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C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES P. M ICHOR Manifolds of smooth maps Cahiers de topologie et géométrie différentielle catégoriques, tome 19, no 1 (1978), p. 47-78. <http://www.numdam.org/item?id=CTGDC_1978__19_1_47_0> © Andrée C. Ehresmann et les auteurs, 1978, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Vol. XIX-1 (1978) CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE MANIFOLDS OF SMOOTH MAPS by X, Y Let P. MICHOR be smooth finite-dimensional be the space of all smooth maps from X Coo ( X , Y ) on a in Sections 1, 2, and smooth manifold modelled pact support of vector on use manifolds, and let Coo ( X, Y ) Y . We introduce the to it then spaces D(F) to [7], of Keller which be can ture to a bundles F over X with the nuclear applied rather strong to some the tangent bundle of dle behaves in a one. %°°-topology C°° (X , Y) of smooth sections with of L. Schwartz. The notion of differentiation which C’ show that We obtain a we use is com- ( LF ) -topology is the concept weak inversion theorem stability. We give a manifold strucCoo ( X, Y ) , and we show that the tangent bunnotions of nice functional way. only finite dimensional manifolds X , Y , since topology Doo depends heavily on it. It may be possible to extend the We have considered the results for some infinite dimensional manifolds Y . We considered theory to the case ory coincides with I week to am very help me been able to only smooth maps, but it is clear how to adapt C’, supposed existing theories, e. g. Leslie [8]. r grateful > 0 . If X is to to E. Binz and A. be compact, then Frblicher, with Lemma 3.8 and without whose prove it. I thank J . Eells vice. 47 who devoted help I would our a the the- whole never have for encouragement and valuable ad- D-TOPOLOGY 1. THE ON X, Y 1.1. NOTATION. Let Coo ( X, Y ) be the Jn (X , Y ) be the fibre let C°°(X, Y). set be smooth finite dimensional of smooth bundle of the canonical manifold mappings from X to of maps from X to n-j ets manifolds, Y ; f or n 6 N , let Y , equipped with makes j n f : X -> ]n (X, Y) f ( Coo ( X, Y ) , where jn f (x) is the n-j et which structure smooth section for each and into a f at of x E X . Let be the components of the fibre bundle proj ection, i. e. and if Q See is a n-j et at x E X of a function [5]. 1.2. DEFINITION. Let K = (Kn), n = 0,1,... be a fixed sequence of com- pact subsets of X such that Then consider sequences is such that nonnegative integer and pair (m , U ) of sequences define mn The S-topology on basis for its open a For each such a C°° ( X , Y ) sets. by: a set is It is easy given by taking to all M (m, U) as a actually a base for sets check that this is topology. 1.3. Let which is compatible Jn ( X, Y), n = 0 , 1, 2, with the topology of the manifold. Consider families 0 = (dn), n = 0, 1, ... of continuous nonnegative functions on X such that the family of the sets (supp On) is locally finite. dn be a metric on ... , f E Coo (X, Y). Then each farnily d = (4)n) open neighborhood Vd ( f ) of f by LEMMA. Let defines an 48 as described above PROOF. We claim that each in such inductively a vd (f) is open. Determine R defined Then consider the continuous maps and let Wl where 17 1,k k > 1 . Then it is (mn)’ way that be the open s et Al-1 ((-1, 1)). Set denotes the canonical easily by checked up by writing out 1.4. Consider sequences of X such that proj ection for the definitions that compact subsets is locally finite on X , and sequences U of open subsets : LEMMA. The Each pair ( L , U ) of sequences family {M’ (L ,U)}, as where L and U above are as defines a above, is set a basis for the 2J-topolo gy. be the fixed sequence of compact subsets of X of PROOF I.2 ; then set as in (XBKno ) clearly locally finite. Let M(m, U) 1.2 ; define L = (Ln) by is define Then clearly 49 be a basic open {M’ ( L , U )) So the system itself a f we can M’ ( L , U) Given of basis if and contains f E M’ (L , U) will we construct a neighborhood Wn = (j n f r1(Un) XBLn° C Wn since in f(XBLno) C Un . For Wn define with X and V0 (f ) C M’(L, U). For each and it is is open. M’ ( L , U) show that each