The diffeomorphism groups of the real line are pairwise bihomeomorphic

THE DIFFEOMORPHISM GR OUPS OF THE REAL LINE ARE P AIR WISE BIHOMEOMORPHIC T ARAS BANAKH AND T A TSUH IKO Y AGASAKI Abstra ct. F or an r = 0 , 1 , · · · , ∞ , by D r ( R ), D r + ( R ), D r c ( R ) w e denote respectively the groups of C r diffeomorphisms, ori entation-preserving C r diffeomorphisms, and compactly su pp orted C r diffeomor- phisms of the real line. W e think of th ese groups as bitop ologies spaces endow ed with the compact- open C r top ology and the Whitney C r top ology . W e pro ve that all the t riples ( D r ( R ) , D r + ( R ) , D r c ( R )), 0 ≤ r ≤ ∞ , are pairwise bitop ologically equiv alen t, whic h allo ws to apply known results o n the top o- logical structure of homeomorphism groups of the real l ine to recognizi ng the topological structure of the diffeomorphisms groups of R . 1. Introduction Groups of homeomorphisms and diffeomorph isms of smo oth manifolds are classical ob jects in Differen tial Geometry and T op ol ogy , see [12]. In this pap er w e shall study the compact-op en and Whitney C r top ologies on the group D r ( R ) of C r diffeomorphisms of the real line for r = 0 , 1 , · · · , ∞ . The group D r ( R ) will be considered toget her with the subgroup D r + ( R ) ⊂ D r ( R ) of o rienta tion- preserving C r diffeomorphisms of R , and the subgroup D r c ( R ) ⊂ D + ( R ) of C r diffeomorphisms h : R → R that ha v e compact s upp o rt supp( h ) = c l R { x ∈ R : h ( x ) 6 = x } . In such a w a y , w e obtain the triple of diffeomorphism groups ( D r ( R ) , D r + ( R ) , D r c ( R )). The group D r ( R ) carries t w o natural top ologies: the compact-op en C r top ology an d the Whit- ney C r top ology , see [12], [13]. It will be con v enien t to consider those t w o top ologies on D r ( R ) sim ultaneously b y thin k in g o f D r ( R ) as a bito p ologic al space. By a bitopological space w e understand a trip le ( X , w, s ) consisting of a se t X and t w o to p ologies w ⊂ s on X , called the w e ak a nd str ong top olo gies of the bitop ological space. A fun ction f : X → Y b et we en b itop ological spaces is called we akly (resp. str ongly ) c ontinuous if f is con tin uous with r esp ect to the w eak (resp. strong) top ologies on X and Y . A fun ction f : X → Y is bic ontinuous if f is b oth w eakly and strongly co nti nuous. Tw o bitop ological spaces are called bihome omorph ic (resp. we akly home omorphic , str ongly home omorphic ) if there is a bijectiv e map f : X → Y suc h that b oth f and f − 1 are bicon tin uous (resp. w eakly con tin uous, strongly contin uous). Examples of bitop ological spaces v ery often arise in T op ology and F unctional Analysis. F or exam- ple, for tw o top ological spaces X , Y the space C ( X, Y ) of con tin uous fun ctions fr om X to Y can b e seen as a bitopological sp ace wh ose wea k and strong top ologies are the compact-open and Whitney top ologies. W e recall that the Whitney to p ology on C ( X , Y ) is generated b y the base consisting of the sets U Γ = { f ∈ C ( X, Y ) : { ( x, f ( x )) : x ∈ X } ⊂ U } where U runs o v er op en subsets of X × Y . Another example of a bitop ologica l sp ace is the Cartesian pro duct Π i ∈I X i of topological spaces X i , i ∈ I . Its w eak top ology is th e T yc hono v pro d uct top ology while the s tr ong top ology is the b o x-pro d uct top ology . The latter is generated by the base consisting of pro du cts Q i ∈I U i of op en subsets U i ⊂ X i , i ∈ I . The pr o duct Q i ∈I X i endo w ed with the b ox- pro duct top ology is denoted by 2000 Mathematics Subje ct Cl assific ation. 57 S05, 58D05, 58D15. Key wor ds and phr ases. The diffeomorphism group, the Whitn ey topology , the b ox produ ct. 1 2 T. BANA KH AND T. Y AGASAKI  i ∈I X i . If eac h sp ace X i , i ∈ I , has a distinguished p oint ∗ i ∈ X i , then the pro duct Π i ∈I X i con tains an im p ortant subs p ace Σ i ∈I X i = { ( x i ) i ∈I ∈ Y i ∈I X i : |{ i ∈ I : x i 6 = ∗ i } < ℵ 0 } called the σ -pr o duct of th e p ointed spaces X i . If all the spaces X i , i ∈ I , coincide w ith some fixed space X , then the spaces Π i ∈I X and Σ i ∈I X are d en oted by Π I X and Σ I X , resp ectiv ely . It is inte resting to remark that the bitop ological sp ace Π I X is bihomeomorphic to the fun ction sp ace C ( I , X ) wh er e I is endo w ed with the discrete top ology . The spaces Π I X and Σ I X endo w ed with the Tyc hono v pro duct to p ology are d enoted b y Π I X and Σ I X . Th e same spaces endow ed with the b o x-pro duct top ology are d enoted by  I X and ⊡ I X , resp ectiv ely . So, in a sen se, the bitop ological s p ace Π I X decomp oses in to tw o top ological sp aces: Π I X and  I X . Th e same r emark concerns the b itop ological sp ace Σ I X that d ecomp oses int o t wo top ological spaces Σ I X and ⊡ I X . Bitop ologica l spaces that are (lo cally) homeomorphic to Σ ω R were c haracterized in the pap er [4] that also presen ts man y exa mples of suc h spaces app