ) is the locally convex space Γ C ℓ (f * (T N )) of all C ℓ -sections in the pullback bundle f * (T N ) → M , which can be identified with if M is not compact, the locally convex space of compactly supported C ℓ -sections of f * (T N ) is used. Let L be a smooth manifold modeled on locally convex spaces (possibly with rough boundary), and k ∈ N 0 ∪{∞}. For compact M , it is known from [4, Proposition 1.23 and Definition 1.17] that a map is C k if and only if the corresponding map of two variables, is C k,ℓ in the sense of [3], i.e., a continuous map which in local charts admits up to ℓ directional derivatives in the second variable, followed by up to k directional derivatives in the first variable, with continuous dependence on point and directions (see 2.11 and 2.12 for details). We thus obtain a bijection Φ : C k (L, C ℓ (M, N )) → C k,ℓ (L × M, N ), g → g ∧ .
As our first result, for compact L we construct a smooth manifold structure on C k,ℓ (L × M, N ) which turns Φ into a C ∞ -diffeomorphism. More generally, analogous to the n = 2 case of C k,ℓ -maps, we consider N -valued C α -maps on an n-fold product M 1 × · · · × M n of smooth manifolds for any n ∈ N and α = (α 1 , . . . , α n ) ∈ (N 0 ∪ {∞}) n . With terminology explained presently, we get: Theorem 1.1 Given α = (α 1 , . . . , α n ) ∈ (N 0 ∪{∞}) n , let M j for j ∈ {1, . . . , n} be a compact smooth manifold with rough boundary. Let N be a smooth manifold modeled on locally convex spaces such that N can be covered by local additions. Then C α (M 1 ×· · ·×M n , N ) admits a smooth manifold structure which is canonical. The following hold for this canonical manifold structure: (a) C α (M 1 × · · · × M n , N ) can be covered by local additions. If N admits a local addition, then also C α (M 1 × · · · × M n , N ) admits a local addition.
The following terminology was used: We say that a smooth manifold N can be covered by local additions if N is the union of an upward directed family (N j ) j∈J of open submanifolds N j which admit a local addition. For instance, any (not necessarily paracompact) finite-dimensional smooth manifold has this property, e.g. the long line. We also used canonical manifold structures.
Canonical manifold structures are essentially unique whenever they exist, and so are pre-canonical ones (see Lemma 4.3 (b) for details).
We address two further topics for not necessarily compact domains: (i) We formulate criteria ensuring that C α (M 1 ×· · ·×M n , G) admits a canonical smooth manifold structure (making the latter a Lie group), for a Lie group G modeled on a locally convex space; (ii) Manifold structures on C α (M 1 ×· · ·×M n , N ) which are modeled on certain spaces of compactly supported T N -valued functions, in the spirit of [21].
To discuss (i), we use a generalization of the regularity concept introduced by John Milnor [22] (the case r = ∞). If G is a Lie group modeled on a locally convex space, with neutral element e, we write λ g : G → G, x → gx for left translation with g ∈ G and consider the smooth left action of G on its tangent bundle. We write g := T e G for the Lie algebra of G. Let r ∈ N 0 ∪ {∞}. The Lie group G is called C r -semiregular if, for each C r -curve γ : [0, 1] → g, the initial value probleṁ is smooth, then G is called C r -regular (cf. [12]). If s ≤ r and G is C s -regular, then G is C r -regular (see [12]). We show: Theorem 1.3 Let G be a C r -regular Lie group modeled on a locally convex space with r ∈ N 0 ∪ {∞}. For some n ∈ N, let M 1 , . . . , M n be locally compact smooth manifolds with rough boundary and α ∈ (N 0 ∪ {∞}) n . For each j ∈ {1, . . . , n} such that M j is not compact, assume that α j ≥ r + 1 and M j is 1-dimensional with finitely many connected components. Then we have: (a) C α (M 1 × · · · × M n , G) admits a canonical smooth manifold structure; (b) The canonical manifold structure from (a) makes C α (M 1 × · · · × M n , G) a C r -regular Lie group.
The Lie algebra of C α (M 1 × · · · × M n , G) can be identified with the topological Lie algebra C α (M 1 × · · · × M n , L(G)) in a standard way (Proposition 6.6). Of course, we are most interested in the case that the non-compact 1-dimensional factors are σ-compact and hence intervals, or finite disjoint unions of such. But we did not need to assume σ-compactness in the theorem, and thus M j with α j ≥ r + 1 might well be a long line, or a long ray.
Disregarding the issue of being canonical, the Lie group structure on C ∞ (M 1 × · · · × M n , G) = C α (M 1 × · · · × M n , G) with α 1 := · · · := α n = ∞ was first obtained in [24], for smooth manifolds M j without boundary which are compact or diffeomorphic to R. The Lie group structure for n = 1 was first obtained in [2] for domains diffeomorphic to intervals, together with a sketch for the case n = 2 (assuming additional conditions, e.g. α 1 ≥ r + 3 and α 2 ≥ r + 1 if M 1 = M 2 = R). Our approach differs: While the studies in [24] and [2] assume regularity of G from the start to enforce exponential laws, and build it into a notion of Lie group structures on mapping groups that are "compatible with evaluations," we take canonical and pre-canonical manifold structures as the starting point (independent of regularity) and combine them with regularity or compatibility with evaluations (adapted to C α -maps in Definition 6.2) only when needed.
As to topic (b), our constructions show: Theorem 1.4 Given α = (α 1 , . . . , α n ) ∈ (N 0 ∪{∞}) n , let M j for j ∈ {1, . . . , n} be a paracompact, locally compact smooth manifold with rough boundary; abbreviate M := M 1 × · · · × M n . Let N be a smooth manifold modeled on locally convex spaces such that N admits a local addition. Let π T N : T N → N be the canonical map. For f ∈ C α (M, N ) and a compact subset K ⊆ M , the set is a vector subspace of x∈M T f (x) N , and a locally convex space in the topology induced by C α (M, T N ). Give Γ f = K Γ f,K the locally convex direct limit topology. Then C α (M, N ) admits a unique smooth manifold structure modeled on the set E := {Γ f : f ∈ C α (M, N )} of locally convex spaces such that, for each f ∈ C α (M, N ) and local addition Σ : T N ⊇ U → N of N , the map In the case that n = 1, k = ∞ and M := M 1 is a smooth manifold with corners, we recover the smooth manifold structure on C ∞ (M, N ) discussed by Michor [21].
Using manifold structures on infinite cartesian products of manifolds making them "fine box products" (a concept recalled in Section 7), Theorem 1.4 turns into a corollary to Theorem 1.1.
In the case n = 1, for compact M and ℓ ∈ N 0 ∪ {∞}, canonical manifold structures on C ℓ (M, N ) as in Theorem 1.1 have already been considered in [4], in a weaker sense (fixing m = 1 in Definition 1.2). Parts of our discussion adapt arguments from [4] to the more difficult case of C α -maps.
Acknowledgement. The authors would like to thank the mathematical institute at NTNU Trondheim for its hospitality while conducting the work presented in this article, as well as Nord universitet Levanger.

Consider locally convex spaces E, F and a map
exist for all j ∈ N 0 such that j ≤ k, x ∈ U and y 1 , . . . , y j ∈ E, and the maps d j f : U × E j → F are continuous, then f is called C k . If U may not be open, but has dense interior U o and is locally convex in the sense that each x ∈ U has a convex neighbourhood in U , following [15] The C ∞ -maps are also called smooth.

Remark 2.2
If E = R n and U is relatively open in [0, ∞[ n , then f as above is C k if and only f has a C k -extension to an open set in R n (see [13], cf. [17]).

2.3
Let k ∈ N∪{∞}. A manifold with rough boundary modeled on a non-empty set E of locally convex spaces is a Hausdorff topological space M , together with a set A of homeomorphisms ("charts") φ : . Let ∂M be the set of all x ∈ M such that φ(x) ∈ ∂V φ for some (and hence any) chart φ around x. If E is a singleton, M is called pure. If M is a C k -manifold with rough boundary and ∂M = ∅, then M is called a C k -manifold or a C k -manifold without boundary, for emphasis. (See [15] for all of this in the pure case; cf. [4] for modifictions in the general case).