d-topology basis for the a n, V d(f) is open in x c We claim that subset C of is bounded below an This is Wn . seen as and contained in the open and so For x E Wn set by choose a positive constant on follows: j n f (C) t/ . So Un for all xEC. Now an (x)>E a we continuous function contains each compact In ( X, Y) is compact in a set construct a 6x : Wn -> continuous function: R such that d x ( x ) > 0, 0 dx an ; this is possible since an is bounded below on a compact neighborhood of x in W n . Then use a partition of unity on X subordinate and to the cover and obtain Now W’ n bn . (XBLn° ) of open is sets locally finite, such that each XBLno XBLno C I/( C Wn is closed. So there is and W’ 71 is a again locally family finite. Choose continuous functions un : X - [ 0 , 1] such that un = 1 on XBL no , Un = 0 off W’n and let Then dn: x-> [ 0 , oo ) is continuous and d = (dn) is (supp dn)’ C ’ (W’n) is locally finite, so by 1.3 Vd(f) Given ge V d (f), th en f or all n > 0 and for all x E X, i.e. 50 a family is 3-open. such that : so Since for all this implies j n g (x ) fUn get g f M’ ( L , U ) . We have x f XIL no by the definition of an , so we fE f V d (f)C M’ (L,U). Q.E.D. 1.5. CO ROL L AR Y. D-topology on a) Fix a Coo following equivalent descriptions of the (X, Y ) : sequence K Then the systems base We have the the = (Kn) of compact sets in X such that of sets of the form 9-topology COO (X, Y), where through (mn ) all sequences of integers and U =(U n ), U n open in J mn(X, Y ). The D-topology is independent of the choice of the sequence (Kn). J (X, Y), compatible with b) Fix sequence (dn ) o f metrics dn is a for on m = runs n a the is on manifold topologies. a neighborhood Then the system for f c coo (X, Y) cp = (On) runs through base open sets, where : X -> [0,oo ] such that independent of the c) The system (supp d n) is of sets of the form in the 9-topology, consisting of all sequences locally finite. o f continuous The a D-topology is of the metrics dn . of sets of the form choice of open sets for the D-topology on Coo (X , Y ), runs through all sequences of compact sets Ln C X such locally finite and Un open in j n ( X , Y ) . is maps base 51 where L = that (Ln) (XBLn o ) is 1.6. REMARKS. a) The °° in [9], D-topology e. g. described very as Whitney C’°°-topology, explicitly in [5], II - 3. It is is finer than the the the topology topology °° of Morlet [3]. b) If one , produced by applying then the is the on the seminorms of does Hence the 9-topology not have compact support. The set D (of all functions with compact support in the name. Coo ( Rn , R)) is the maximal linear subspace of Coo ( Rn , R ) which is a topological linear space with the topology induced from the S-topology (and from the Whitney C°°-topology too, but 9 with the Whitney topology has no merits from the point of view of functional analysis and this is our reason for introducing the 2-topology). c) C° (X, Y ) with the D-topology is a Baire space. This is proved by Morlet [3 . A quite straightforward proof is possible using description space 1.5 b, which and are can Coo (X , Y) chosen to also be used is be complete complete. 1.7. L EMMA. A sequence to f iff there exists a of the fn ’s equal f off to in this define pseudo-metrics for uniformity if the metrics dn (fn) in Coo (X, Y) compact K set uniformity, on In ( X, F. a converges in the K C X such that all but and j 1 fn , j 1 f «uniformly. on finitely many K, for all 1 c N . Sufficiency is obvious by looking at the neighborhood Necessity follows since fn -+ f in the 2-topology implies fn -> f PROOF. ney topology and there it is well known 1,8. LEMMA. For each is continuous in the PROOF. For each is defined as k ) 0 the S-topology ( [5], II- 3, page 43 base 1.5 b. in the Whit- ). map 9-topology. 