earing in T op ology , T op ologica l An gebra or Analysis. In p articular, it is prov ed in [4] that a bitop ological linear space X is b ih omeomorphic to Σ ω R if and only if X has infinite algebraic d imension, th e w eak top ology of X is metrizable and X endo w ed with its strong top ology is the direct limit of fin ite-dimensional compacta, see [4, 2.2]. Another natural example o f a bitop ologica l space wh ose bitop ologica l structure has b een recognized is the function space C ( R ) endo w ed with the compact -op en and Whitney top ologies. According to the (pro of of th e) main result in [10] the function space C ( R ) is bihomeomorphic to the coun table p o we r Π ω l 2 of the separable Hilb ert sp ace l 2 . The same result is true for a bitop ologica l space C r ( R ) of C r -differen tiable fun ctions on the real line. The we ak and strong top ologies of C r ( R ) are indu ced from the w eak and strong top ologies of the p ro du ct Π m ∈ Z C r ( I m ) via th e em b edd ing C r ( R ) ֒ → Π m ∈ Z C r ( I m ) , f 7→ ( f | I m ) m ∈ Z . Here C r ( I m ) is the F r ´ ec het space of all C r differen tiable fu nctions f : I m → R on the closed int erv al I m = [ m, m +1]. The (metrizable) top ology of C r ( I m ) is generated b y the seminorms k f k n = max 0 ≤ k ≤ n max x ∈ I m | f ( k ) ( x ) | ( n ∈ ω , n ≤ r ) . The we ak top ology of the bitop ological space C r ( R ) is referred to as the c omp act-op en C r top olo gy while the strong top ology as the Whitney C r top olo gy , see [12], [13]. In the space C r ( R ) consider the sub space C r c ( R ) = { f ∈ C r ( R ) : f | R \ K ≡ 0 f or some compact subs et K ⊂ R } consisting of C r differen tiable fu nctions w ith compact supp ort. The follo w ing theorem describing the bitop ologic al structure of th e p air ( C r ( R ) , C r c ( R )) w as im- plicitly prov ed in [10], [11]. Theorem 1.1 (Gur an, Zaric hn yi) . F or every r ∈ ω ∪ {∞} the p air ( C r ( R ) , C r c ( R )) is bihome omorphic to the p air ( Π ω l 2 , Σ ω l 2 ) . W e sa y that for b itop ological spaces X , Y and their subsets X ′ ⊂ X and Y ′ ⊂ Y the p airs ( X , X ′ ) and ( Y , Y ′ ) are b ihome omorphic if there is a bihomeomorphism h : X → Y su c h that h ( X ′ ) = Y ′ . In this pap er we concen trate at stud y in g the bitop ological str u cture of the diffeomorphism group D r ( R ) ⊂ C r ( R ) and its su bgroups D r + ( R ) and D r c ( R ). F or r = 0 the group D 0 ( R ) turns in to the DIFFEOMORPHISM GROUPS OF THE REAL LINE 3 homeomorphism group H ( R ) of the r eal line. In this case the triple ( D 0 ( R ) , D 0 + ( R ) , D 0 c ( R )) is d enoted b y ( H ( R ) , H + ( R ) , H c ( R )). In [3] it was pro v ed that the pair ( H + ( R ) , H c ( R )) is we akly and strongly homeomorphic to the pair ( Π ω l 2 , Σ ω l 2 ) (surprisingly , but w e d o not kno w if these t w o p airs are bihomeomorphic!) In this pap er we shall extend th e mentioned result to the pairs ( D r + ( R ) , D r c ( R )) for all r ≤ ∞ . Our principal result is: Theorem 1.2. F or every r ∈ ω ∪ {∞} the triple ( D r ( R ) , D r + ( R ) , D r c ( R )) is bihome omo rphic to the triple ( H ( R ) , H + ( R ) , H c ( R )) . W e sa y that for bitop ologica l spaces X, Y and their subsets X 1 , X 2 ⊂ X and Y 1 , Y 2 ⊂ Y the triples ( X, X 1 , X 2 ) and ( Y , Y 1 , Y 2 ) are bihome omorphic if there is a b ihomeomorphism h : X → Y suc h that h ( X 1 ) = Y 1 and h ( X 2 ) = Y 2 . By [3] the p airs of homeomorph ism groups ( H + ( R ) , H c ( R )) and ( H ( R ) , H c ( R )) endow ed with th e Whitney top ology are h omeomorphic to the pair (  ω l 2 , ⊡ ω l 2 ). This fact com bined with T heorem 1.2 implies Corollary 1.1. F or every r ∈ ω ∪ {∞} the p airs ( D r ( R ) , D r c ( R )) and ( D r + ( R ) , D r c ( R )) of diffe omor- phism gr oups endow e d with the Whitney C r top olo gy ar e ho me omorphic to the p air (  ω l 2 , ⊡ ω l 2 ) . On the other hand, th e pair o f homeo morph ism groups ( H + ( R ) , H c ( R )) end o w ed with the compact- op en top ology is homeomorphic to the pair ( Π ω l 2 , Σ ω l 2 ), s ee [3]. Com bining th is f act with Theo- rem 1.2 we get another Corollary 1.2. F or every r ∈ ω ∪ {∞} the p air ( D r + ( R ) , D r c ( R )) of diffe omorphism gr oups e ndowe d with the c omp act-op en C r top olo gy is home omorphic to the p air ( Π ω l 2 , Σ ω l 2 ) . 