Math 519 ? Spring 2014
Erin Compaan
1 Introduction
Manifolds at any regularity level, from continuous to smooth, can be studied. It’s natural to wonder how distinct these concepts are – if there is a fundamental difference between, say, a C1 manifold and a smooth one. A function space topology helps answer this question. With a topology in place, one can also prove various useful approximation results.
In the first section, the weak and strong topologies are introduced using coordinate- based definitions, and some of their basic properties discussed. In the next section, we give some of the major approximation results, e.g. that every Cr map can be approximated by a smooth map and every Cr manifold is Cr-diffeomorphic to a smooth manifold. The last section introduces jets to give equivalent definitions of the weak and strong topologies. As an application, this defintion is used to prove another approximation result.
This exposition is based on that in Hirsch’s Differential Topology, Chapter 2. Munkres is a useful reference for topological background. In the following, M and N are Cr manifolds, with r ∈ [0, ∞] depending on the context. All manifolds are without boundary.
2 Coordinate-Based Definitions
We begin by defining the weak topology on Cr(M,N).
Definition 1 Let f ∈ Cr(M,N). Take a chart (φ, U) on M, a compact set K ⊂ U, a chart (ψ, V ) on N such that f(K) ⊂ V and > 0. The corresponding weak subbasic neighborhood of f is the set
Br(f;(φ, U),K, (ψ, V ), ) of functions g ∈ Cr(M,N) such that g(K) ⊂ V and
k −1 k −1 D (ψfφ )(x) − D (ψgφ )(x) < for all x ∈ φ(K) and each k = 0, 1, . . . , r.
Here k · k is a standard Euclidean norm.
Using these sets as a subbasis, we obtain a topology on Cr(M,N). The resulting topo- r logical space is denoted CW (M,N). This topology has the advantage of being quite easy to define and understand, but there is a possible drawback: A basic neighborhood of f in this topology consists of functions g which are close to f on some compact set. If M is
1 not compact, this may not be a good measure of how close the functions are. The strong topology defined below remedies this issue.
Definition 2 Let f ∈ Cr(M,N) and take
• Φ = {(φα,Uα)}α a family of charts on M such that {Uα}α is locally finite,
• K = {Kα}α a family of compact sets with Kα ⊂ Uα,
• Ψ = {(ψα,Vα)}α a family of charts on N such that f(Kα) ⊂ Vα, and
• E = {α}α a collection of positive numbers. The corresponding strong basic neighborhood of f is the set
Br(f;Φ,K, Ψ,E)
r of functions g ∈ C (M,N) such that for every α, we have g(Kα) ⊂ Vα and
k −1 k −1 D (ψαfφα )(x) − D (ψαgφα )(x) < α for all x ∈ φ(Kα)and each k = 0, 1, . . . , r.
To see that this defines a basis, suppose g ∈ Br(f;Φ,K, Ψ,E) ∩ Br(f 0;Φ0,K0, Ψ0,E0). Then the neighborhood Br(g;Φ ∪ Φ0,K ∪ K0, Ψ ∪ Ψ0, E˜) is contained in the intersection pro- vided the distances in E˜ are sufficiently small. The set Cr(M,N) with the strong topology r is denoted CS(M,N).
We can now define the strong and weak topologies on C∞(M,N). The strong topol- ∞ r ogy is the union of all the topologies induced by the inclusions C (M,N) → CS(M,N) for r ∈ N, and similarly the weak topology is the union of the topologies induced by the ∞ r inclusions C (M,N) → CW (M,N).
To validate the use of the strong topology, one can show that various important sets, such r as submersions, embeddings, proper maps, etc., are open in CS(M,N) for any 1 ≤ r ≤ ∞:
r Theorem 3 The following sets are open in CS(M,N) for any r ≥ 1: i) Cr submersions, iv) proper Cr maps, ii) Cr immersions, v) closed Cr embeddings, iii) Cr embeddings, vi) Cr diffeomorphisms.
Proof of i). It is enough to prove this in the case r = 1 since
{Cr(M,N) submersions} = {C1(M,N) submersions } ∩ Cr(M,N)
r 1 and the inclusion CS(M,N) → CS(M,N) is continuous.
1 Let f ∈ C (M,N) be an immersion. Choose an atlas Φ = {φα,Uα}α of M such that each Uα has compact closure and f(Uα) ⊂ Vα for some chart (ψα,Vα) on N. Let Ψ = {(ψα,Vα)},
2 and let K = {Kα}α be a compact cover of M such that Kα ⊂ Uα.
The set −1 Aα = {D(ψαfψα (x): x ∈ φ(Kα)} m n n m is then compact in the set L(R , R ) of linear maps from R to R . Each map is surjective n m and surjectivity is an open condition in L(R , R ), so there is α such that if S ∈ Aα, n m T ∈ L(R , R ), and kT − Sk < α, then T is surjective. Set E = {α}. Then each element 1 in B (f;Φ, Ψ,K,E) is a submersion.