1 the follows : if mapping a rep- 52 d, then resents It is a end on fact, routine task special embedding ). By for each definition Now basic open set ak,l (a ) is well defined, i. e. the representative f , and that it show that choice of the an to we we use not dep- is smooth (in have the basis be a in and open in Now does set and Set which is open in a basic open which can be set and denote is and in seen Then by writing out the definitions. Q.E.D. 1.9. If are A , B, P are topological continuous maps, with the topology we Hausdorff spaces and consider the set A xpB defined by : induced from A X B . Then A x B is the pullback of the uA , 77B in the category of Hausdorff topological spaces. A continuous map is called proper if the inverse image of a compact subset is compact. The subset C° prop (X, Y ) of proper maps of Coo ( X, Y) mappings is open in the 9-topology, since it is open in the (see [9], 2, Proposition 4 ). 53 coarser Whitney topology ([9], 2, L emma 1). Let A , B and P be Hausdorff topological spaces. Suppose P is locally compact and paracompact. Let TT A : A , P and rrB : B - P be continuous mappings. Let K C A and L C B be such that tt A/K and rrB/L are proper. Let U be an open neighborhood of KxPxL in A fr B . Then there exist open neighborhoods V of K in A and Wof L in B LEMMA such that KxP L C VxP C U . 1.10. PROPOSITION. 1 f X, Y, given by ( g, f)- go f , Z are smooth is continuous in the manifolds, then composition : 2-topology. (g , f) E Coo ( Y , Z ) x Coo prop (X , Y) and let M’(L,K) be a 2-open ne ighborhood of g o f in Coo ( X , Z) ( c f . 1.5 c ), i. e . L ( L n ) PROOF. Let basic and U ite, = (Un), where each open in Jn(X, Z) = Un Now consider the Ln is compact in X with (XBLno ) locally fin- and topological pullback for each n > 0 , as in I.9 : The maps are and well defined, since they without are smooth, since they constant term, followed are locally j ust composition by truncation 54 to of polynomials order n . We have So ) is proper, since it is is proper. So all the there exist homeomorphism, inverse is compact in is proper since f a hypotheses to then of Lemma 1.10 are fulfilled, so neighborhoods and 1 in , for each n > 0 such that Since family pact (f sets ite. Then and locally compact and XBLno is locally finite, the (XBLno)) i s locally finite too. So there exists a sequence of comK in Y such that f (XBLno) C YBK n and (YBKno) is locally fin- is proper, Y is f we have is open. is open in then we have , and if and is open Moreover Now let since and then for since all n > 0 and x for all ( xBL n 0 we have 55 we set Now if so so hence g’o f’ E M’ (L, U). Note that 2. THE 2.1. DEFINITION. Let relatively COO ( X, Y ) and the compact This is are used Y locally compact in the proof. Doo- TOPOLOGY ON Coo (X, Y). f , g E Coo (X , Y ) is we is above relation an we call f equivalent equivalence D-topology to relation. The the weakest among all now are be smooth finite dimensional manifolds. If set in X , clearly finer than the X, Y g g ). Doo-topology topologies and for which all (f~ on on the Coo ( X, Y ) equivalence set which classes of the open. 2.2. REMARK. The 9°°-topology on Coo ( X, Y ) is given by the following topology induced from the equivalence classes with the 2-topology and take their dis j oint union. It is clear how to translate the different descriptions of the 19-topology given in 1.5 : In 1.5 a and c , j ust take all intersections of basic 2-open sets with equivalence classes. In 1.5 process : take all b , add f-g to the definition of 2.3. COROLL ARY. A sequence Td ( f ) . (fn) in Coo (X , Y) converges in the Doo- topology iff there exists a compact set K c X such that all but a finite number o f the fn ’s equal f o f f K and j 1 fn -> jl f «uniformly on K>> for all 1. 2.4. COROLLARY. For each k> 0 the map 56 is continuous in the topology. PROOF. j k respects the 2.5. X, Y, Z be smooth manifolds. Then the subset of COO prop (X , Y ) is 2’-open in Coo ( X , Y ) and composition COROLLARY. proper maps is continuous in the equivalence relation. Let Doo-topology. PROOF. If where fl , f2 are proper, then glo fl - g2 f2 ’ o 2.6. REMARKS. a) By 2.3 and 1.7 convergence of sequences in C’(X, Y) is equival- Whitney C°-topology, the S-topology and the Doo -topology. Therethe Thom Transversality-Theorem and the Multijet-Transversality-Theo( cf. [3L II, Theorems 4.9 and 4.13 ) hold for the 2-topology and the for the ent fore rem %°-topology too, since in the proofs of these, convergent sequences are constructed. b) reason C°°(X, Y) with the g-topology is in general no Baire space. The for this will become clear in Section 3. However, it is paracompact and normal (cf. 3.9 ). 