2. Preliminaries In the sequ el n, m, k , i, s will denote elemen ts of the set ω of all non-negativ e in tegers. On the other hand, r ∈ ω ∪ {∞} can also accept the infi nite v alue ∞ . Eac h n umber n ∈ ω will b e identified with the discrete top ological sp ace n = { 0 , . . . , n − 1 } . By R + = (0 , ∞ ) we denote the op en half-line. F or an r ∈ ω ∪ {∞} b y C r ( R ) w e denote the space of al l C r differen tiable functions f : R → R . This sp ace has a ric h algebraic stru ctur e. First, it is an algebra with r esp ect to the addition f + g and the m ultiplication f g o f functions f , g ∈ C r ( R ). On the other hand, C r ( R ) is a monoid w ith resp ect to the op eration f ◦ g of comp osition of functions. Th e identit y map id R of R is the tw o-sided unit of this monoid, while D r ( R ) is the subgroup of inv ertible elemen t of C r ( R ). Th e group D r + ( R ) of orien tation-preserving C r diffeomorphisms of R lie s in th e submonoid C r + ( R ) ⊂ C r ( R ) consisting of fun ctions f : R → R with strictly p ositiv e d er iv ativ e f ′ . If r = 0 then C r + ( R ) is the space of strictly increasing con tin uous fun ctions on R . The classical theorem of Inv erse F unction im p lies that the group D r + ( R ) coincides with the set of sur jectiv e f unctions f ∈ C r + ( R ). F or a sub set B ⊂ R w e put C r ( R , B ) = { f ∈ C r ( R ) : f ( R ) ⊂ B } . If r = 0, then we write C ( R ) instead of C 0 ( R ). F unctions f ∈ C ∞ ( R ) w ill b e called smo oth . F or eac h f unction f ∈ C r ( R ) and ev ery n ∈ ω , n ≤ r w e can calculate th e (fin ite or in finite) norm k f k n = max 0 ≤ k ≤ n sup x ∈ R | f ( k ) ( x ) | . If no other top ology on C r ( R ) is s p ecified, then w e think of C r ( R ) as a bitop ological space endo w ed with the compact open and Whitney C r top ologies. Those are ind u ced from the T yc hono v pr o duct 4 T. BANA KH AND T. Y AGASAKI and b ox-prod u ct top ologies of Π n ∈ Z C r ( I n ) under the em b edd ing e : C r ( R ) → Π n ∈ Z C r ( I n ) , e : f 7→ ( f | I n ) n ∈ Z . Here C r ( I n ) is the F r´ ec h et sp ace of C r differen tiable functions on the int erv al I n = [ n, n +1], endo w ed with the top ology generated by the semin orms k f k m = m ax 0 ≤ k ≤ m max x ∈ I n | f ( k ) ( x ) | for m ∈ ω , m ≤ r . 3. The diff erential opera tors ∂ D A Giv en a p oin t a ∈ R and a num b er n ∈ ω , consider the differen tial op erator ∂ ( n ) a : C ∞ ( R ) → R , ∂ ( n ) a : f 7→ f ( n ) ( a ) assigning to a C ∞ differen tiable fu nction its n -th deriv ativ e at the p oint a . F or a subset D ⊂ ω , the op erators ∂ ( n ) a , n ∈ D , comp ose an op erator ∂ D a : C ∞ ( R ) → R D , ∂ D a : f 7→ ( f ( n ) ( a )) n ∈ D . The p ow er R D is considered as a bitop ological space wh ose weak and strong top ologies coincide with the Tyc honov pr o duct top ology . No w let A b e a closed discrete subsp ace of R . T he op erators ∂ D a , a ∈ A , comp ose a bicon tin uous op erator ∂ D A : C ∞ ( R ) → Π A R D , ∂ D A : f 7→ ( ∂ D a f ) a ∈ A = ( f ( n ) ( a )) n ∈D a ∈ A . Here Π A R D is a bitop ological space whose w eak top ology is the Tyc h ono v pro du ct top ology while the strong top ology is the b o x-top ology of the space  A R D . No w we shall try to lo cate the image ∂ D A ( D ∞ + ( R )) of D ∞ + ( R ) in Π A R D . F or the singleton D = 1 = { 0 } ⊂ ω , this is easy: the image ∂ 1 A ( D ∞ + ( R )) = { ( f ( a )) a ∈ A ∈ Π A R : f ∈ D ∞ + ( R ) } coincides with the set D 1 + ( A ) of all sequences ( f 0 a ) a ∈ A suc h that • f 0 a < f 0 b for an y a < b in A ; • inf a ∈ A f 0 a = −∞ if inf A = −∞ , and • sup a ∈ A f 0 a = + ∞ if sup A = + ∞ . F or a subset D ⊂ ω co nta ining 0 and 1 we put D D + ( A ) = { ( f n a ) n ∈ D a ∈ A ∈ Π A R D : ( f 0 a ) a ∈ A ∈ D 1 + ( A ) and ( f 1 a ) a ∈ A ∈ R A + } and observ e that ∂ D A ( D ∞ + ( R )) ⊂ D D + ( A ). The pro of of Theorem 1.2 will hea vily r ely on the fact th at the different ial op erator ∂ ω Z : D ∞ + ( R ) → D ω + ( Z ) has a biconti nuous sectio n E Z : D ω + ( Z ) → D ∞ + ( R ). Giv en a double sequence f = ( f n a ) n ∈ ω a ∈ Z ∈ D ω + ( Z ) ⊂ Π Z R D the op erator E Z returns a smo oth fun ction E Z f ∈ D ∞ + ( R ) with deriv ative s ( E Z f ) ( n ) ( a ) = f n a , n ∈ ω , at eac h p oin t a ∈ Z . The constru ction of the section E Z will rely on the sections E 1 and E 2 of the op erators ∂ ω 1 and ∂ ω 2 for the singleton 1 = { 0 } ⊂ R and the doubleton 2 = { 0 , 1 } ⊂ R . By a se ction of a function f : X → Y w e und er s tand an y function g : Y → X suc h that f ◦ g = id Y . 4. A section E 1 of the oper a tor ∂ ω 1 In this section w e shall constru ct a weakly con tin uous section E 1 of the differen tial op erator ∂ ω 1 : D ∞ + ( R ) → D ω + ( 1 ) ⊂ R ω , ∂ ω 1 : f 7→ ( f ( n ) (0)) n ∈ ω for the singleton 1 = { 0 } ⊂ R . Here D ω + ( 1 ) = { ( f n 0 ) n ∈ ω ∈ R ω : f 1 0 > 0 } ⊂ R ω is a bitop ological space whose w eak and strong top ologies coincide with the T yc hono v pro d u ct top ology inh erited f rom R ω . DIFFEOMORPHISM GROUPS OF THE REAL LINE 5 Prop osition 4.1. The op er ator ∂ ω 1 : D ∞ + ( R ) → D ω + ( 1 ) has a we akly c ontinuous se ction E 1 : D ω + ( 1 ) → D ∞ + ( R ) . Conse quently, ∂ ω 1 ( D ∞ + ( R )) = D ω + ( 1 ) . The pr o of of this prop osition is rather long and r equires a lot of preliminary work. Cho ose a smo oth fun ction γ : R → [0 , 1] su c h that γ ( x ) = 1 if | x | ≤ 1 / 2 and γ ( x ) = 0 if | x | ≥ 1. Lemma 4.1. (1) The map R ∋ c 7− → γ ( cx ) ∈ C ∞ ( R ) is we akly c ontinuous. (2) F or any c > 0 and n, s ∈ ω we have     x n γ ( cx )  ( s )    ≤ (2 n ) s c s − n k γ k s ( x ∈ R ) Pr o of. (2)  x n γ ( cx )  ( s ) = s X k =0  s k  ( x n ) ( k ) γ ( cx ) ( s − k ) = min { n,s } X k =0  s k  n ! ( n − k )! x n − k γ ( s − k ) ( cx ) c s − k . Here, as exp ected,  s k  = s ! k !