2.4
All manifolds and Lie groups considered in the article are modeled on locally convex spaces which may be infinite-dimensional, unless the contrary is stated. Finite-dimensional manifolds need not be paracompact or σ-compact, unless stated explicitly. As we are interested in manifolds of mappings, consideration of pure manifolds would not be sufficient.

If
U is an open subset of a locally convex space E (or a locally convex subset with dense interior), we identify its tangent bundle T U with U × E, as usual, with bundle projection (x, y) → x. If M is a C k -manifold with rough boundary and f : M → U a C k -map with k ≥ 1, we write df for the second , using the bundle projection π T M : T M → M .

2.6
If G is a Lie group with neutral element e, we write L(G) := T e G (or g) for its tangent space at e, endowed with its natural topological Lie algebra structure.
If ψ : G → H is a smooth homomorphism between Lie groups, we let L(ψ) := T e ψ : L(G) → L(H) be the associated continuous Lie algebra homomorphism.

2.9
Let M be a smooth manifold (without boundary)

2.11
Let F and E 1 , . . . , E n be locally convex spaces, U j ⊆ E j be an open subset for j ∈ {1, . . . , n} and f : U → F be a map on U := U 1 × · · · × U n . Identifying E := E 1 × · · · × E n with E 1 ⊕ · · · ⊕ E n , we can identify each E j with a vector subspace of E, and simply write D y f (x) for a directional derivative with x ∈ U , y ∈ E j (rather than D (0,...,0,y,0,...,0) f (x) with j − 1 zeros on the left and n − j zeros on the right-hand side). For y = (y 1 , . . . , y k ) ∈ E k j , abbreviate Let α ∈ (N 0 ∪ {∞}) n . Following [1], we say that f is C α if f is continuous, the iterated directional derivatives exist for all β ∈ N n 0 with β ≤ α, x ∈ U and y j = (y j,1 , . . . , y j,βj ) ∈ (E j ) βj for j ∈ {1, . . . , n}, and d β f : U × E β1 1 × · · · × E βn n → F is continuous. If U j may not be open but is a locally convex subset of E j with dense interior, we say that f :

2.12
Let M 1 , . . . , M n be C ∞ -manifolds with rough boundary, α ∈ (N 0 ∪ {∞}) n and N be a C k -manifold with k ≥ |α|. We say that a map f : is C α . The latter then holds for any such charts, by the Chain Rule for C α -maps (as in [1,Lemma 3.16]).
(b) If F is the projective limit of a projective system ((F a ) a∈A , (λ a,b ) a≤b ) of locally convex spaces F a and continuous linear mappings λ a,b : Lemma 2.15 Let M , N , and L 1 , . . . , L n be smooth manifolds with rough boundary, F be a locally convex space, ψ : M → F × N be a C ∞ -diffeomorphism, and f : L 1 × · · · × L n → M be a map. Assume that F is the projective limit of a projective system ((F a ) a∈A , (λ a,b ) a≤b ) of locally convex spaces F a and continuous linear mappings λ a,b : F b → F a , with limit maps λ a : F → F a . For a ∈ A, let M a be a smooth manifold and ρ a : M → M a be a C ∞ -map. Assume that there exist C ∞ -maps ψ a : M a → F a × N making the diagram we shall write (α, β) as a shorthand for (α 1 , . . . , α n , β 1 , . . . , β m ) and abbreviate C (α,β) as C α,β . Likewise for higher numbers of multiindices.
Let r ∈ N 0 ∪{∞}, E 1 , . . . , E n and F be locally convex spaces and U j be a locally convex subset of E j with dense interior, for j ∈ {1, . . . , n}. We mention that a map f : U 1 × · · · × U n → F is C r if and only if it is C β for all β ∈ (N 0 ∪ {∞}) n such that |β| ≤ r. More generally, the following is known (as first formulated and proved in the unpublished work [18]): Then a map f : The compact-open C α -topology As a further preliminary, we introduce a topology on C α (M 1 × · · · × M n , N ) which parallels the familiar compact-open C k -topology on C k (M, N ). Basic properties are recorded, with proofs in Appendix A.
As usual, T 0 M := M , T 1 M := T M and T k M := T (T k−1 M ) for a smooth manifold M with rough boundary and integers k ≥ 2 (see [15]).

Definition 3.4
The compact-open C α -topology on C α (M, N ) is the initial topology with respect to the mappings Pushforwards and pullbacks are continuous.

Lemma 3.5
Using compact open C α -topologies, we have: (a) If L is a smooth manifold with rough boundary and g : N → L a smooth map, then the following map is continuous: (b) Let L j be a smooth manifold with rough boundary for j ∈ {1, . . . , n} and g j : L j → M j be a smooth map. Abbreviate L := L 1 × · · · × L n and g := g 1 × · · · × g n . Then the following map is continuous: Remark 3.6 If L j is a full submanifold of M j for j ∈ {1, . . . , m}, then the inclusion map g j : L j → M j , x → x is smooth. By Lemma 3.5 (b), the map    (a) If F is a locally convex space whose topology is initial with respect to a family (λ i ) i∈I of linear mappings λ i : F → F i to locally convex spaces F i , then the compact-open C α -topology on C α (M, F ) is initial with respect to the mappings ((λ i ) * ) i∈I : C α (M, F ) → C α (M, F i ).
(b) If F is a locally convex space and F = i∈I F i for a family (F i ) i∈I of locally convex spaces, let pr i : F → F i be the projection onto the ith component and (pr i ) * : is an isomorphism of topological vector spaces.
(c) Assume that all of M 1 , . . . , M n are locally compact. Let N i be a smooth manifold with rough boundary for i ∈ {1, 2} and pr i : N 1 × N 2 → N i be the projection onto the ith component. Using the compact-open C α -topology on sets of C α -maps, we get a homeomorphism Using the multiplication R × T N → T N , (t, v) → tv with scalars, we have: is continuous.
In [1], Exponential Laws were provided for function spaces on products of pure manifolds. The one we need remains valid for manifolds which need not be pure: Lemma 3.13 Let N 1 , . . . , N m and M 1 , . . . , M n be smooth manifolds with rough boundary (none of which needs to be pure). Let α ∈ (N 0 ∪{∞}) m , β ∈ (N 0 ∪{∞}) n and E be a locally convex space. Abbreviate N := N 1 × · · · × N m and M := M 1 ×· · ·×M n . For f ∈ C α,β (N ×M, E), we then have f x := f (x, ·) ∈ C β (M, E) for each x ∈ N and the map f ∨ : is linear and a homeomorphism onto its image. If M j is locally compact for all j ∈ {1, . . . , n}, then Φ is a homeomorphism. The inverse map Φ −1 sends g ∈ C α (N, C β (M, E)) to the map g ∧ defined via g ∧ (x, y) := g(x)(y).
We mention that the C α -topology on C α (U, F ) can be described more explicitly.
Lemma 3.14 Let E j be a locally convex space for j ∈ {1, . . . , n} and U j ⊆ E j be a locally convex subset with dense interior. Let F be a locally convex space, α ∈ (N 0 ∪ {∞}) n , and U := U 1 × · · · × U n . Then the compact-open C α -topology on C α (U, F ) is initial with respect to the maps for β ∈ (N 0 ) n with β ≤ α, using the compact-open topology on the ranges.

(Pre-)Canonical manifold structures
In this section, we establish basic properties of canonical manifolds of mappings, and pre-canonical ones. We begin with examples. (a) Let M 1 , . . . , M n be locally compact smooth manifolds with rough boundary and E a locally convex space. Then C α (M 1 ×· · ·×M n , E) is a canonical manifold due to Lemma 3.13. The same holds for C α (M 1 × · · · × M n , N ) if N is a smooth manifold diffeomorphic to E, endowed with the C ∞manifold structure making ϕ * : Familiar examples of mapping groups turn out to be canonical, notably loop groups C k (S 1 , G) for G a Lie group, and certain Lie groups of the form C k (R, G) discussed in [2,24]. We extend these constructions in Section 6.
We will now establish general properties of canonical manifolds.