2.7. PROPOSITION. Let E - X be rr : a smooth finite dimensional vector Let D (E ) denote the space of all smooth sections with compact support o f this bundle. Then D(E), with the topology induced from the Dootopology on Coo (X, E ), is a locally convex topological linear space, in fact a dually nuclear (LF)-space. Furthermore it is a Lindelöf sp ace, hence bundle. paracompact and normal. Coo (E), the space Coo ( X , E ), and 19 ( E ) is clearly 2-closed PROOF. of all space is closed and open in sections, bundle 17 I : F - X such that the Whitney space X X Rn . Then Coo ( E O F ) = c:o (X, a vector thus on a it is exactly the topology induced from the 57 Coo ( E ). sum Rn) in the There exists EO F -> X is trivial, and the Doo -topology gm-topology on C°°(X, E (9 F ). Then l%(E ) and the is a topology b with paring 1.5 in fact, topological subspace, is exactly one topology the of L. direct summand of a Schwartz, as is of the well known systems of seminorms seen by com- for D (X x R (compare [6] page 170). Therefore it is a (LF)-space, nuclear and dually nuclear, and locally convex inductive limit of countably separable Frechet spaces (which where K = of these is its closed ular, can (Kl) a be identified with is a sequence of compact Lindel8f space subspace D ( E ) a of X as in 1.2. Each of metrizable ), so D (X x R)n Lindel6f space, and clearly completely ( separable is sets and and reg- hence paracompact and normal. Q. 3. THE MANIFOLD STRUCTURE ON Coo(X, Y) EQUIPPED WITH E. D. THE Doo-TOPOLOGY. 3.1. DEFINITION. tubular a neighborhood submersion a) Let X be rr : b) the rr : submanifold of the smooth manifold Y . A of X in Y is an open subset Z of Y together with manifolds, let Z -> X such that : Z -> X is a vector embedding X-> 3.2. PROPOSITION. Let f E Coo (X, Y ) a Z is the X, and denote the Then there exists bundle ; zero section of this bundle. Y be smooth finite dimensional graph Z f of X f in X X Y with vertical proj ection, i. the submersion Zf -31 X f is j ust the restriction to Zf o f the mapping (x , y ) -> ( x , f (x ) ) from X x Y onto X f. PROOF. We have Xf C X X Y , so TX (X f) is subbundle of TX f (X X Y). We claim that for each (x, f (x)) E X f the space T (x,f (x))(Xf) is transverby X f. a tubular neighborhood e. 7T: a sal to 58 in : is where unit for vector in ’ smooth a any . Such a is of the form path with path iff so if has is the and path with smooth a has the form vector iff then non-zero first coordinate and Thus and is transversality follows since a vector dle u XxY is a bundle over Xf x}x Y) T(x,y)({ x,y since it over X X is j ust exponential maps, defined tions respectively. Then an on to So the Y via the realization of the normal bundle be is dim X f = dim X . pullback embedding neighborhoods a vector X Y, bun- and it Y ). Now let : U and V of the zero sec- in diffeomorphism 59 X f -> X TXf (Xf) TX (X exponential for X X Y , given explicitly by gives of the of X onto our open neighborhood of Zf if vx E V . We denote it by exp . Clearly there is a diffeomorphism of the section of E zero onto Xf cp in X from the proj ection. It is easily commutes, where p : Z - So have we making p: a fibre Zf -> X f Xf neighborhood tt o d = tt/W, checked that the is the restriction into where preserving 77; E + Xf Z, thus diagram to preserving diffeomorphism a vector which is fibre E, the whole of (cf. [5], II, Lemma 4.7 ), i. e. such that is the Y , given by X Z of the map r = exp o d-1:E-> bundle . Q. E. D . 