Proof of iv). This property in fact holds for r = 0 also, as the proof will show. Let f : M → N be a proper map. Take an atlas Φ = {(φα,Uα)}α on M such that the Uα are locally finite and have compact closure. Let K = {Kα}α be a compact cover of M with Kα ⊂ Uα. Using the fact that f is proper, it is possible to ensure, by decomposing the Uα and Kα into smaller sets if necessary, that there is an atlas Ψ = {(ψα,Vα)}α of N such r that the Vα are locally finite and f(Uα) ⊂ Vα. We claim that B (f;Φ, Ψ,K,E) consists of proper maps, for arbitrary E = {α}α. Indeed, if g is in this neighborhood of f and L ⊂ N is compact, then g(Kα) ⊂ Vα for each α and L intersects only finitely many Vα. Then −1 −1 g (L) is a closed set which intersects only finitely many Kα, so g (L) is a finite union of compact sets.
The proofs of the remaining statements are omitted. The proof of ii) is parallel to that of i); the proof of iii) is more technical. Property iv) follows since the set of closed embeddings is the set of proper embeddings, while vi) holds by reducing to the case of connected manifolds and noting that diffeomorphisms are exactly the proper embeddings which are submersions.
3 Approximations
With a topology on Cr(M,N) established, we can make precise statements about the re- lationship between Cr and Cs manifolds. The first results deal with approximating Cr functions by higher-regularity maps.
Suppose first that M and N are subsets of Euclidean space. In this case, smooth approximations are constructed using the convolution: If f ∈ Cr(M,N) and K ⊂ M is compact, choose a radius δ such that B(x, δ) ⊂ M for each x ∈ K and such that kDkf(y) − Dkf(x)k ≤ for all x, y ∈ K such that |x − y| ≤ δ. Let θ be a smooth bump function with supp θ ⊂ B(0, δ) and R θ = 1. Then the convolution of f with θ is a smooth r function which C approximates f on the neighborhood ∪x∈K B(x, δ).
A partition of unity is used to extend this result beyond compact sets.
m n ∞ r Theorem 4 Let M ⊂ R and N ⊂ R be open. Then C (M,N) is dense in CS(M,N) for any r ≥ 0.
3 Proof. Let f ∈ Cr(M,N). Since M and N are subsets of Euclidean space, a basis of r r CS(M,N) at f consists of sets B (f; K,E), where K = {Kα}α is a locally finite collection of compact subsets of U and E = {α}α is a collection of positive numbers. The elements of Br(f; K,E) are the functions g ∈ Cr(M,N) such that
k k kD g(x) − D f(x)k < α for all x ∈ Kα and each k = 0, 1, . . . , r.
Given K and E, we will construct a smooth function which lies in Br(f; K,E). Let {λα}α be a smooth partition of unity on M such that Kα ⊂ Int (supp λα) and supp λα is compact. For any δα > 0, there is a smooth function gα which approximates f on a neighborhood of the compact set supp λα satisfying
k k kD gα(x) − D f(x)k < δα for all x ∈ Kα ⊂ supp λα, k = 0, 1, . . . , r.
We will choose the δα later. Define g : M → N by X g(x) = λα(x)gα(x). α
k j Note that for each k, the derivative D (λαgα) is a sum of bilinear functionals in the D λα j and D gα. This functional depends only on r and the dimensions of the Euclidean spaces – not on λα or gα. Thus for some Ck > 0, we have the estimate
k j j kD λα(x)gα(x)k ≤ Ck max kD λα(x)k max kD gα(x)k. 0≤j≤k 0≤j≤k
Let C = max{C0,C1,...,Cr}, and let Aα = {β : Kα ∩ Kβ 6= ∅}. Local finiteness of K = {Kα}α implies that this a finite set.
Now let x ∈ Kα. We have
k k X k kD g(x) − D f(x)k = D λβ(x) gβ(x) − f(x)
β∈Aα X j ≤ Ck max kD λβ(x)k · δβ. 0≤j≤k β∈Aα
We can make this as small as we like, i.e. smaller than α, by choosing the δα sufficiently r small. This choice ensures g ∈ B (f; K,E) as desired.
As one might expect, exactly the same statement holds for general manifolds M and N. The idea of the proof is to take a countable atlas {(φn,Un)}n of M and apply the Euclidean space result to define an approximating function on each coordinate domain. By repeating this process inductively, one can obtain an approximating function on the entire manifold M. There’s a fair bit of work involved in making this precise though, as one must show that the cobbled-together function is sufficiently smooth and remains a good approximation.