Conventions
We denote by α, β multiindices in (N 0 ∪ {∞}) n for some n ∈ N. Likewise we will usually adopt the shorthand M := M 1 × M 2 × · · · × M n where the M i are locally compact manifolds (possibly with rough boundary). If M is the domain of definition of the function space C α (M, N ) we will assume that the number of entries of the multiindex α coincides with the number of factors in the product M . (b) Pre-canonical manifold structures are unique in the following sense: If we write C α (M, N ) ′ for C α (M, N ) with another pre-canonical manifold structure, then id : (c) Let S ⊆ N be a submanifold such that the set C α (M, S) is a submanifold of C α (M, N ). Then the submanifold structure on C α (M, S) is pre-canonical.
Proof. (a) Since id : be a product of smooth manifolds (possibly with rough boundary) modeled on locally convex spaces and f : We now turn to smoothness properties of the composition map.
is a C ∞,s,α -map. The formula shows that comp ∧ (f, g, x) = ev(f, ev(g, x)), where the outer evaluation map is C ∞,|α|+s and the inner one C ∞,α , by Lemma 4.3 (a), as C |α|+s (N, L) and C α (M, N ) are pre-canonical manifolds. Using the chain rule [1, Lemma 3.16], we deduce that comp ∧ is C ∞,s,α . ✷ Corollary 4.7 Let C |α|+s (N, L) and C α (M, L) be endowed with pre-canonical manifold structures. For a C α -map g : M → N the pullback g * : The chain rule also allows the following result to be deduced.
Proof. Due to the chain rule, the pullback g * makes sense.
The key point was the differentiability of the evaluation map together with a suitable chain rule. Thus, by essentially the same proof, one obtains from the chain rule [1, Lemma 3.16] the following statement whose proof we omit.

Proposition 4.9
Assume that all the manifolds of mappings occurring in the following are endowed with pre-canonical manifold structures. Further, we let . Then for σ = (σ 1 , . . . , σ n ) and α = (α 1 , . . . , α n ), the composition map The above discussion shows that composition, pushforward, and pullback maps inherit differentiability and continuity properties. The following variant will be used in the construction process of canonical manifold structures. Proposition 4.10 Let K be a compact smooth manifold such that C α (K, M ) and C α (K, N ) admit canonical manifold structures. If Ω ⊆ K × M is an open subset and f : Ω → N is a C |α|+k -map, then is an open subset of C α (K, M ) and Proof. By compactness of K, the compact-open topology on C(K, M ) coincides with the graph topology (see, e.g., [15,Proposition A.6.25]). Thus Lemma 3.16] shows that For later use we record several observations on stability of (pre-)canonical structures under pushforward by diffeomorphisms.
(c) Let C α (M, N ) be endowed with a pre-canonical manifold structure and assume that both C β (L, C α (M, N )) and C β,α (L × M, N ) are smooth manifolds making the bijection and only if f 1 and f 2 are C β . As the manifold structures are (pre-)canonical, this holds if and As ψ is a smooth diffeomorphism we deduce from the chain rule that this is the case if and only if f ∧ is of class C β,α . Thus C α (M, N 1 ) is precanonical. If C α (M, N 2 ) is even canonical, the C α -topology is transported by the diffeomorphism ψ * to the C α -topology on C α (M, N 1 ). Hence the manifold C α (M, N 1 ) is also canonical in this case.
(c) By construction, a map f : Exploiting the Theorem of Schwarz [1, Proposition 3.5], this is equivalent to f ∧ being C β,α•σ . Thus C α•σ (Q, N ) is pre-canonical. The converse assertion for C α•σ (M, N ) follows verbatim by replacing φ σ with its inverse. Note that if one of the manifolds is even canonical, it follows directly from the definition of the C α -topology, Definition 3.4, that reordering the factors induces a homeomorphism of the C α -and C α•σ -topology. Hence we see that one of the manifolds is canonical if and only if the other is so.
(b) Note that the inverse of φ * σ is (φ −1 σ ) * whence the situation is symmetric and it suffices to prove that φ * σ (and by an analogous argument also its inverse) ) being a C ∞,α -mapping. This follows from Lemma 4.3 (a), the chain rule, and Lemma 2.17.
(c) Replacing φ σ with ψ 1 × · · · × ψ n , the argument is analogous to (b). If C α (M, N ) is canonical, then the C α -topology pulls back to the C α -topology under the diffeomorphism, by Lemma 3.5. ✷ An exponential law is available for pre-canonical smooth manifold structures.
Proposition 4.13 Let L 1 , . . . , L m and N be smooth manifolds with rough boundary, and M 1 , . . . , M n be locally compact smooth manifolds with rough boundary. Assume that C α (M, N ) is endowed with a pre-canonical smooth manifold structure and also C β (L, C α (M, N ) and C β,α (L × M, N ) are endowed with precanonical smooth manifold structures. Then the bijection Proof. If we give C β (L, C α (M, N )) the smooth manifold structure making Φ a C ∞ -diffeomorphism, then this structure is pre-canonical by Lemma 4.11 (c). It therefore coincides with the given pre-canonical smooth manifold structure thereon, up to the choice of modeling spaces (Lemma 4.3 (b)). ✷ There is a natural identification of tangent vectors for pre-canonical manifolds, in good cases.
corresponds to an equivalence class of curves γ v : Under additional assumptions, one can show that Ψ is a diffeomorphism, allowing tangent vectors v ∈ T C α (M, N ) to be identified with Ψ(v). We will encounter a setting in which this statement becomes true in the next section (see Theorem 5.14).

Constructions for compact domains
We now construct and study manifolds of C α -mappings on compact domains. The results of this section subsume Theorem 1.1. They generalize constructions for C k,ℓ -functions in [4, Appendix A].

5.1
Let N be a smooth manifold, α ∈ (N 0 ∪ {∞}) n and M = M 1 × · · · × M n be a locally compact smooth manifold with rough boundary. If π : E → N is a smooth vector bundle over N and f : M → N is a C α -map, then we define Pointwise operations turn Γ f into a vector space. Let us prove that Γ f is a locally convex space. To this end, we cover N with open sets (U i ) i∈I on which the restriction Combining continuity of f and local compactness of M we can find families K j of full compact submanifolds of M j with the following properties: The interiors of the sets in K j cover M j .
There is a set K ⊆ 1≤j≤n K j such that for every K = K 1 × · · · × K n ∈ K we have f (K) ⊆ U iK for some i K ∈ I and the interiors of the submanifolds in K cover M . Hence we deduce from Lemma 3.7 that the map Restricting Ψ to this subset we obtain a topological embedding where the identification exploits Lemma 3.11 and the fact that pushforwards with smooth diffeomorphisms induce homeomorphisms of the C α -topology (see Lemma 3.5). The image of e are precisely the mappings which coincide on the intersections of the compact sets K (see (10) and the explanations there).
Hence we can exploit that point evaluations are continuous on C α (K, E iK ) by [2,Proposition 3.17] to see that the image of e is a closed vector subspace As the space on the right hand side is locally convex, we deduce that the co-restriction of e onto its image is an isomorphism of locally convex spaces. Thus Γ f is a locally convex topological vector space. We will sometimes write Γ f (E) instead of Γ f to emphasize the dependence on the bundle E.
The previous setup allows an essential Exponential Law to be deduced.
Proof. With the notation as in 5.1 we identify Γ f via e with a closed subspace of the locally convex space K∈K C α (K, E iK ) (the identification will be suppressed in the notation). Thus Lemma 2.14 (a) implies that the map g is C β if and only if the components g K : L → C α (K, E iK ) are C β -maps. By the Exponential Law [1,Theorem 4.4], the latter holds if and only if the mappings are of class C β,α . Since the interiors of sets K ∈ K cover M , we deduce that this is the case if and only if g ∧ is of class C β,α . ✷

Remark 5.3
If all fibres of E are Fréchet spaces and K is σ-compact and locally compact, then Γ F is a Fréchet space; if all fibres of E are Banach spaces, K is compact, and |α| < ∞, then Γ f is a Banach space. To see this, note that we can choose the family K in 5.1 countable (resp., finite). Supressing again the identification, is linear and a topological embedding with closed image. If all F j are Fréchet spaces, so is each C α (K j , F j ) (cf., e.g., [15]) and hence also Γ f . If all F j are Banach spaces and |α| as well as J is finite, then each C α (K j , F j ) is a Banach space (cf. loc. cit.) and hence also Γ f .
Observe that the exponential law for Γ f gives this space the defining property of a pre-canonical manifold (and the only reason we do not call it pre-canonical is that it is only a subset of C α (M, E)). In particular, the proof of Lemma 4.3 (a) carries over and yields: In the situation of 5.1, the evaluation map Lemma 5.5 Let π 1 : E 1 → N and π 2 : E 2 → N be smooth vector bundles over a smooth manifold N . Let α ∈ (N 0 ∪ {∞}) m and f : M → N be a C α -map on a product M = M 1 × · · · × M n of smooth manifolds with rough boundary. Then the following holds: . Evaluating at points we see that the map Γ f (ψ) is linear; being a restriction of the continuous map is a continuous linear map which is a homeomorphism as it has the continuous

Construction of the canonical manifold structure
Having constructed spaces of C α -sections as model spaces, we are now in a position to construct the canonical manifold structure on C α (K, M ), assuming that M is covered by local additions and K is compact.
we say that the local addition Σ is normalized.
Until Lemma 5.9, we fix the following setting, which allows a canonical manifold structure on C α (K, M ) to be constructed.