3.3. For further reference Coo (X, Y ) , an open ection and a we have a vector neighborhood p f: fibre commutes. we Zf of repeat the situation in detail : For each bundle Xf ?7f: E (f) -> in X X Y Xf together f given by with the vertical proj - Zf 4 X f, preserving diffeomorphism t f: E (f ) -i’ For f, g (Coo (X, Y) we have the and 60 in Zf, i. e. the diffeomorphisms diagram They satisfy 3.4. Since the main idea of the construction of the manifold will be blurred give an outline of it now, disregarding ologies, continuity and differentiability. For f E Coo (X, Y ) we have the setting of 3.3. Let up by technicalities later and denote Then Zf . by Coo (Zf) on, we the space of smooth sections of the define df: Uf -> Coo (Z f) vector top- bundle by and for is the canonical where 77, uses We now ping. declare that is the vertical a chart for We will check the coordinate cond map such that with VI V f n Vg # 0 . Xh C Zf n Zg . Let us proj ection. proj ection f, and that change That check the now. means Then of ’ Vif =d-1 f since and CPt is the coordinate map- Let that g E Coo (X, Y) be a sethere is h E Coo (X, Y) map df o wg /dg(Uf n Ug). If then So Of 0 09 (s) structure = s oaf in the notation of 3.3. But, of course, the changes . 61 linear REMARK. If 3.5. duce topology a in 3.4 should be Uf Coo ( X, Y) on it is clear that such topology a such that must f, chart for a then we have to intro- is open. From the form of Uf be finer than the Whitney Uf C 0-topology. topology (and the VJhitney CO -topology too) the space Coo(Zf) is not a topological vector space, only a topological module over the topological ring Coo (Xf, R). One could choose this solution and use differential calculus on such modules, as sketched by F. Berquier [1]. The other But with this possible solution, presented here, of Coo (Zf ) logy ; this is the which is is space D(Zf) look at the maximal linear linear space with the topological a to Whitney Whitney C°°-topology on it has no merits from the point functional Analysis (cf. the topological conclusions of 2.7), so to introduce the g-topology. The equivalence relation 2.1 is we want how to to modify tioned, and model the manifold the construction most of the to topological on obtain proofs which we 3.6. THEOREM. Let X, Y be smooth Doo-topology is a smooth of view of So Uf is set compact support of Let will give choose necessary, spaces. It is clear of the other models just men- remain valid. manifolds. discussion of up in 3.3. For 2’-open. one vector we Then C’ (X, Y) with the manifold. PROOF. We postpone the the notation CO -topo- of smooth sections with compact support. But the if subspace f E Coo (X , Y) D(E (f)) rr f: E( f ) -> differentiability we to 3.7, and define the chart we use Uf by: be the space of smooth sections with X . We define 95f: U f -* D(E ( f )) and Then and 62 by: both maps so verse to If on Now are Doo-continuous by 2.5. We claim each other. Note that rrY the other we study hand g c U f , open to = s E D (E ( f )). g . Let change. Let g f Coo ( X , Y) h f Coo (X, Y ) with check the map in D(E( f )) . Let are in- Then then the coordinate be non-void. So there is We have oj0g that d f and qjf such that UfnUg d f o wg/d g(Ug n Uf). Clearly dg (Ug n Uf) is s E dg(Ug n Uf) C D(E(g)). So or Coo (dfg*, E(g )): pullback So we s -> s o df isust carrying (dfg ) * E ( g ) , have j ust to a vector check the bundle over over Xf, differentiability 63 sections of and so E(g) is linear : of the map to the t-1f given by s-> o r o s , and this is assured Lemma 3.8. by Q. E. D. 3.7. We a have now to fix the notion of rather strong concept, the notion locally convex spaces expansion to hold. and strong Let E and F be continuous we enough locally f convex is of class of Keller [7], It is valid for arbitrary the Taylor for the chain rule and f : E - F be a x, y E E and X 6 R linear spaces, let C1c, if for all x , and the at F. to so on. Keller If f mapping E , F for each x E E , the derivative of mapping (x, y) -> D f (x )y is continuous as a mapping from is Df (x) E X E linear a is of class f [7] is of class fulfills the following f (HL) C1tt , then it is actually Coott , then even For each seminorm p 3.8. LEMMA. Let X be bundles and differentiable in an apparent- Keller [7], 1.2.8 ) : on F there is over a following on F’ there is smooth X. Let U be smooth section with compact support seminorm q on E such that a seminorm q on E Taylor with series. E -> X and p : F -> X be open neighborhood of the image of a manifold, an a stronger condition holds : the Similar conditions hold for the remainders in vector C1c, is of class C; == Coott . condition (see For each seminorm p is of class (x,y) -> Df (x)y if the remainder sense : (HL’) Cc has shown that ly stronger If Then Coott