From this approximation result for general manifolds and the openness theorem stated in the previous section, its clear that the set of Cr(M,N) diffeomorphisms is dense in the
4 set of Cs diffeomorphisms for any 1 ≤ r ≤ s. From this it follows that two manifolds are Cr diffeomorphic if and only if they are Cs diffeomorphic. One direction of this statement is obvious, of course, but the other is more surprising.
Furthermore, the following result tells us that if M is a Cr manifold, we can make it into a Cs manifold by eliminating some of the charts in its atlas.
Theorem 5 Let M be a manifold with a Cr differential structure β. Then for any s ≥ r, there is a Cs differential structure γ on M such that every chart of γ is a chart of β.
The proof uses Zorn’s lemma to find a maximal subset of B ⊂ M on which there is such a Cs structure. One then proves by contradiction that B must in fact be the entire manifold M. Using this theorem, we conclude that for 1 ≤ r ≤ s ≤ ∞, any Cr manifold is Cr diffeomorphic to a Cs manifold. This indicates that generally it suffices to study C∞ manifolds – the results for Cr manifolds with r < ∞ will be parallel.
4 Topologies through Jets
This section introduces an alternative, more abstract, definition of the topologies on Cr(M,N). The construction is based on identifying Cr(M,N) with a set of continuous maps from M into a new manifold J r(M,N), called the set of r-jets from M to N. Though constructing the space of jets takes some work, the resulting definition of the topology on Cr(M,N) is elegant and yields some useful properties.
Definition 6 Let M and N be Cr manifolds with r < ∞. The set of r-jets from M to N is the set of equivalance classes
r r r J (M,N) = jxf : x ∈ M and f ∈ C (U, N) for some open set U containing x . r r 0 0 The equivalence relation is jxf = jx0 f if and only if x = x and for any charts (φ, V ) of M and (ψ, W ) of N such that x ∈ V and f(V ) ⊂ W , the derivatives of ψfφ−1 and ψf 0φ−1 at φ(x) agree up to order r.
r The set of r-jets based at a point x is denoted Jx(M,N).
r m n It is useful to consider the set of jets between two Euclidean spaces, J (R , R ). In this r case, there is an obvious choice of representative for the jet jxf, namely the r-th order Taylor approximation of f at x. This is a polynomial completely determined by the derivatives r m n of f at x, a finite list of numbers. Therefore it makes sense to identify Jx(R , R ) with a vector space:
r m n n 1 m n 2 m n r m n Jx(R , R ) = R × Lsym(R , R ) × Lsym(R , R ) × · · · × Lsym(R , R ).
k m n m n Here Lsym(R , R ) is the space of symmetric k-linear maps from R to R , the space containing the k−th derivative at x. Allowing the basepoint x to vary yields an identification r m n m n 1 m n 2 m n r m n J (R , R ) = R × R × Lsym(R , R ) × Lsym(R , R ) × · · · × Lsym(R , R ) .
5 The same construction works for J r(U, V ) when U and V are open subsets of Euclidean space, and we see that J r(U, V ) is an open subset of a finite-dimensional vector space. This is used to construct a topology on J r(M,N) for general Cr+s manifolds M and N as follows. Let (φ, U) and (ψ, V ) be charts on M and N respectively. There is a bijection
r r Θφψ : J (U, V ) → J (φ(U), ψ(V )) r r −1 jxf 7→ jφ(x)(ψfφ ).
Furthermore, note that Θ Θ−1 : jrg 7→ jr (ψψ˜ −1)g(φφ˜ −1) . φ˜ψ˜ φψ y (φφ˜ −1)(y)
˜ −1 ˜ −1 r+s −1 r The transition functions ψψ and φφ are C diffeomorphims, so Θφ˜ψ˜Θφψ is a C map between Euclidean spaces if we identify J r(φ(U), ψ(V )) and J r(φ˜(U), ψ˜(V )) with Euclidean r spaces as described above. Thus the maps Θφψ can be taken as charts for J (M,N), where J r(M,N) is endowed with the topology induced by these maps.
We are now prepared to identify Cr(M,N) with a set of continuous maps from M to J r(M,N) via the following injection:
jr : Cr(M,N) → C0(M,J r(M,N)) r f 7→ [x 7→ jxf].
r r r r To see that this is indeed an injective map, note that j f = j g if and only is jxf = jxg for every x in M, which in particular implies that f(x) = g(x) for each x.
Through this map, a topology on C0(M,J r(M,N)) yields a topology on Cr(M,N). Recall the compact-open topology on C(X,Y ), which is generated by subbasic sets of the form {f ∈ C(X,Y ): f(K) ⊂ V }, where K ⊂ X is compact and V ⊂ Y is open. This topology on C0(M,J r(M,N)) leads to the weak topology on Cr(M,N) defined previously.