( s − k )! stands for the binomial co efficien t. Therefore, w e ha ve     x n γ ( cx )  ( s )    ≤ min { n,s } X k =0  s k  n ! ( n − k )! | x | n − k | γ ( s − k ) ( cx ) | c s − k ≤ min { n,s } X k =0  s k  n ! ( n − k )! c − ( n − k ) k γ ( s − k ) k 0 c s − k ≤ c s − n min { n,s } X k =0  s k  n k k γ ( s − k ) k 0 ≤ c s − n n s k γ k s s X k =0  s k  = (2 n ) s c s − n k γ k s .  Let n ∈ ω . F or b ∈ R and c > 0, consid er the function g n ( x ) = b n ! x n γ ( cx ) ∈ C ∞ ( R ). Lemma 4.2. (1) The map R × R + ∋ ( b, c ) 7− → g n ∈ C ∞ ( R ) is we akly c ontinuous. (2) Supp ose s ≥ 0 and n ≥ s + 1 . (i) If c ≥ 1 , then    g ( s ) n    0 ≤ | b | (2 n ) s c − 1 k γ k n . (ii) If d > 0 and c = n 2 2 n | b | d k γ k n + 3 , then    g ( s ) n    0 ≤ n s − 1 d 2 n . Pr o of. (2) Sin ce g ( s ) n ( x ) = b n !  x n γ ( cx )  ( s ) , from L emm a 4.1 it follo ws that    g ( s ) n ( x )    = | b | n !     x n γ ( cx )  ( s )    ≤ | b | (2 n ) s c s − n k γ k s ≤ | b | (2 n ) s c s − n k γ k n . (i) Sin ce c ≥ 1 and n ≥ s + 1, we h a v e    g ( s ) n ( x )    ≤ | b | (2 n ) s c − 1 k γ k n ( x ∈ R ) . (ii) If b = 0, then the assertion is tr ivial. If b 6 = 0, since c ≥ 1 and c > n 2 2 n | b | d k γ k n , by (i) w e ha v e    g ( s ) n ( x )    ≤ | b | (2 n ) s k γ k n c ≤ | b | (2 n ) s k γ k n n 2 2 n | b | d k γ k n = 2 s 2 n n s − 1 d 2 n ≤ n s − 1 d 2 n ( x ∈ R ) .  Let us recall that D ω + ( 1 ) = { ( f n 0 ) n ∈ ω ∈ R ω : f 1 0 > 0 } . Give n a sequence f = ( f n 0 ) n ∈ ω ∈ D ω + ( 1 ), consider the sequence of fun ctions g n ∈ C ∞ ( R ) d efined by g 0 ( x ) = f 0 0 , g 1 ( x ) = f 1 0 · x, and g n ( x ) = f n 0 n ! x n γ ( c n x ) , wh ere c n = n 2 2 n | f n 0 | f 1 0 k γ k n + 3 for n ≥ 2 . 6 T. BANA KH AND T. Y AGASAKI The follo wing lemma implies Prop osition 4.1. Lemma 4.3. (1) The series ∞ X n =0 g ( s ) n c onver ges uniformly for any s ∈ ω . (2) The f unction g := ∞ X n =0 g n satisfies the fol lowing c ondition s: (i) g ∈ C ∞ ( R ) , (ii) g ( s ) = ∞ X n =0 g ( s ) n ( s ∈ ω ) , (iii) ∂ ω 1 g = f , (iv) g ′ ( x ) > 0 , ( v) g ( x ) = f 0 0 + f 1 0 · x ( | x | ≥ 1 / 3) and (vi) g ∈ D ∞ + ( R ) . (3) The map E 1 : D ω + ( 1 ) → D ∞ + ( R ) ⊂ C ∞ ( R ) , E 1 : f 7→ g , is we akly c ontinuous. (4) ∂ ω 1 ( D ∞ + ( R )) = D ω + ( 1 ) . Pr o of. By L emma 4.2 (2-ii), f or s ≥ 0 and n ≥ s + 2 w e ha ve    g ( s ) n    0 ≤ n s − 1 f 1 0 2 n . (1) S ince ∞ X n =0 n s 2 n < ∞ , w e ha v e ∞ X n =2    g ( s ) n    0 < ∞ , whic h implies the assertion. (2) T he assertions (i), (ii) follo w from (1) and the basic prop erties of un iformly con v ergen t series. (iii) Since g 0 ( x ) = f 0 0 , g 1 ( x ) = f 1 0 · x , g n ( x ) = f n 0 n ! x n ( | x | < 1 2 c n ) ( n ≥ 2), it follo ws that g ( s ) n (0) = f n 0 δ n,s ( n ≥ 0) and g ( s ) (0) = ∞ X n =0 g ( s ) n (0) = f s 0 . (iv) Since g ′ ( x ) = ∞ X n =0 g (1) n ( x ) = f 1 0 + ∞ X n =2 g (1) n ( x ), it follo ws that | g ′ ( x ) − f 1 0 | =      ∞ X n =2 g (1) n ( x )      ≤ ∞ X n =2   g (1) n ( x )   ≤ ∞ X n =2   g (1) n   0 ≤ ∞ X n =2 f 1 0 2 n = f 1 0 2 , and so g ′ ( x ) ≥ 1 2 f 1 0 > 0. (v) Th e conclusion follo ws from the equalit y g n ( x ) = 0 holding for all n ≥ 2 and | x | ≥ 1 / 3. (vi) By (v) g is surjectiv e. Th us, by (i) and (iv) w e ha v e g ∈ D ∞ + ( R ). (3) The w eak con tin uit y of the map E 1 : D ω + (1) → D ∞ + ( R ), E 1 : f 7→ g , will foll o w as so on as we c hec k that for eve ry s ∈ ω the map ∂ ( s ) ◦ E 1 : D ω + (1) → C ( R ) , ∂ ( s ) ◦ E 1 : f 7→ g ( s ) , is weakly con tin uous. By (2)(ii) w e hav e g ( s ) = P ∞ n =0 g ( s ) n . No te that th e follo wing maps are we akly con tin uous: D ω + (1) ∋ f 7− → g n ∈ C ∞ ( R ), D ω + (1) ∋ f 7− → g ( s ) n ∈ C ( R ). The w eak conti nuit y of g n in f f ollo ws from the d efinition of g n and the wea k con tin uit y of g ( s ) n in f follo ws from the weak cont inuit y of the m ap C ∞ ( R ) ∋ h 7− → h ( s ) ∈ C ( R ). Since    g ( s ) n    0 ≤ n s − 1 f 1 0 2 n ( n ≥ s + 2), we see that the m ap ∂ ( s ) ◦ E 1 is weakly con tin uous. (4) T he equalit y ∂ ω 1 ( D ∞ + ( R )) = D ω + (1) f ollo ws from (2-iii).  DIFFEOMORPHISM GROUPS OF THE REAL LINE 7 5. A section E 2 of the oper a tor ∂ ω 2 In t his section for the d oubleton 2 = { 0 , 1 } ⊂ R w e co nstru ct a conti nuous sect ion E 2 of the differen tial op erator ∂ ω 2 : D ∞ + ( R ) → D ω + ( 2 ) , ∂ ω 2 : f 7→ ( f ( n ) ( a )) n ∈ ω a ∈ 2 . Here D ω + ( 2 ) = { ( f n a ) n ∈ ω a ∈ 2 : f 0 0 < f 0 1 , ( f 1 a ) a ∈ 2 ∈ R 2 + } ⊂ Π a ∈ 2 R ω is a bitop ological s pace wh ose we ak and strong top ologies coincide w ith the T yc hono v pro d uct top ol- ogy inherited from Π a ∈ 2 R ω . Prop osition 5.1. The op er ator ∂ ω 2 : D ∞ + ( R ) → D ω + ( 2 ) has a we akly c ontinuous se ction E 2 : D ω + ( 2 ) → D ∞ + ( R ) . Conse quently, ∂ ω 2 ( D ∞ + ( R )) = D ω + ( 2 ) . The pro of of this prop osition requires some preparatory wo rk. First, let us prov e a lemma that will allo w to join smo oth increasing f unctions. Lemma 5.1. If f , g ∈ C r + ( R ) , f ≤ g , and α ∈ C ∞ ( R , [0 , 1]) , α ′ ≥ 0 , then h := (1 − α ) f + αg ∈ C r + ( R ) . Pr o of. F or r ≥ 1 we ha v e h ′ = ( − α ′ ) f + (1 − α ) f ′ + α ′ g + αg ′ = α ′ ( g − f ) + (1 − α ) f ′ + αg ′ > 0 . If r = 0, then for any real num b ers x < y we get α ( x ) ≤ α ( y ) and h ( x ) = (1 − α ( x )) f ( x ) + α ( x ) g ( x ) < (1 − α ( x )) f ( y ) + α ( x ) g ( y ) = = f ( y ) + α ( x ) ( g ( y ) − f ( y )) ≤ f ( y ) + α ( y ) ( g ( y ) − f ( y )) = h ( y ) , witnessing that h ∈ C 0 + ( R ).  