we want to use. have where f mapping. differentiability s 64 of E let n : and let a : U , F be a smooth fibre preserving is of class Coott, map. Then the map where in D (E). Furthermore, derivative o f a , D D (a)=D (df a), is op en PROOF. If we can holds in the since the so D (a ) D (a) , so So we continuity condition C1tt, it is of class (" holds in the converges for s is of class to to show and D D (a ) = D(dfa ) again, and on of d fa And for this it suffices form same use of Corollary 2.3. For x as to c expressions give zero. to x converge uniformly to So only «partial dewe those of E supp s all partial neighborhood of x, . So we can show that for each xo uniformly on locally trivializing chart X, to the compact support of the section 9 all rivatives >> of ( 2 ) with respect a is of the so on. will make we by the definition vatives » converge fibre that, for y E RB{ 0 }: 3T.topology ; show that is the E V, s E D(E), then D(a ) is of class Coott , for D(d fa ) is automatically fulfilled by 2.5, Outside of the compact support of 9 both have d fa show that Doo -topology have where some about xo 65 and have now the situation ( 3 ). deritake In this set up we have and for x c where t,t: W -> Rr W, Step uniformly the we use on some R r, Rs . smooth and ax : 1. We will show that for h - 0 the converges For that are neighborhood expression of xo to [4] value theorem in the form of Dieudonn6 mean ( 8. 5 .4 ) : Let E and F be into F of If f We In is set two neighborhood differentiable on S , a f - f ’(d ) our case for f f be a continuous mapping S j oining two points a, b of E . Banach spaces, let of a segment then and get this looks like sup and Step 2. "We uniformly on this by reducing verges do and now ( 5 ). But by the uniform show that the differential of some neighborhood of xo to continuity of the (4) with respect mapping to x con- the differential of (5 ). We Step 1. So we compute first the differentials of (4 ) first some preliminaries : Let p : Rn x Rs - Rs be the canoto 66 let j x ; Rr -> Rn+r be have ax p a o j x , so the partial derivative. is nical and for x E X jx( Then proj ection, y ) = (x , y). where to it d2 designs corresponds given by we the second = d2 a (x, y, z)= d2 a ( x, y ) z . designs evaluation, i. e. e(f, y) = d2 a If f (y ), then we have since So we can since Now e compute the derivative of is we are bilinear, ready to such that o mapping a inj ection d2 a : so compute the derivative of 67 (5 ) : a mapping: Now we set then p* is again linear and so for any function f into L (Rr , Rn x RS ), have So: since ( 5 ) is the following expression by (7). So the derivative of Now compute the derivative with respect we 68 to x of (4): we Now the 1 we put together pieces expression we can ). First we in the square bracket is of the conclude that this converges for h - 0 to Then consider we from ( 11 consider same uniformly form on a again of the same form as ( 2 ), so by Step uniformly on a neighborhood of xo for y -> 0 to This is 69 ( 2 ), by Step neighborhood of xo as 1 this so again converges Then is half of (10). The one converges for h - 0 Step uniformly 3: The higher to x (11), by uniform continuity remaining member of (10). of xo neighborhood on a the derivatives. In Step 2 we of dfa have shown that the der- of the convergence situation convergence situations two member of to df a (x, t (x)) d t ( x) y , ivative with respect of remaining (4) -> ( 5 ) is the sum (4)-> (5) and something which converges clearly uniformly in all derivatives. So for the apply Step 2 to the two parts, and we continue in second derivative that way for the we j ust higher der- ivatives. Q. E. D. 3.9. REMARKS. (a) In the proof of Lemma 3.8, considered have compact support ; 2.1 so brought advantage. (b) By 2.7 each chart C°° (X , Y) pact. with the But D (Zf) is no longer be an absolute lopped a is so used of : Uf -> D(Zf ) 2’-topology not a heavily that all sections we introducing the equivalence relation we is of locally Baire space, if X is Coo (X , Y) paracompact, thus paracomnot neighborhood compact, so Coo ( X , Y ) Baire space. One would hope that retract, but the is paracompact, theory is not Coo ( X , Y ) turns out to very much deve- for non-metrizable spaces. 