5.7
We consider a product K = K 1 ×K 2 ×· · ·×K n of compact smooth manifolds with rough boundary, a smooth manifold M which admits a local addition Σ : T M ⊇ U → M , and α ∈ (N 0 ∪ {∞}) n .

Manifold structure on
is a homeomorphism with inverse g → θ −1 • (f, g). By the preceding, if also h ∈ C α (K, M ), then ψ : is a C ∞,α -map (exploiting that the evaluation map ε : Γ f × K → T M is C ∞,α , by Lemma 5.4). Hence C α (K, M ) endowed with the C α -topology has a smooth manifold structure for which each of the maps φ −1 f is a local chart.
We now prove that the manifold structure on C α (K, M ) is canonical. Together with Lemma 4.3 (b), this implies that the smooth manifold structure on C α (K, M ) constructed in 5.8 is independent of the choice of local addition.
Lemma 5.9 The manifold structure on C α (K, M ) constructed in 5.8 is canonical.
Proof. We first show that the evaluation map ev : is an open x-neighborhood in N . As the map (φ −1 is C β,α by [1, Lemma 3.16], the map φ −1 f • h| W : W → Γ f (and hence also h| W ) is C k , by Lemma 5.2. ✷ Proposition 5.10 Let K = K 1 × · · · × K n be a product of compact smooth manifolds with rough boundary and M be a manifold covered by local additions. For every α ∈ (N 0 ∪ {∞}) n , the set C α (K, M ) can be endowed with a canonical manifold structure.
Proof. Let (M j , Σ j ) j∈J be an upward directed family of open submanifolds M j with local additions Σ i whose union coincides with M . As K is compact, we observe that the sets C α (K, , shows that the submanifold structure must coincide with the canonical structure constructed on C α (K, M j ) via 5.8. As C α (K, M ) = j∈J C α (K, M j ) and each step of the ascending union is canonical, the same holds for the union. ✷

The tangent bundle of the manifold of mappings
In the rest of this section, we identify the tangent bundle of C α (K, M ) as the manifold C α (K, T M ) (under the assumption that K is compact and M covered by local additions). To explain the idea, let us have a look at C α (K, T M ).

5.11
Consider a smooth manifold M covered by local additions. Then also T M is covered by local additions, cf. [4, A.11] for the construction. Thus for K a compact manifold C α (K, M ) and C α (K, T M ) are canonical manifolds. If we denote by π : T M → M the bundle projection, Corollary 4.6 shows that the pushforward π * : C α (K, T M ) → C α (K, M ) is smooth. The fibres of π * are the locally convex spaces π −1 * (f ) = Γ f from 5.1. We deduce that π * : C α (K, T M ) → C α (K, M ) is a vector bundle (see Theorem 5.14 for a detailed proof).
We will first identify the fibres of the tangent bundle.

5.12
The tangent space where the equivalence relation c ∼ c ′ holds for two such curves if and only ifċ(0) =ċ ′ (0). Since the manifold structure is canonical (Lemma 5.10) we see that c is C 1 if and only if the adjoint map c ∧ : ]−ε, ε[ ×K → N is a C 1,α -map. The exponential law shows that the derivative of c corresponds to the (partial) derivative of c ∧ , i.e. the mapping Ψ from (2) restricts to a bijection We wish to glue the bijections on the fibres to identify the tangent manifold as the bundle from 5.11. To this end, we recall a fact from [4, Lemma A.14]:

5.13
If a manifold M admits a local addition, it also admits a normalized local addition.
Hence we may assume without loss of generality that the local additions in the following are normalized. Moreover, we will write ε x : C α (K, M ) → M for the point evaluation in x ∈ K. Then the tangent bundle of C α (K, M ) can be described as follows.
Theorem 5.14 Let K = K 1 × · · · × K n be a product of compact smooth manifolds with rough boundary and M be covered by local additions. Then is a smooth vector bundle with fibre Γ f over f ∈ C ℓ (K, M ). For each v ∈ T (C ℓ (K, M )), we have Ψ(v) := (T ε x (v)) x∈K ∈ C α (K, T M ) and the map (2), is an isomorphism of smooth vector bundles (over the identity).
If we wish to emphasize the dependence on M , we write Ψ M instead of Ψ. Proof. Since M is covered by local additions, there is a family of open submanifolds (ordered by inclusion) (M j ) j∈J which admit local additions Σ j . Now by compactness of K the image of f ∈ C α (K, M ) is always contained in some M j and similarly for τ ∈ Γ f we then have τ (K) ⊆ π −1 (M j ) = T M j , where π := π T M is the bundle projection of T M . As the family (M j ) j of open manifolds exhausts M , we have C α (K, M ) = j∈J C α (K, M j ) and all of these subsets are open. Hence it suffices to prove that Ψ restricts to a bundle isomorphism for every M j . In other words we may assume without loss of generality that M admits a local addition Σ. Given f ∈ C α (K, M ), the map M )) and Ψ takes T f (C α (K, M )) bijectively and linearly onto Γ f . Now the manifolds T (C α (K, M )) and C α (K, T M ) are the disjoint union of the sets T f (C α (K, M )) and Γ f = π −1 * ({f }), respectively, we see that Ψ is a bijection. If we can show that Ψ is a C ∞ -diffeomorphism, π * : C α (K, T M ) → C α (K, M ) will be a smooth vector bundle over C α (K, M ) (like T (C α (K, M ))). Finally, Ψ will then be an isomorphism of smooth vector bundles over id M . Θ .
Here for f ∈ C α (K, M ) we have considered the composition 0•f ∈ C α (K, T M ).
Then the sets Hence it suffices to prove that the bijective map Ψ restricts to a C ∞ -diffeomorphism on these open sets. In other words it suffices to show that for each f ∈ C ℓ (K, M ) (as all other mappings in the formula are smooth diffeomorphisms). Now for all (σ, τ ) ∈ O f × Γ f , and thus we can rewrite Ψ(T φ f (σ, τ )) as Thus the desired formula holds and shows that Ψ is a C ∞ -diffeomorphism. This concludes the proof. ✷

Remark 5.15
Assume that the local additions Σ : U i → M i covering M are normalized. Then the proof of Theorem 5.14 shows that (5)).
Using canonical manifold structures, we have: Corollary 5.16 Let K = K 1 × · · · × K n be a product of compact smooth manifolds with rough boundary, α ∈ (N 0 ∪ {∞}) n and g : M → N be a C |α|+1 -map between smooth manifolds M and N covered by local additions. Then the tangent map of the C 1 -map which is continuous linear and corresponds to T f (g * ).
Moreover, the identification of the tangent bundle allows us to lift local additions (cf. [4, Remark A.17]).
Proof. We apply Proposition 5.10 to obtain canonical manifold structures on C α (K, M ) and C β,α (L × K, M ). By Lemma 5.17, C α (K, M ) is covered by local additions. Hence we may apply Proposition 5.10 again to obtain a canonical manifold structure on C β (L, C α (K, M )). By Proposition 4.13, the bijection C β,α (L × K, M ) → C β (L, C α (K, M )) is a diffeomorphism. ✷