Lemma 5.2. Ther e exists a we akly c ontinuous map E 0 : D ω + ( 2 ) → C ∞ + ( R ) such that for every f = ( f n a ) n ∈ ω a ∈ 2 ∈ D ω + ( 2 ) the f unction E 0 f has the fol lowing pr op erties: (1) ( E 0 f ) ( n ) (0) = f n 0 for al l n ∈ ω ; (2) E 0 f ( R ) =  − ∞ , ( f 0 0 + f 0 1 ) / 2  . Pr o of. Cho ose an y smo oth function α : R → [0 , 1] s u c h that α ′ ≥ 0, ( −∞ , 1 3 ] ⊂ α − 1 (0) and [ 2 3 , + ∞ ) ⊂ α − 1 (1). Al so fix a smo oth fun ction β ∈ C ∞ + ( R ) suc h that β ( R ) = ( 1 3 , 1 2 ). By Pr op osition 4.1, the differentia l op erator ∂ ω 1 : D ∞ + ( R ) → D ω + ( 1 ) , ∂ ω 1 : f 7→ ( f ( n ) (0)) n ∈ ω , has a w eakly con tin uous section E 1 : D ω + ( 1 ) → D ∞ + ( R ). Giv en a d ouble sequence f = ( f n a ) n ∈ ω a ∈ 2 ∈ D ω + ( 2 ), consider the sequence f 0 = ( f n 0 ) n ∈ ω ∈ D ω + ( 1 ) and the smo oth function E 1 f 0 ∈ D ∞ + ( R ). Since E 1 f 0 (0) = f 0 0 < f 0 1 , th e num b er c f = ( E 1 f 0 ) − 1 ( 2 3 f 0 0 + 1 3 f 0 1 ) is p ositiv e. It f ollo ws that for ev ery x ∈ [0 , c f ] w e get E 1 f 0 ( x ) ≤ 2 3 f 0 0 + 1 3 f 0 1 < β f ( x ), where β f ( x ) = f 0 0 + ( f 0 1 − f 0 0 ) · β ( x ) ∈  2 3 f 0 0 + 1 3 f 0 1 , 1 2 f 0 0 + 1 2 f 0 1  . By Lemma 5.1, the function E 0 f : x 7→ (1 − α ( c f x )) · E 1 f 0 ( x ) + α ( c f x ) · β f ( x ) b elongs to C ∞ + ( R ). The c hoice of the functions α guarant ees that E 0 f ( x ) = β f ( x ) if x ≥ 2 3 c f and E 0 f ( x ) = E 1 f 0 ( x ) if x ≤ 1 3 c f . Consequen tly , E 0 f ( R ) = ( −∞ , su p x ∈ R β f ( x )) =  − ∞ , f 0 0 + f 0 1 2  8 T. BANA KH AND T. Y AGASAKI and ( E 0 f ) ( n ) (0) = ( E 1 f 0 ) ( n ) (0) = f n 0 for all n ∈ ω . The constru ction of the fun ction E 0 f imp lies that th e map E 0 : D ω + ( 2 ) → C ∞ + ( R ) , E 0 : f 7→ E 0 f is weakly con tin uous.  By analogy we can pro ve Lemma 5.3. Ther e exists a we akly c ontinuous map E 1 : D ω + ( 2 ) → C ∞ + ( R ) such that for every f = ( f n a ) n ∈ ω a ∈ 2 ∈ D ω + ( 2 ) the f unction E 1 f has the fol lowing pr op erties: (1) ( E 1 f ) ( n ) (1) = f n 1 for al l n ∈ ω ; (2) E 1 f ( R ) =  f 0 0 + f 0 1 2 , + ∞ ) . Pro of of Pro p osition 5.1. Let E 0 , E 1 : D ω + ( 2 ) → C ∞ + ( R ) b e the weakly cont inuous maps giv en by Lemmas 5.2 and 5.3. Fix a s m o oth function α ∈ C ∞ ( R , [0 , 1]) suc h that α ′ ≥ 0, ( −∞ , 1 3 ] ⊂ α − 1 (0), and [ 2 3 , + ∞ ) ⊂ α − 1 (1). Giv en a double sequence f = ( f n a ) n ∈ ω a ∈ 2 ∈ D ω + ( 2 ), consid er the map E 2 f : x 7→ (1 − α ( x )) · E 0 f ( x ) + α ( x ) · E 1 f ( x ) , and observ e that E 2 f ( x ) = ( E 0 f ( x ) if x ≤ 1 3 , E 1 f ( x ) if x ≥ 2 3 Consequent ly , ( E 2 f ) ( n ) (0) = ( E 0 f ) ( n ) (0) = f n 0 and ( E 2 f ) ( n ) (1) = ( E 1 f ) ( n ) (1) = f n 1 , witnessing that ∂ ω 2 ( E 2 f ) = f . It f ollo ws from sup x ∈ R E 0 f ( x ) = f 0 0 + f 0 1 2 = inf x ∈ R E 1 f ( x ) and Lemma 5.1 that the map E 2 f = α E 0 f + (1 − α ) E 1 f b elongs to C ∞ + ( R ). Being a sur j ectiv e smo oth fun ction with positiv e d eriv ativ e, the fun ction E 2 f b elongs to the diffeomorphism group D ∞ + ( R ). The we ak con tin uit y of the op erators E 0 and E 1 imply the w eak cont inuit y of the extension operator E 2 : D ω + ( 2 ) → D ∞ + ( R ) , E 2 : f 7→ E 2 f .  6. A s ection E Z of the oper a tor ∂ ω Z In this section w e shall constru ct a b icon tin uous section E Z of the differentia l op erator ∂ ω Z : D ∞ + ( R ) → D ω + ( Z ) , ∂ ω Z : f 7→ ( f ( n ) ( a )) n ∈ ω a ∈ Z . Here D ω + ( Z ) = { ( f n a ) n ∈ ω a ∈ Z ∈ Π a ∈ Z R ω : ( f 0 a ) a ∈ Z ∈ D 1 + ( Z ) , ( f 1 a ) a ∈ Z ∈ R Z + } , where D 1 + ( Z ) = { ( f 0 a ) a ∈ Z ∈ Π a ∈ Z R : f 0 a < f 0 a +1 ( ∀ a ∈ Z ) , inf a ∈ Z f 0 a = −∞ , sup a ∈ Z f 0 a = + ∞} . DIFFEOMORPHISM GROUPS OF THE REAL LINE 9 The sp ace D ω + ( Z ) is a su bspace of the bitopological space Π Z R ω whose w eak top ology is the T yc hono v pro du ct top ology while th e strong top ology is the top ology of the b o x-pro du ct  Z R ω . The same concerns th e sub space D 1 + ( Z ) of the bitop ological space Π Z R endow ed with the T yc hono v and b ox- pro du ct top ologies. It is clear that the image ∂ 1 Z ( D ∞ c ( R )) of the group D ∞ c ( R ) of compactly supp orted C ∞ diffeomor- phisms of the real line lies in the s u bset D 1 c ( Z ) = { ( f 0 a ) a ∈ Z ∈ D 1 + ( Z ) : |{ a ∈ Z : f 0 a 6 = a }| < ℵ 0 } ⊂ D 1 + ( Z ) . By analogy , the image ∂ ω Z ( D ∞ c ( R )) of D ∞ c ( R ) lies in the su b set D ω c ( Z ) = { ( f n a ) n ∈ ω a ∈ Z ∈ D ω + ( Z ) : |{ a ∈ Z : ( f n a ) n ∈ ω 6 = ∂ ω a id R }| < ℵ 0 } ⊂ D ω + ( Z ) . Prop osition 6.1. The differ ential op er ator ∂ ω Z : D ∞ + ( R ) → D ω + ( Z ) has a bic ontinuous se c tion E Z : D ω + ( Z ) → D ∞ + ( R ) such th at E Z ( D ω c ( Z )) ⊂ D ∞ c ( R ) . Conse quently, ∂ ω Z ( D ∞ + ( R )) = D ω + ( Z ) and ∂ ω Z ( D ∞ c ( R )) = D ω c ( Z ) . W e sh all pro v e this prop osition after s ome preparatory w ork. Giv en a p oin t a ∈ R and t w o function f , g : R → R with f ( a ) = g ( a ) let f ∪ a g b e th e function equal to f on th e r ay ( −∞ , a ] and to g on the ray [ a, + ∞ ). It is clear that f ∪ a g is con tin uous if so are the functions f a nd g . In order to describ e the sm o othness prop erties of f ∪ a g , for eve ry n ≤ r ≤ ∞ consider the differen tial op erator ∂ 6 n a : C r ( R ) → n Y k =0 R , ∂ 6 n a : f 7→ ( f ( k ) ( a )) n k =0 . The next lemma follo ws from the defi nition of the d eriv ative and th e ind uction on n ∈ ω . Lemma 6.1. If f , g ∈ C n ( R ) and ∂ 6 n a f = ∂ 6 n a g , then h = f ∪ a g ∈ C n ( R ) and h ( n ) = f ( n ) ∪ a g ( n ) . Lemma 6.2. Su pp ose f , g , φ, ψ ∈ D n + ( R ) , and let a ∈ R and b = f ( a ) . (1) If ∂ 6 n a f = ∂ 6 n a g and ∂ 6 n b φ = ∂ 6 n b ψ , then ∂ 6 n a ( φ ◦ f ) = ∂ 6 n a ( ψ ◦ g ) . (2) If ∂ 6 n a f = ∂ 6 n a g , then ∂ 6 n b f − 1 = ∂ 6 n b g − 1 . In p articular, (i) ∂ 6 n a f = ∂ 6 n a id R iff ∂ 6 n a f − 1 = ∂ 6 n a id R , and (ii) ∂ 6 n a ( ϕ ◦ f ) = ∂ 6 n a ϕ iff ∂ 6 n a f = ∂ 6 n a id R . Pr o of. (1) Lemma 6.1 imp lies f ∪ a g , φ ∪ b ψ ∈ D n + ( R ) and h := ( φ ◦ f ) ∪ a ( ψ ◦ g ) = ( ψ ∪ b ψ ) ◦ ( f ∪ a g ) ∈ D n + ( R ) and hence ∂ 6 n a ( φ ◦ f ) = ∂ 6 n a h = ∂ 6 n a ( ψ ◦ g ) . (2) Since f ∪ a g ∈ D n + ( R ), we ge t h := f − 1 ∪ b g − 1 = ( f ∪ a g ) − 1 ∈ D n + ( R ) and hence ∂ 6 n b f − 1 = ∂ 6 n b h = ∂ 6 n b g − 1 .  W e sh all use Lemma 6.2 to pro ve the follo wing self-generaliza tion of Pr op osition 5.1. Prop osition 6.2. F or any doubleton A = { a, b } ⊂ R the differ ential op er ato r ∂ ω A : D ∞ + ( R ) → D ω + ( A ) , ∂ ω A : g 7→  g ( n ) ( x )  n ∈ ω x ∈ A , admits a we akly c ontinuous se ction E A : D ω + ( A ) → D ∞ + ( R ) such that E A ◦ ∂ ω A (id R ) = id R . 10 T. BANA KH AND T. Y AGASAKI Pr o of. According to Prop osition 5.1, the differenti al op erator ∂ ω 2 : D ∞ + ( R ) → D ω + ( 2 ) has a weakly con tin uous section E 2 : D ω + ( 2 ) → D ∞ + ( R ). W rite A = { a, a + c } for some c > 0 and consider the affine map ℓ ( x ) = a + cx of the real line wh ic h maps the doubleton 2 = { 0 , 1 } on to the d oubleton A . This map induces t w o wea k homeomorphisms: L : D ∞ + ( R ) → D ∞ + ( R ) , L : f 7→ f ◦ ℓ, and Λ : D ω + ( A ) → D ω + ( 2 ) , Λ : ( f n a ) n ∈ ω a ∈ A 7→  c n · f n ℓ ( a ) ) n ∈ ω a ∈ 2 . F or ev ery f ∈ C ∞ ( R ) and n ∈ ω we get ( f ◦ ℓ ) ( n ) ( x ) = c n f ( n ) ( ℓ ( x )). Th is yields the comm utativit y of the follo wing diagram: D ∞ + ( R ) L − − − − → D ∞ + ( R ) ∂ ω A   y   y ∂ ω 2 D ω + ( A ) Λ − − − − → D ω + ( 2 ) It follo ws that th e map ˜ E A = L − 1 ◦ E 2 ◦ Λ is a w eakly cont inuous section of the op erator ∂ ω A . How ever this section can map the d ouble sequence ∂ ω A id R ∈ D ω + ( A ) on to s ome function g = ˜ E A ( ∂ ω A id R ) that is not equal to id R . T o fix this problem, we mo dif y the map ˜ E A to the m ap E A : D ω + ( A ) → D ∞ + ( R ) , E A : f 7→ ( ˜ E A f ) ◦ g − 1 . This m ap is weakly cont inuous b ecause D ∞ + ( R ) endow ed w ith the compact-op en C ∞ is a top ological group. It is clear that E A ∂ ω A id R = ( ˜ E A ∂ ω A id R ) ◦ g − 1 = g ◦ g − 1 = id R . According to Lemma 6.2, the equalit y ∂ ω A g = ∂ ω A id A implies that ∂ ω A g − 1 = ∂ ω A id R . Then for ev ery f ∈ D ω + ( A ) w e get ∂ ω A E A f = ∂ ω A   ˜ E A f  ◦ g − 1  = ∂ ω A ˜ E A f = f , witnessing that E A : D ω + ( A ) → D ∞ + ( R ) is a section of the op erator ∂ ω A .  Pro of of Pro p osition 6.1. W e need to construct a b icon tin uous section E Z : D ω + ( Z ) → D ∞ + ( R ) of the differenti al op erator ∂ ω Z : D ∞ + ( R ) → D ω + ( Z ) such that E Z ( D ω c ( Z )) ⊂ D ∞ c ( R ). F or ev ery m ∈ Z consider the closed in terv al I m = [ m, m +1] and its b oundary ∂ I m = { m, m + 1 } in R . It is clear that the pro jection π m : D ω + ( Z ) → D ω + ( ∂ I m ) , π m : ( f n a ) n ∈ ω a ∈ Z 7→ ( f n a ) n ∈ ω a ∈ ∂ I m is b icon tin uous. By Prop osition 6.2, for eve ry m ∈ Z the differenti al op erator ∂ ω ∂ I m : D ∞ ( R ) → D ω + ( ∂ I m ) has a w eakly cont inuous section E ∂ I m : D ω + ( ∂ I n ) → D ∞ + ( R ) su c h that ( E ∂ I m ◦ ∂ ω ∂ I m )id R = id R . Define the map E Z : D ω + ( Z ) → D ∞ + ( R ) assigning to eac h doub le sequence f = ( f n a ) n ∈ ω a ∈ Z ∈ D ω + ( Z ) th e function E Z f ∈ D ∞ + ( R ) d efined by ( E Z f ) | I m = ( E ∂ I m π m f ) | I m for ev ery m ∈ Z . Let us show that this defin ition is correct. F or every m ∈ Z consider the smo oth function g m = E ∂ I m ( π m f ) ∈ D ∞ + ( R ). It f ollo ws that g ( n ) m ( m ) = f n m and g ( n ) m ( m +1) = f n m +1 for ev ery n ∈ ω , and hence g ( n ) m ( m ) = f n m = g ( n ) m − 1 ( m ). Consequen tly , ∂ ω m g m = ∂ ω m g m − 1 and then the function g m − 1 ∪ m g m is smo oth with ∂ ω m ( g m − 1 ∪ m g m ) = ( f n m ) n ∈ ω . Sin ce ( E Z f ) | [ m − 1 , m + 1] = ( g m − 1 ∪ m g m ) | [ m − 1 , m +1], we conclude that E Z f is smo oth on the op en inte rv al ( m − 1 , m +1) and ∂ ω m E Z f = ( f n m ) n ∈ ω . This im p lies DIFFEOMORPHISM GROUPS OF THE REAL LINE 11 that ∂ ω Z E Z f = f and hence E Z : D ω + ( Z ) → D ∞ + ( R ) is a required section of the differen tial op er ator ∂ ω Z : D ∞ + ( R ) → D ω + ( Z ). Using the fact that for ev ery f ∈ D ω ( Z ) th e v alues of the fu nction E Z f on eac h int erv al I m , m ∈ Z , dep end only on the pro j ection π m f ∈ D ω + ( ∂ I m ), w e can show that th e map E Z : D ω + ( Z ) → D ∞ ( R ) is bicon tin uous. Finally we chec k that E Z ( D ω c ( Z )) ⊂ D ∞ c ( R ). T ak e any f = ( f n a ) n ∈ ω a ∈ Z ∈ D ω c ( Z ). By the d efinition of D ω c ( Z ), there exists M ∈ ω such that ( f n a ) n ∈ ω = ∂ ω a id R for ev ery a ∈ Z with | a | ≥ M − 1. Then for ev er y m ∈ Z with | m | > M we get π m f = ∂ ω ∂ I m id R and E ∂ I m π m f = id R . Consequen tly , E Z f | I m = E ∂ I m π m f | I m = id R | I m for all m > | M | , whic h means that E Z f ∈ D ∞ c ( R ).  7. The pr oof of Theorem 1.2 W e need to p ro v e that for eve ry r ∈ ω ∪{∞} the triple of diffeomorphism groups ( D r ( R ) , D r + ( R ) , D r c ( R )) endo w ed with the compact-op en and Whitney C r top ologies is b ihomeomorphic to th e triple of h ome- omorphism groups ( H ( R ) , H + ( R ) , H c ( R )) endo w ed w ith th e compact-op en and Whitney top ologies. First we sh o w that the couple ( D r + ( R ) , D r c ( R )) is bihomeomorphic to ( H + ( R ) , H c ( R )). Consider th e bicon tin uous differen tial op erator ∂ 6 r Z : D r + ( R ) → D ω + ( Z ) assigning to eac h C r diffeomorphism f ∈ D r + ( R ) the double sequence ( f n a ) n ∈ ω a ∈ Z where f n a =      f ( n ) ( a ) if n ≤ r , 1 if r < n = 1, 0 if r < n 6 = 1 . It is clear that ∂ 6 r Z maps D r + ( Z ) and D r c ( R ) into the s u bspaces D 6 r + ( Z ) = { ( f n a ) n ∈ ω a ∈ Z ∈ D ω + ( Z ) : f n a = ∂ ( n ) a id R for all n > r and a ∈ Z } and D 6 r c ( Z ) = D 6 r + ( Z ) ∩ D ω c ( Z ) of the bitop ological space D ω + ( Z ) = { ( f n a ) n ∈ ω a ∈ Z : ( f 0 a ) a ∈ Z ∈ D 1 + ( Z ) , ( f 1 a ) a ∈ Z ∈ Π Z R + } . By the definition of the spaces D 6 r + ( Z ) and D 6 r c ( Z ), the corresp ondence ( f n a ) n ∈ ω a ∈ Z ← →  ( f 0 a ) a ∈ Z , (log f 1 a , ( f n a ) n =2 ,...,r ) a ∈ Z  yields the follo wing conclusion: Claim 7.1. The p air ( D 6 r + ( Z ) , D 6 r c ( Z )) is bihome omorphic to the p air ( D 1 + ( Z ) × Π Z R r , D 1 c ( Z ) × Σ Z R r ) . Lemma 6.2 implies that the sub set D r + ( R ; Z ) = { f ∈ D r + ( R ) : ∂ 6 r Z f = ∂ 6 r Z id R } is a (w eakly) closed su bgroup of D r + ( R ). The bitop ologica l structure of the subgroup D r ( R ; Z ) can b e describ ed as f ollo ws. Using Lemma 6.1, one can c hec k that the m ap Ψ : D r ( R ; Z ) → Π m ∈ Z C r ( I m ) , Ψ : f 7→ ( f | I m ) m ∈ Z 12 T. BANA KH AND T. Y AGASAKI determines a bihomeomorphism of the group D r ( R ; Z ) on to the pro duct Π m ∈ Z D r ∂ ( I m ) of the groups D r ∂ ( I m ) = { f ∈ C r + ( I m ) : ∂ 6 r ∂ I m f = ∂ 6 r ∂ I m id R } ⊂ C r ( I m ) . The pro du ct Π m ∈ Z D r ∂ ( I m ) is consid ered as a bitop ological space carrying the Tyc hono v and b ox- pro du ct top ologies. The bihomeomorphism Ψ maps the subgroup D r c ( R ; Z ) = D r ( R ; Z ) ∩ D r c ( R ) of D r ( R ; Z ) on to the subspace Σ m ∈ Z D r ∂ ( I m ) = { ( f m ) m ∈ Z ∈ Π m ∈ Z D r ∂ ( I m ) : |{ m ∈ Z : f m 6 = id I m }| < ℵ 0 } of Π m ∈ Z D r ∂ ( I m ). Observe that for ev ery in teger num b er m ∈ Z th e group D r ∂ ( I m ) is a conv ex su bset of the F r´ ec het space C r ( I m ). Consequent ly , this group is an absolute retract acco rdin g to th e Borsuk-Dugundji Th e- orem [9], [7]. Being a Polish non-locally compact absolute r etract, the group D r ∂ ( I m ) is homeomorphic to the separable Hilb ert sp ace l 2 b y the Dobro w olski-T oru ´ nczyk Theorem [8]. No w we see that th e pair ( Π m ∈ Z D r ∂ ( I m ) , Σ m ∈ Z D r ∂ ( I m )) is bih omeomorph ic to the pair ( Π Z l 2 , Σ Z l 2 ). T aking in to accoun t that the former pair is bihomeomorphic to ( D r ( R ; Z ) , D r c ( R ; Z )), we get Claim 7.2. Th e p air ( D r ( R ; Z ) , D r c ( R ; Z )) i s bihome omorphic to ( Π Z l 2 , Σ Z l 2 ) . By Prop osition 6.1, the op erator ∂ ω Z : D ∞ + ( R ) → D ω + ( Z ) has a bicon tin uous sec tion E Z : D ω + ( Z ) → D ∞ + ( R ) suc h that E Z ( D ω c ( Z )) ⊂ D ∞ c ( R ). It is easy to see that E Z restricted to D 6 r + ( Z ) ⊂ D ω + ( Z ) is a bicon tin uous section of th e op erator ∂ 6 r Z : D r + ( R ) → D 6 r + ( Z ). No w consider the map Φ : D 6 r + ( Z ) × D r ( R ; Z ) → D r + ( R ) , Φ : ( f , g ) 7→ ( E Z f ) ◦ g . The