4. MISCELLANY. 4.1. LEMMA. Let serving at rr : smooth map. E - X be a derivative I f the s0 E D (E) is surj ective, then bundle and E - E be fibre preD D (a) (s0) : D (E)-> 2(E) of D (a ) there is an open neighborhood U of the vector 70 a : a image of and s, U -> V is a : a of the image of a V neighborhood diffeomorphism. an open o so , such that PROOF. by 3.8. Fix x E X . Let ex Then by hypothesis there is a s s E D (E) and let Ex E3(E) such that s (x ) = ex such that i.e. So the linear map The Jacobi df a (x, so (x)): Ex -+ Ex differential matrix of a (in a is surj ective, thus invertible. local trivialization of E ) looks like so it is invertible Theorem there is neighborhood Vx phism. If we take then a : U -> v is have to force Let that be a r(x) is a sets of a open a o so local diffeomorphism E , i. e. positive definite, symmetric metric of the on the bundle where W is open in X and we we and c > of So we each fibre a section r : X - S2 ( E *) such bilinear form for all x E X .We Ex = Rn r ( tt (p ) ) ( p - S0 tt (p ), p - S0 tt (p ) ) E} , 0 . We neighborhoods consisting choose a locally trivializing equip trivially surj ective. form : following M ( W , E ) = I p E E | tt (p ) E W of and is inj ectivity. r consider an image of so , and by the classical inversion neighborhood Ux of so (x ) in E and an open (x ) in E such that a : Ux ->Vx is a diffeomor- the too on sets assert that each so (x ) has chart S about xo , with the inner 71 a basis M (W, f ). To prove that, so E/ S S X R n and Let product r (x). k (x ) be of the form = the linear automorphism which carries r(x) into the standard inner product of Rn . is homeomorphism (in a is then WC homeomorphism S exactly on an open these So a are obviously we assume We fact now assert diffeomorphism). which carries each The map set of the form M (W,E) with set base of a without that a a : neighborhoods. loss of generality that each Ux U - V is inj ective. If is of the form M(W,E). then If p :;-p Then in U and 7T (p ) = tt (p) = tt (p), tt (p) E then W n W and if e.g. c > i, then so since a/ M (W, () is a diffeomorphism. Q. E.D. for special mappings). Let tt: E , X be a vector bundle and a : E , E be a fibre preserving smooth map. I f the derivative D D(a) (s0): D(E)-> D(E) is surjective, then there are open neighborhoods U of so in D (E ) and V of a s, in D ( E) such that D (a): U -> V is a dif feomorphism. 4.2. COROLLARY (Inversion Theorem o PROOF. a so (X) By 4.1, there are in E such that a : neighborhoods Uo of so (X) and Vo Uo - Vo is a diffeomorphism. If open 72 of and then 3(a): U- V has the smooth inverse Q. 4.3. If fields f f Coo ( X, Y ) , let us denote by Df along f >> with compact support, i. e, (X , TY ) the space of E. D. rector- the space of smooth maps such that relatively We have that D f (X , T Y) = D ( f*T Y), compact in X . the space of smooth sections with compact support of the pullback f*T Y. Comparing D(f*T Y) = D(E( f)) via the linear map s - s with 3.3 o af . Thus we we see that have: PROPOSITION. of all smooth mappings a and I x E X I a (x) # 0} is relatively compact in X . proj ection, then the space such that If a : X -> T Y 77 y: T Y -> Y is the is the proj ection of in the obvious T Coo and T Coo (X, c show that the tangent bundle properties with the only exception that may only be applied to proper mappings. note tor a vector bundle and torial PROOF. becomes a -+ a 0 g. REMARK. b and tor Y) sense. given by a -> (T f)o a , given by (X, Y) a is clear from the discussion T Coo (X , Y) has nice func- the contravariant preceding the partial proposition. -We func- only that the tangent space to so E D(E) is again D (E) where E is a vecbundle over X . We will show in 4.4 that the definition using smooth paths is equivalent. 