Lie groups of Lie group-valued mappings
We now prove Theorem 1.3, starting with observations. Lemma 6.1 Let M 1 , . . . , M n be locally compact smooth manifolds with rough boundary, G be a Lie group, and α ∈ (N 0 ∪{∞}) n . Setting M := M 1 × · · ·× M n , the following holds: (a) C α (M, G) is a group.
Another concept is useful, with notation as in 2.6.
so obtained is an isomorphism of topological vector spaces. × · · · × M n , N := N 1 × · · · × N m , and G be a Lie group. Assume that C β (M, G) is endowed with a pre-canonical smooth manifold structure which is compatible with evaluations and that C α (N, C β (M, G)), whose definition uses the latter structure, is endowed with a pre-canonical smooth manifold structure which is compatible with evaluations. Endow C α,β (N × M, G) with the smooth manifold structure turning the bijection Then the preceding smooth manifold structure on C α,β (N × M, G) is pre-canonical and compatible with evaluations.
Proof. By Lemma 4.11 (c), the C ∞ -manifold structure on C α,β (N × M, G) is pre-canonical, whence the latter is a Lie group. The C ∞ -diffeomorphism Φ is a homomorphism of groups. Hence Then also is an isomorphism of topological Lie algebras and so is by the Exponential Law (Lemma 3.13). Hence is an isomorphism of topological Lie algebras. Regard v ∈ L(C α,β (N ×M, G)) as a geometric tangent vector [γ] for a smooth curve γ : We deduce that (L(ε (x,y) )(v)) (x,y)∈N ×M = φ(v) ∈ C α,β (N × M, L(G)). Since φ is an isomorphism of topological Lie algebras, the Lie group structure on C α,β (N × M, G) is compatible with evaluations. ✷ Lemma 6.5 Let M 1 , . . . , M n be locally compact smooth manifolds with rough boundary, M := M 1 ×· · ·×M n , α ∈ (N 0 ∪{∞}) n , and G be a Lie group. Assume that C α (M, G) is endowed with a pre-canonical smooth manifold structure which is compatible with evaluations. If the Lie group G is C r -regular for some r ∈ N 0 ∪ {∞}, then also the Lie group C α (M, G) is C r -regular.
Proof. Consider the smooth evolution map Evol: The structures being pre-canonical, and The final assertion is clear: Starting with any canonical structure on C α (M, G) and a chart φ : U φ → V φ → E φ around the constant map e, using left translations (which are C ∞ -diffeomorphisms) we can create charts around every f ∈ C α (M, G) which are modeled on the given E φ . We can therefore select a subatlas making C α (M, G) a pure smooth manifold. Since E φ is isomorphic to L (C α (M, G)), which is isomorphic to E := C α (M, L(G)) as a locally convex space (by compatibility with evaluations), we can replace E φ with E. The pure canonical structure modeled on E is unique, since id C α (M,G) is a C ∞diffeomorphism for any two canonical structures (cf. Lemma 4.3 (b)).
Lemma 6.7 Let M 1 , . . . , M n be compact smooth manifolds with rough boundary, G be a Lie group and α ∈ (N 0 ∪ {∞}) n . Abbreviate M := M 1 × · · · × M n . Then C α (M, G) admits a canonical smooth manifold structure which is compatible with evaluations. If G is C r -regular for r ∈ N 0 ∪ {∞}, then so is C α (M, G).
Proof. By Theorem 1.1, C α (M, G) admits a canonical smooth manifold structure. Let θ : M → G be the constant map x → e. By Theorem 5.14, the By Lemma 3.9, C α (M, T G) induces on C α (M, L(G)) the compact-open C αtopology. Thus, the Lie group structure on C α (M, G) is compatible with evaluations. For the last assertion, see Lemma 6.5. ✷ Lemma 6.8 Let M be a 1-dimensional smooth manifold with rough boundary, such that M has only finitely many connected components (which need not be σcompact). Let r ∈ N 0 ∪{∞}, G be a C r -regular Lie group, and k ∈ N∪{∞} such that k ≥ r + 1. Then C k (M, G) admits a canonical smooth manifold structure which makes it a C r -regular Lie group and is compatible with evaluations.
Proof. We first assume that M is connected. Let g := L(G) be the Lie algebra of G. If N is a full submanifold of M , we write Ω 1 C k−1 (N, g) ⊆ C k−1 (T N, g) for the locally convex space of g-valued 1-forms on N , of class C k−1 . Using the Maurer-Cartan form (N, G), called its left logarithmic derivative. Fix x 0 ∈ M . For every σ-compact, connected, full submanifold N ⊆ M such that x 0 ∈ N , there exists a C ∞ -diffeomorphism ψ : I → N for some nondegenerate interval I ⊆ R, such that 0 ∈ I and ψ(0) = x 0 . Then the diagram is commutative, where ψ * : C k (N, G) → C k (I, G), f → f • ψ and the vertical map θ on the right-hand side, which takes ω to ω •ψ, are bijections. For each ω ∈ Ω 1 C k−1 (N, g), there is a unique f ∈ C k (N, G) such that f (x 0 ) = e and δ N (f ) = ω: In fact, Lemma 2.8 yields a unique η ∈ C k (I, G) with η(0) = e and δ ℓ (η) = θ(ω); then f := (ψ * ) −1 (η) is as required. We set Evol N (ω) := f .
. Let K be the set of all connected, compact full submanifolds K ⊆ M such that x 0 ∈ K. By the preceding, [16, Theorem 3.5] provides a smooth manifold structure on C k (M, g) which makes it a C r -regular Lie group, is compatible with evaluations, and turns ) into a C ∞ -diffeomorphism. It remains to show that the smooth manifold structure is canonical. To prove the latter, we first note that K is directed under inclusion. In fact, if K 1 , K 2 ∈ K, then K 1 ∪ K 2 is contained in a σ-compact, connected open submanifold N of M (a union of chart domains diffeomorphic to convex subsets of R, around finitely many points in the compact set K 1 ∪ K 2 ). Pick a C ∞ -diffeomorphism ψ : I → N as above. Then ψ −1 (K 1 ) and ψ −1 (K 2 ) are compact intervals containing 0, whence so is their union. Thus K 1 ∪ K 2 is a connected, compact full submanifold of N and hence of M .
For K, L ∈ K with K ⊆ L, let r K,L : Ω 1 C k−1 (L, g) → Ω 1 C k−1 (K, g) be the restriction map. As a consequence of Lemma 3.7 and (8), holds as a locally convex space, using the restriction maps r K : Ω 1 C k−1 (M, g) → Ω 1 C k−1 (K, g) as the limit maps. For K ∈ K, let ρ K : C k (M, G) → C k (K, G) be the restriction map; endow C k (K, G) with its canonical smooth manifold structure (as in Lemma 6.7), which is compatible with evaluations (the "ordinary" Lie group structure in [16]). Then [16,proof of Theorem 3.5]). Note that ρ K = ψ −1 K • (r K × id G )• ψ is smooth on C k (M, G), using the above Lie group structure making ψ a C ∞ -diffeomorphism. Let α ∈ (N 0 ∪ {∞}) m , L 1 , . . . , L m be smooth manifolds with rough boundary, L := L 1 × · · · × L m and f : (K, g), and N := G, we see that f is C α . The smooth manifold structure on C k (M, G) is therefore pre-canonical. The topology on the projective limit Ω 1 C k−1 (M, g) is initial with respect to the limit maps r K , whence the topology on Ω 1 C k−1 (M, g) × G is initial with resspect to the maps r K × id G . Since ψ is a homeomorphism, we deduce that the topology O on the Lie group C k (M, G) is initial with respect to the maps (r K × id G )• ψ = ψ K • ρ K . Since ψ K is a homeomorphism, O is initial just as well with respect to the family (ρ K ) K∈K . But also the compact-open C k -topology T on C k (M, G) is initial with respect to this family of maps (see Lemma 3.7), whence O = T and C k (M, G) is canonical.
If M has finitely many components M 1 , . . . , M n , we give C k (M, G) the smooth manifold structure turning the bijection into a C ∞ -diffeomorphism. Let ρ j be its jth component. Since ρ is a homeomorphism for the compact-open C k -topologies (cf. Lemma 3.7) and an isomorphism of groups, the preceding smooth manifold structure makes C k (M, G) a Lie group and is compatible with the compact-open C k -topology. As each of the Lie groups C k (M j , G) is C r -regular, also their direct product (and thus C k (M, G)) is C r -regular. Since ρ = (ρ j ) n j=1 is an isomorphism of Lie groups, is an isomorphism of topological Lie algebras. For x ∈ M j , the point evaluation g). Let us show that the Lie algebra homomorphism φ : L(C k (M, G)) → C k (M, g) is a homeomorphism. Lemma 3.7 entails that the map is a homeomorphism. By the preceding, r • φ = (φ 1 × · · · × φ n ) • (L(ρ j )) n j=1 is a homeomorphism, whence so is φ. Thus, the Lie group structure on C k (M, G) is compatible with evaluations. If α, L = L 1 × · · · × L m and f : L → C k (M, G) are as above and f is C α , then f ∧ is C α,k by the above argument. If, conversely, , . . . , n}. As a consequence, ρ • f is C α and thus also f . We have shown that the smooth manifold structure on C k (M, G) is pre-canonical and hence canonical, as compatibility with the compact-open C k -topology was already established. ✷ Another lemma is useful. Assume that C β (M, G) has a pre-canonical smooth manifold structure, using which C α (N, C β (M, G)) has a canonical smooth manifold structure. Endow C α,β (N × M, G) with the pre-canonical smooth manifold structure turning Assume that there exists a family (K i ) i∈I of compact full submanifolds K i of N whose interiors cover N , with the following properties: (a) For each i ∈ I, we have K i = K i,1 × · · · × K i,m with certain compact full submanifolds K i,ℓ ⊆ N ℓ ; and (b) C β (M, C α (K i , G)) admits a canonical smooth manifold structure for each i ∈ I, using the canonical smooth manifold structure on C α (K i , G) provided by Theorem 1.1.