We also have the strong topology on C(X,Y ), defined using the graph: If Gf ⊂ X × Y is the graph of f and W ⊂ X × Y is a neighborhood of Gf , then {g ∈ C(X,Y ): Gg ⊂ W } is a basic neighborhood of f. This topology on C0(M,J r(M,N)) gives the strong topology on C0(M,N).
We now show that Cr(M,N) is identified with a weakly closed set in C0(M,J r(M,N), which has some useful consequences.
Theorem 7 The image of jr in C0(M,J r(M,N) is a weakly closed set.
Proof. The image is weakly closed if it is closed under uniform convergence on compact sets (by the definition of weak convergence when the target space is metric). Any compact set in M can be decomposed into a collection of compact convex coordinate neighborhoods. m Thus, in the local representation, it suffices to show that if U ⊂ R is an open set and
6 r n k {fn} ⊂ C (U, R ) is a sequence such that the derivative sequences {D fn}, k = 0, 1, 2, . . . , r k m n converge uniformly to gk : U → L (R , R ) on U, then the gk came from a single function r m k in C (U, R ). In particular, we must show that gk = D g0. For k = 1, write
g0(x + y) = lim fn(x + y) since fn → g0 n→∞ Z 1 = lim fn(x) + Dfn(x + ty) y dt n→∞ 0 Z 1 = g0(x) + g1(x + ty) y dt by uniform convergence. 0
This implies that g1 = Dg0 as desired. The cases k = 2, . . . , r are parallel.
Defining the topology on C∞(M,N) using jets is a bit more complicated, since we must first define J ∞(M,N). Repeating the above construction won’t work; instead, J ∞(M,N) is defined to be the inverse limit (in the category of topological spaces) of the sequence J 0(M,N) ← J 1(M,N) ← J 2(M,N) ← · · · . The maps jr yield a map j∞ : C∞(M,N) → C0(M,J ∞(M,N)). We can use this corre- spondence as above to construct weak and strong topologies on C∞(M,N). Note that the image in C0(M,J ∞(M,N) is still weakly closed.
r With this established, the properties of the weak and strong topologies give that CS(M,N) is a Baire space in the strong topology, i.e. countable intersections of open dense sets are r dense. This means that the space CS(M,N), though it is not second-countable and has infinitely many components, is well-behaved in some sense. More concretely, one can use the Baire property to prove approximation results like those given in the previous section, as follows.
Theorem 8 Let M and N be Cr manifolds, r ≥ 1. If dim N ≥ 2 dim M + 1, then embeddings are dense in the set of proper maps {f ∈ Cr(M,N): f is proper}.
Proof. Let f ∈ Cr(M,N) be a proper map. Given a neighborhood B = Br(f;Φ,K, Ψ,E), we want find an embedding in B. We may assume, by taking a smaller neighborhood if needed, that B consists of proper maps (since the set of proper maps is open) and that the collection {Kα}α covers M. For each α, let
Xα = {g ∈ B : g|Kα is an embedding}.
This is open since the set of embedding is open in C(Kα,N) (this holds for manifolds r r with boundary such as Kα, and the restriction map C (M,N) → C (Kα, N) is continuous. We claim that Xα is also dense in B. To check this, take an open set Wα with compact closure such that Kα ⊂ Wα ⊂ Wα ⊂ Uα and fix g ∈ B. By the proof of [a version of] the r Whitney embedding theorem, we can approximate g by a C embedding hα on Wα. Then let {λ, θ} be a partition of unity subordinate to {M\Kα,Wα}. The function λg + θhα is an element of Xα which approximates g. Similarly, if Kα ∩ Kβ 6= ∅, then the collection
Xαβ = {g ∈ B : f|Kα∪Kβ is an embedding}
7 is open and dense in B.
Now let K1,K2,... be a sequence of locally finite refinements of the cover K such that j j j j for any pair of distinct points x, y ∈ M, we have x ∈ Kα ∈ K and y ∈ Kβ ∈ K for j j j some disjoint Kα,Kβ ∈ K , i.e. points eventually have disjoint neighborhoods. Since M is second-countable, the Kj can be chosen to be countable.
j Let X ⊂ B be the set of functions g such that f| j j is an embedding for any disjoint Kα∪Kβ j j j j Kα,Kβ ∈ K . Each X is a countable intersection of dense open sets in B (using the above j T j arguement and the fact that K is countable), and hence so is j X . Since B is Baire, the intersection is an open dense set in B. Elements of this intersection are functions which j j are embeddings when restricted to Kα ∪ Kβ for any j, α, and β. This means that they are j immersions, and the separating property of the refinements K implies injectivity.
References
[1] Hirsch, Morris W., Differential Topology. Springer-Verlag, New York, 1994.
[2] Munkres, James R., Topology: A First Course. Prentice-Hall, New Jersey, 1975.
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