73 b ) By the usual definition of the tangent bundle give D (X , T Y ) We an open subset of ection, a vector on Y . We is a is a for structure Zf f E Coo (X, Y) bundle. Let again c . Now and let is the we be the operate with the data from 3.3. It is easily be to chart for so small that for each x E X the f (x) E Y , by being canonical proj - prove that Oy is chosen trivializing which it inherits COO (X, T Y ). Clearly D (X , ttY) and that it is smooth follows from is and if the differentiable zero seen vectorfield that set then neighborhood with vertical proj ection E (OY f ) we can choose the Whitney sum tubular of X 0 Y of C X X T Y and o and then and the latter space is D(X , tt Y) linearly homeomorphic becomes the proj ection under these identifications. If f E Ug ( i. e. Zf small to we enough ), choose another then 74 trivializing chart Ug with so D(f* T Y) is to D(g* T Y). linearly homeomorphic c ) Let h f Coo ( X, Y ) and consider again the data from 3.3: behave to check that is smooth. For s so the (Compare the we have mapping is j ust composition followed by tion E D(E(h)) pulling back to another with a fibre preserving smooth function bundle, and so is smooth by 3.8. proof of 3.6 .) Under the identifica- vector with the last argument in the or D(E(h)) with Dh (X , T Y) and of D (E ( f o h )) with D fo h ( X , T Y’) mapping T Coo ( X, f) just coincides with D(X , T f ) which is seen by writing out the definitions. Likewise is j ust pulling back sections, [(dh )-1 od ah 0 g] * E (h). easily seen to coincide one can check that linear, if thus E (h o g ) is chosen to be Coo(g, Y) is smooth too and T Coo(g, Y) is with D ( g, T Y) under the appropriate identificaSo tions. Q. E. D. 4.4. Given U open in as the space of all d (0) = a space D (E), equivalence then we can define classes of smooth s, , where 75 T So paths 0 U for so E U : R -> U with (L) Via 0 -+ (d d)0 cides and s - (t -> so it is +ts) easily seen that Ts0 U coin- with D(E ). Furthermore LE MMA. If d : we have : COO (X, R , Y) is a smooth path then iff for all x E X , where ex : C°° ( X , Y ) -> Y is evaluation at x. That is, two paths through f E Coo (X , Y) are equivalent iff they are pointwise equivalent at f ( x ) in Y for all x c X . P ROO F. locally. Q. E. D. sketchy development of an application of the inversion to stability. The result is probably well known to specialists. Let us denote by Diff (X ) the open subset of diffeomorphisms of Coo (X , X). It is a group and there is a right action of it on Coo ( X, Y ) . A mapping f E Coo (X , Y) is called source-g’ -stable if the orbit of f under 4.5. We give principle 4.1 Diff (X) is is smooth now a Doo-open Coo (X , Y). by 4.3, and infinitesimally source-S’ -stable if surj ective. Following the lines of [5], V- 5 - 6 it is possible if f is source-Doo -stable, then it is infinitesimally source- is its derivative Df *(IdX) to in is show that at IdX . f is called s-stable. Now let 95f : Uf-> D (Zf) dIdv: UId X -> D (ZIdX) be a chart of f in be a chart of COO (X, Y) . 76 Then IdX in Diff(X), and is given by s -> (I dX xf) o s , as is followed linear isomorphism (pullback) proof f infinitesimally source-Doo-stable, then f * is in a neighborhood of the image of IdX ( = X ) given by composition with the fibre preserving map IdX x f , and the derivative is surj ective. As in 4.1 we conclude that the fibre maps are locally subseen in the of 4.3. If by a is (IdX x f )/ (ZIdX)* mersions, i. e. f So we is a submersion, and submersions are source-2c*- stable. obtain the result : The source-Doo-stable in Coo mappings (X, Y) are exactly the sub- mersions. 4.6. In pings analogous way one can characterize the image-2’-stable mapCoo prop (X, Y ) as the proper immersions, but one has to replace an in the argument of 4.5 consider blem in on vector a later by one using the adj oint bundle valued distributions. article, as we of D ( f *)(IdY), Maybe we will do with the canonical Diff(X). 77 and has to will tackle this pro- Lie-group structure REFERENCES. 1. F. BERQUIER, Calcul différentiel dans les modules quasi-topologiques. Variétés Esquisses Mathématiques 24, Amiens ( 1976). différentiables. 2. C. BESSAGA and A. ology, PELCZYNSKI, Selected topics Publishers, 1975. Topologie différentielle, 3. H. CARTAN, 4. J. DIEUDONNE, Foundations infinite dimensional top- Séminaire E. N. S. Paris ( 1961- 62). of modern Analysis I, 5. M. GOLUBITSKY and V. GUILLEMIN, Stable Springer G.T.M. 14, 1973. 6. J. HORVATH, 7. H. 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