Then the pre-canonical manifold structure on C α,β (N × M, G) is canonical.
Proof. Let O be the topology on C α,β (N × M, G), equipped with its precanonical smooth manifold structure. Using Theorem 1.1, for i ∈ I we endow C α (K i , C β (M, G)) with a canonical smooth manifold structure; the underlying topology is the compact-open C α -topology. The given smooth manifold structure on C α (N, C β (M, G)) being canonical, its underlying topology is the compact-open C α -topology, which is initial with respect to the restriction maps for i ∈ I. We have bijections using in turn the Exponential Law (in the form (1)), a flip in the factors (cf. Lemma 4.12 (a)), and again the Exponential Law. If, step by step, we transport the smooth manifold structure from the left to the right, we obtain a precanonical smooth manifold structure in each step (see Lemmas 4.11 (c) and 4.12 (a)). As pre-canonical structures are unique, the pre-canonical structure obtained on C β (M, C α (K i , G)) must coincide with the canonical structure which exists by hypothesis. Hence, using this canonical structure, the map Let L k be the set of compact full submanifolds of M k for k ∈ {1, . . . , n}. Write L 1 × · · · × L n =: J. If j ∈ J, then j = (L j,1 , . . . , L j,n ) with certain compact full submanifols L j,k ⊆ M k ; we define L j := L j,1 × · · · × L j,n . By Lemma 3.7, the topology on C β (M, C α (K i , G)) is initial with respect to the restriction maps using the compact-open C α -topology on the range which underlies the canonical smooth manifold structure given by Theorem 1.1. Let Θ i,j be the composition of the bijections . As each of the domains and ranges admits a canonical smooth manifold structure (by Theorem 1.1), all of the maps have to be homeomorphisms (see Proposition 4.13 and Lemma 4.12 (b)). Thus Θ i,j is a homeomorphism. By transitivity of initial topologies, O is initial with respect to the mappings which are the restriction maps C α,β (N × M, G) → C α,β (K i × L j , G). Also the compact-open C α,β -topology on C α,β (N × M, G) is initial with respect to the maps ρ i,j , and hence coincides with O. The given pre-canonical smooth manifold structure on C α,β (N × M, G) therefore is canonical. ✷ Lemma 6.10 Let M 1 , . . . , M n be locally compact, smooth manifold with rough boundary, M := M 1 × · · · × M n α ∈ (N 0 ∪ {∞}) n , and G be a Lie group. Assume that the group C α (M, G) is endowed with a smooth manifold structure which makes it a Lie group and is compatible with evaluations. Let σ be a permutation of {1, . . . , n} and Q := M σ(1) × · · ·× M σ(n) . Consider φ σ : M → Q, x → x • σ. Then the smooth manifold (and Lie group) structure on the group C α•σ (Q, G) making the bijective group homomorphism Proof. The map ψ : C α•σ (Q, L(G)) → C α (M, L(G)), f → f • φ φ is an isomorphism of topological vector spaces, by Example 4.1 and Lemma 4.12 (b). Writē ε y : C α•σ (Q, G) → G for the point evaluation at y ∈ Q and ε x : . As a consequence, is an isomorphism of topological vector spaces, being a composition of such. ✷ Proof of Theorem 1.3 and Proposition 6.6.
Step 1. We first assume that M j is 1-dimensional with finitely many components for all j ∈ {1, . . . , n}, and prove the assertions by induction on n. The case n = 1 was treated in Lemma 6.8. We may therefore assume that n ≥ 2 and assume that the conclusions hold for n − 1 factors. We abbreviate k := α 1 , β := (α 2 , . . . , α n ), and L := M 2 × · · · × M n . By the inductive hypothesis, C β (L, G) admits a canonical smooth manifold structure which makes it a C r -regular Lie group and is compatible with evaluations. By the induction base, C k (M 1 , C β (L, G)) admits a canonical smooth manifold structure making it a C r -regular Lie group. Since C β (L, G) is canonical, the group homomorphism is a bijection (see (4.13)). We endow with the smooth manifold structure turning Φ into a C ∞ -diffeomorphism. By Lemma 6.4, this structure is pre-canonical, makes C α (M, G) Lie group, and is compatible with evaluations. The Lie group C α (M, G) is C r -regular, as Φ is an isomorphism of Lie groups. Let C 1 , . . . , C ℓ be the connected components of M 1 . Let K be the set of compact, full submanifolds K of M 1 . Then the interiors K o cover M 1 (as the interiors of connected, compact full submanifolds cover each connected component of M 1 , by the proof of Lemma 6.8). Now C k (K, G) admits a canonical smooth manifold structure making it a C r -regular Lie group, by Lemma 6.7. Thus C β (L, C k (K, G)) admits a canonical smooth manifold structure, by the inductive hypothesis. By Lemma 6.9, the pre-canonical smooth manifold structure on C α (M, G) is canonical.
Step 2 (the general case). Let M 1 , . . . , M n be arbitrary. Using Lemma 4.12 (a), we may re-order the factors and assume that there exists an m ∈ {0, . . . , n} such that M j is compact for all j ∈ {1, . . . , n} with j ≤ m, while M j is 1-dimensional with finitely many components for all j ∈ {1, . . . , n} such that j > m. If m = 0, we have the special case just settled. If m = n, then all conclusions hold by Lemma 6.7. We may therefore assume that 1 ≤ m < n. We abbreviate K := M 1 × · · · × M m and N := M m+1 × · · · × M n . Let γ := (α 1 , . . . , α m ) and β := (α m+1 , . . . , α n ). By Step 1, C β (N, G) admits a canonical smooth manifold structure which makes it a C r -regular Lie group and is compatible with evaluations. By Lemma 6.7, C γ (K, C β (N, G)) admits a canonical smooth manifold structure which makes it a C r -regular Lie group and is compatible with evaluations. We give C α (M, G) = C γ,β (K × N, G) the smooth manifold structure making the bijection C β (N, G)), f → f ∨ a C ∞ -diffeomorphism. By Lemma 6.4, this smooth manifold structure is precanonical, makes C α (M, G) a Lie group, and is compatible with evaluations.
The Lie group C α (M, G) is C r -regular as Φ is an isomorphism of Lie groups. Now C γ (K, G) admits a canonical smooth manifold structure, which makes it a C r -regular Lie group (Lemma 6.7). By Step 1, C β (N, C γ (K, G)) admits a canonical smooth manifold structure. The pre-canonical smooth manifold structure on C α (M, G) is therefore canonical, by Lemma 6.9.
The following result complements Theorem 1.3. Under a restrictive hypothesis, it provides a Lie group structure without recourse to regularity.
Proposition 6.11 Let M 1 , . . . , M n be locally compact smooth manifolds with rough boundary, α ∈ (N 0 ∪{∞}) k and G be a Lie group that is C ∞ -diffeomorphic to a locally convex space E. Abbreviate M := M 1 × · · · × M n . Then C α (M, G) admits a canonical C ∞ -manifold structure, which is compatible with evaluations.
If G is C r -regular for some r ∈ N 0 ∪ {∞}, then also C α (M, G) is C r -regular.
Proof. By Example 4.1, H := C α (M, G) admits a canonical smooth manifold structure and this structure makes it a Lie group (see Lemma 6.1). Let ψ : G → E be a C ∞ -diffeomorphism such that ψ(e) = 0. Abbreviating g := L(G) and h := L(H), the map α := dψ| g : g → E is an isomorphism of topological vector spaces. Then also φ := α −1 • ψ : G → E is a C ∞ -diffeomorphism such that φ(e) = 0; moreover, dφ| g = id g . Now is a C ∞ -diffeomorphism, and thus β := d(φ * )| h : h → C α (M, g) is an isomorphism of topological vector spaces. For x ∈ M , let ε x : H → G and e x : C α (M, g) → g be the respective point evaluation at x. We show that β(v) = (L(ε x )(v)) x∈M for each v ∈ h, whence the Lie group structure on H is compatible with evaluations. Regard v = [γ] as a geometric tangent vector. As L(ε x )(v) ∈ g, we have and e x is continuous and linear. For the final assertion, see Lemma 6.5. ✷

Manifolds of maps with finer topologies
We now turn to manifold structures on C α (M, N ) for non-compact M , which are modeled on suitable spaces of compactly supported C α -functions. Notably, a proof for Theorem 1.4 will be provided. Such manifold structures need not be compatible with the compact-open C α -topology, and need not be pre-canonical. But we can essentially reduce their structure to the case of canonical structures for compact domains, using box products of manifolds as a tool. We recall pertinent concepts from [14].

7.1
If I is a non-empty set and (M i ) i∈I a family of C ∞ -manifolds modeled on locally convex spaces, then the fine box topology O fb on the cartesian product P := i∈I M i is defined as the final topology with respect to the mappings for φ := (φ i ) i∈I ranging through the families of charts φ i : = i∈I E i is endowed with the locally convex direct sum topology, and the left-hand side V φ of (9), which is an open subset of E φ , is endowed with the topology induced by for only finitely many i ∈ I .
Note that the projection pr i : P → M i is continuous for each i ∈ I, entailing that the fine box topology is Hausdorff. In fact, using the continuous linear projection π i : E φ → E i onto the ith component, we deduce from the continuity of pr i • Θ φ = φ −1 i • π i | V φ for each φ that pr i is continuous.

7.2
Let φ be as before and ψ be an analogous family of charts ψ i : for all but finitely many i ∈ I, then which is an open 0-neighbourhood in i∈I E i . The transition map is C ∞ (as follows from [11,Proposition 7.1]) and in fact a C ∞ -diffeomorphism, and hence a homeomorphism, since for infinitely many i ∈ I, then (Θ φ ) −1 (U φ ∩ U ψ ) = ∅ and the transition map trivially is a homeomorphism. Using a standard agrument, we now deduce is open in (P, O fb ) for all φ and Θ φ is a homeomorphism onto its image (see, e.g., [15,Exercise A.3.1]). By the preceding, the maps Φ φ := (Θ φ | U φ ) −1 : U φ → V φ ⊆ E φ are smoothly compatible and hence form an atlas for a C ∞ -manifold structure on P . Following [14], we write P fb for P , endowed with the topology O fb and the smooth manifold structure just described, and call P fb the fine box product.
Some auxiliary results are needed. We use notation as in 5.8 and Theorem 1.4.
. . , M n are paracompact, L ⊆ M is a compact subset and K := K 1 × · · · × K n with compact full submanifolds K j ⊆ M j for j ∈ {1, . . . , n}, then the linear map Γ f,L → Γ f |K , τ → τ | K is continuous. . , n} such that K x := K x,1 × · · · × K x,n ⊆ M \ K and x ∈ K o x . Lemma 3.7 implies that the compact-open C α -topology on Γ f,L is initial with respect to the restriction maps ρ : Γ f,L → C α (K, T N ) and ρ x : Γ f,L → C α (K x , T N ) for x ∈ M \ K. As each ρ x is constant (its value is the function K x ∈ y → 0 f (y) ∈ T f (y) N ), it can be omitted without affecting the initial topology. The topology on Γ f,K is therefore initial with respect to ρ, and hence also with the co-restriction r of ρ. Thus r is a topological embedding and hence an homeomorphism, as r(τ ) = σ can be achieved for σ ∈ Γ f |K ,L if we define τ : Being linear, r is an isomorphism of topological vector spaces. ✷ Proof of Theorem 1.4. For j ∈ {1, . . . , n}, let (K j,i ) i∈Ij be a locally finite family of compact, full submanifolds K j,i of M j whose interiors cover M j . Let I := I 1 × · · · × I n . Then the sets K i := K 1,i1 × · · · × K n,in form a locally finite family of compact full submanifolds of M whose interiors cover M , for i = (i 1 , . . . , i n ) ∈ I. The map In fact, the inclusion "⊆" is obvious. If (f i ) i∈I is in the set on the right-hand side, then a piecewise definition, f (x) : For each i ∈ I, endow C α (K i , N ) with the canonical smooth manifold structure, as in Theorem 1.1, modeled on the set N ). Let Σ : T N ⊇ U → N be a local addition for N ; as in Section 5, write is continuous on Γ f,L for each compact subset L ⊆ M (see Lemma 7.3 (b)) and hence continuous on the locally convex direct limit Γ f . As above, we see that which is a closed vector subspace of i∈I Γ f |K i . We now show that s is a homeomorphism onto its image. In fact, s admits a continuous linear left inverse. To see this, pick a C ∞ -partition of unity (h i ) i∈I on M subordinate to (K o i ) i∈I ; then L i := supp(h i ) is a closed subset of K i and thus compact. The multiplication operator β i : Γ f |K i → Γ f |K i ,Li , τ → h i τ is continuous linear (by Lemma 7.3 (a)). Moreover, the restriction operator s i : Γ f,Li → Γ f |K i ,Li is an isomorphism of topological vector spaces (Lemma 7.3 (c)). Thus By the universal property of the locally convex direct sum, also the linear map is continuous. We easily verify that σ • s = id Γ f .
In fact, for i, j ∈ I and x ∈ K i ∩ K j we have Σ(τ i (x)) = Σ(τ j (x)) if and only if τ i (x) = τ j (x), from which the assertion follows in view of (10) and (11). Thus be the corresponding submanifold chart for im(ρ). Then Hence is a chart for the smooth manifold structure on C α (M, N ) modeled on E (the set of all Γ f ) which makes ρ : C α (M, N ) → im(ρ) a C ∞ -diffeomorphism. Note that the smooth manifold structure on C α (M, N ) which is modeled on E and makes ρ a C ∞ -diffeomorphism is uniquely determined by these properties. Thus, it is independent of the choice of Σ. On the other hand, the (φ f ) −1 form a C ∞ -atlas for a given local addition Σ. As the definition of the φ f does not involve the cover (K i ) i∈I , the smooth manifold structure just constructed is independent of the choice of (K i ) i∈I .

A Details for Sections 2 and 3
In this appendix, we provide proofs for preliminaries in Sections 2 and 3.
Proof of Lemma 2.8. The right-hand side (t, y) → y.γ(t) of the differential equationẏ(t) = y(t).γ(t) is C k , whence its solution η will be C k+1 , if it exists.
The following lemma fills in the details for 3.3.
Proof. We show by induction on k 0 ∈ N that the conclusion holds with k ≤ k 0 for all functions as described in the lemma, for all α with α n ≥ k 0 . Using local charts, we may assume that U j := M j is a locally convex subset of a locally convex space E j for all j ∈ {1, . . . , n} and N a locally convex subset of a locally convex space F ; thus f is a map U := U 1 × · · · × U n → F . The case k 0 = 0 being trivial as h 0 = f is C α . Let 1 ≤ k 0 ≤ α n now. Then is a C (α−en,0) -map. Being linear in the final argument, d en f is C α−en as a map of n variables, i.e., as a map on the domain T en U = U 1 × U n−1 × T U n (see [1,Lemma 3.11]). Let pr 1 : T U n = U n × E n → U n be the projection onto the first component. Then g := f •id U1 × · · ·×id Un−1 × pr 1 : U 1 ×· · ·×U n−1 ×T U n → F is C α by the Chain Rule [1, Lemma 3.16], and hence C α−en . Thus h 1 = (g, d en f ) is C α−en , by [1,Lemma 3.8]. By the inductive hypothesis, the maps are C α−en−jen for all j ∈ {0, . . . , k 0 − 1}. It only remains to observe that this map equals h j+1 . ✷ Proof of Lemma 3.5. (a) For β ∈ N n 0 with β ≤ α, consider the maps N ) and making repeated use of the functoriality of T , we see that Thus τ β • C α (M, g) = C(T β M, T |β| g) • T β , which is a continuous map by [15,Lemma A.6.3]. The topology on C β (M, L) being initial with respect to the maps τ β , we deduce that C α (M, g) is continuous.
Proof of Lemma 3.7. By definition, the compact-open C α -topology O on C α (M, N ) is initial with respect to the maps τ β : the compact-open topology on C(T β M, T |β| N ) is initial with respect to the restriction maps ρ β,i : The compactopen C α -topology on C α (K i , N ) being initial with respect to the mappings τ β,i : that O is initial with respect to the maps ρ i . .
Proof of Lemma 3.8. The case n = 1 is well known. The general case follows as T β S = T β1 S 1 × · · · × T βn S n and T β M = T β1 M 1 × · · · × T βn M n .
Proof of Lemma 3.9. The inclusion map λ : S → N is smooth. By Lemma 3.8, the inclusion map T |β| λ : T |β| S → T |β| N is a topological embedding, for each β ∈ (N 0 ) n such that β ≤ α. Thus (T |β| λ) Proof of Lemma 3.10. For each k ∈ N 0 , T k F = F 2 k is a locally convex space. For each β ∈ (N 0 ) n such that β ≤ α, the map is linear. In fact, T k : C k (N, F ) → C(T k N, T k F ) is linear for each smooth manifold N with rough boundary [15, proof of Proposition 4.1.11] and k ∈ N 0 , establishing linearity if n = 1. If n ≥ 2, the preceding entails that T (0,...,0,βn) f (v) = T βn (f (x 1 , . . . , x n−1 , ·))(v n ) is linear in f for all x j ∈ M j for j ∈ {1, . . . , n − 1} and v n ∈ T βn M n , showing that T (0,...,0,βn) f is linear in f . Likewise, g and T (0,...,0,β k−1 ,...,βn) f is linear in f in the recursive construction in 3.3, which gives the assertion for n ≥ 2. Thus is a linear map. It is a homeomorphism onto its image, which is a locally convex space. Hence also C α (M, F ) is a locally convex space.
Proof of Lemma 3.11. (a) For each k ∈ N 0 , the topology on T k F = F 2 k is initial with respect to the linear maps T k λ i = λ 2 k i : F 2 k → F 2 k i . For each β ∈ N n 0 with β ≤ α, the compact-open topology on C(T β M, T |β| F ) is therefore initial with respect to the mappings C(T β M, T |β| λ i ) : C(T β M, T |β| F ) → C(T β M, T |β| F i ) for i ∈ I, see [15,Lemma A.6.4]. Thus, the compact-open C α -topology O on C α (M, F ) is initial with respect to the maps C(T β M, T |β| λ i ) • T β with T β : C α (M, F ) → C(T β M, T |β| F ). As T β (λ i • f ) = (T |β| λ i ) • (T β f ), writing τ i,β (g) := T β g for g ∈ C α (M, F i ) we have The topology on C α (M, F i ) being initial with respect to the mappings τ i,β : C α (M, F i ) → C(T β M, T |β| F i ) for β ≤ α, we deduce that O is initial with respect to the mappings C α (M, λ i ) = (λ i ) * . (b) By [1, Lemma 3.8], the linear map Θ is a bijection. The topology on F being initial with respect to the maps pr i , (a) shows that the topology on C α (M, F ) is initial with respect to the maps (pr i ) * and hence makes Θ a topological embedding. Hence Θ is a homeomorphism, being bijective. (c) By [1, Lemma 3.8], Ψ is a bijection. By Lemma 3.5, Ψ is continuous. To see that Ψ −1 is continuous, we prove its continuity at a given element (f 1 , f 2 ) in C α (M, N 1 ) × C α (M, N 2 ). For x ∈ M , pick a chart φ x,i : U x,i → V x,i ⊆ E x,i of N i around f i (x), for i ∈ {1, 2}. There exist compact full submanifolds K x,j of M j for j ∈ {1, . . . , n} such that K x := K x,1 ×· · ·×K x,n ⊆ (f 1 , f 2 ) −1 (U x,1 ×U x,2 ) and x ∈ K o x . By Lemma 3.7, the topology on C α (M, N 1 × N 2 ) is initial with respect to the restriction maps It thus suffices to show that ρ x • Ψ −1 is continuous at (f 1 , f 2 ) for all x ∈ M . Now ρ x • Ψ −1 = Ψ −1 x • (ρ x,1 × ρ x,2 ) using the continuous restriction maps ρ x,i : C α (M, N i ) → C α (K x , N i ) for i ∈ {1, 2} and the map taking a function to its pair of components. Thus, it suffices to show that Ψ −1 x is continuous at (f 1 | Kx , f 2 | Kx ). Now f i | Kx is contained in the open subset C α (K x , U x,i ) of C α (K x , N i ), on which the latter induces the compact-open C αtopology, by Lemma 3.9. The map Ψ −1 takes this set onto C α (M, U x,1 × U x,2 ), on which C α (M, N 1 × N 2 ) induces the compact-open C α -topology. It thus suffices to show that Ψ −1 x is continuous at (f 1 | Kx , f 2 | Kx ) as a map C α (K x , U x,1 ) × C α (K x , U x,2 ) → C α (K x , U x,1 × U x,2 ). Now (φ x,j ) * : C α (K x , U x,j ) → C α (K x , V x,i ) is a homeomorphism for i ∈ {1, 2} and also (φ x,1 × φ x,2 ) * : C α (K x , U x,1 × U x,2 ) → C α (K x , V x,1 × V x,2 ) is a homeomorphism, by Lemma 3.5. It thus suffices to show that the mapping is continuous. But this mapping is a restriction of the homeomorphism and R : C α (N, C β (M, E)) → i∈I C α (U i , C β (M, E)), f → (f | Ui ) i∈I are isomorphisms of topological vector spaces. By Lemma 3.5, the mapping C α (U i , r) : C α (U i , C β (M, E)) → C α (U i , j∈J C β (V j , E)) is an isomorphism of topological vector spaces and so is the map taking a map to its family of components (see Lemma 3.11 (b)). Hence is an isomorphism of topological vector spaces. By [1,Theorem B], the map Φ i,j : C α,β (U i ×V j , E)) → C α (U i , C β (V j , E)), f → f ∨ is linear and a topological embedding, whence so is Ψ := (i,j)∈I×J Φ i,j : Evaluating at x ∈ N and then in y ∈ M (say x ∈ U i and y ∈ V j ), we verify that for all f ∈ C α,β (N × M, E), whence f ∨ ∈ C α (N, C β (M, E)) and Φ makes sense as a map to the latter space. We have a commutative diagram E)) where the vertical arrows are homeomorphisms and Ψ is a topological embedding. Hence Φ is a topological embedding. If M is locally compact, then so are the V j , whence each of the maps Φ i,j is a homeomorphism by [1,Theorem 4.4] and hence also Ψ. Then also Φ = Ξ −1 • Ψ • ρ is a homeomorphism.