biconti nuit y of the map Φ follo ws from th e bicontin u it y of the sectio n E Z and the bicon tin uit y of the group op eration on D r + ( R ). In fact, the map Φ is a bihomeomorph ism. Its inv erse is defin ed by Φ ′ : D r + ( R ) → D 6 r + ( Z ) × D r ( R ; Z ) , Φ ′ : h 7→ ( ∂ 6 r Z h, ( E Z ∂ 6 r Z h ) − 1 ◦ h ) . F or ev ery h ∈ D r + ( R ) and g = E Z ∂ 6 r Z h , since ∂ 6 r Z g = ∂ 6 r Z E Z ∂ 6 r Z f = ∂ 6 r Z f , f rom Lemma 6.2 it follo ws th at ∂ 6 r Z g − 1 = ∂ 6 r Z f − 1 and ∂ 6 r Z ( g − 1 ◦ f ) = ∂ 6 r Z ( f − 1 ◦ f ) = ∂ 6 r Z id R , whic h witnesses that g − 1 ◦ f ∈ D r ( R ; Z ). Th e bicon tin uit y of the map Φ ′ follo ws from the bicon tin uit y of the maps E Z , ∂ 6 r Z and the group op erations on D r ( R ). I t also follo ws f rom E Z ( D ω c ( Z )) ⊂ D ∞ c ( R ) that Φ ′ ( D r c ( R )) = D 6 r c ( Z ) × D r c ( R ; Z ). This means that the pair ( D r + ( R ) , D r c ( R )) is bih omeomorphic to the pair ( D 6 r + ( Z ) × D r ( R ; Z ) , D 6 r c ( Z ) × D r c ( R ; Z )). By Claims 7.1 and 7.2, the latter pair is bihomeomorph ic to the pair ( D 1 + ( Z ) × Π Z R r × Π Z l 2 , D 1 c ( Z ) × Σ Z R r × Σ Z l 2 )  , whic h is bih omeomorphic to ( D 1 + ( Z ) × Π Z l 2 , D 1 c ( Z ) × Σ Z l 2 )  . Th us w e hav e p ro v ed Lemma 7.1. F or every r ∈ ω ∪ {∞} the p air ( D r + ( R ) , D r c ( R )) is bihome omorphic to the p air  D 1 + ( Z ) × Π Z l 2 , D 1 c ( Z ) × Σ Z l 2 )  . This lemma imp lies that for ev ery r ∈ ω ∪ {∞} the pair ( D r + ( R ) , D r c ( R )) is b ihomeomorphic to the pair ( D 0 + ( R ) , D 0 c ( R )) = ( H + ( R ) , H c ( R )). Consequently , there is a bihomeomorphism H : D r + ( R ) → H + ( R ) su c h that H ( D r c ( R )) = H c ( R ). DIFFEOMORPHISM GROUPS OF THE REAL LINE 13 It remains to extend the b ihomeomorphism H to a bihomeomorphism ˜ H : D r ( R ) → H ( R ). F or this fix an y diffeomorph ism φ ∈ D ∞ ( R ) \ D ∞ + ( R ). F or example, we can tak e φ ( x ) = − x . Extend the bihomeomorphism H to a bihomeomorphism ˜ H : D r ( R ) → H ( R ) assigning to eac h f ∈ D r ( R ) the diffeomorphism ˜ H f = ( H f if f ∈ D r + ( R ) φ − 1 ◦ H ( f ◦ φ ) if f ∈ D r ( R ) \ D r + ( R ) . It is clea r that the bihomeomorphism ˜ H maps the pair ( D r + ( R ) , D r c ( R )) on to the pair ( H + ( R ) , H c ( R )). 8. Open Problems By Corollaries 1.1 and 1.2, for every r ∈ ω ∪ {∞} the p air ( D r + ( R ) , D r c ( R )) is wea kly and strongly homeomorphic to the pair ( Π ω l 2 , Σ ω l 2 ). Problem 8.1. Are the pairs ( D r + ( R ) , D r c ( R )), r ∈ ω ∪ {∞} , bih omeomorphic to the pair ( Π ω l 2 , Σ ω l 2 )? The answer to this problem is affirmativ e if the answer to the f ollo wing problem is affirmativ e. Problem 8.2. I s the p air ( D 1 + ( Z ) , D 1 c ( Z )) bihomeomorphic to the pair ( Π ω R , Σ ω R )? By [3], the pair ( D 1 + ( Z ) , D 1 c ( Z )) is w eakly and strongly homeomorph ic to the p air ( Π ω R , Σ ω R ). Our next q u estion asks if Theorem 1.2 can b e generalized to all smo oth manifolds. Problem 8.3. Ar e the triples ( D r ( M ) , D r + ( M ) , D r c ( M )) and ( H ( M ) , H + ( M ) , H c ( M )) b ihomeomor- phic for an y connected noncompact orienta ble C r manifold M ? Here D r ( M ), D r + ( M ) and D r c ( M ) are th e groups of C r diffeomorphisms, orien tation-preserving C r diffeomorphisms and compactly supp orted C r diffeomorphisms of M , resp ectiv ely . Those groups are bitop ological spaces whose we ak top ology is the compact-op en C r top ology while the strong top ology is th e Whitney C r top ology (v ery strong C r top ology in the sense of [13]). Finally , let us discuss the prob lem of bitop ological characte rization of spaces of th e form Π ω X or Σ ω X for s im p le top ological spaces X li ke I = [0 , 1], I ω , R , or l 2 . The bitop ological spaces Σ ω I and Σ ω I ω ha v e b een characte rized in [4 ]. This c haracterization implies that the b itop ological spaces Σ ω I and Σ ω R are bihomeomorphic. Problem 8.4. Give a bitop ologica l c haracterization of the bitop ological sp ace Σ ω l 2 . Using the tec hnique of [2 ] and [4] it can b e sho wn that th e bitoplogical space Σ ω l 2 is bihomeomor- phic to l 2 × Σ ω R . So, Problem 8.4 can b e reform ulated as the problem of bitop ological c haracterizat ion of the bitop ological space l 2 × Σ ω R . It sh ould b e mentioned that the top ological c haracterizations of the top ological spaces Σ ω l 2 and ⊡ ω l 2 (comp osing the bitop ologo cal s p ace Σ ω l 2 ) are k n o wn, see [6], [5]. The bitop ologica l equiv alence of the spaces Σ ω l 2 and l 2 × Σ ω R suggests the follo wing qu estion. Problem 8.5. Ar e the bitop ological spaces Π ω l 2 and l 2 × Π ω R bihomeomorphic? In order to answ er this p roblem it w ould b e helpfu l to kn ow the answer to the f ollo wing (a pparently , v ery difficult) problem. Problem 8.6. Give a bitop ologica l c haracterization of the bitop ological sp aces Π ω R and Π ω l 2 . 14 T. BANA KH AND T. Y AGASAKI By Anderson’s Th eorem [1], the spaces Π ω R and Π ω l 2 are w eakly homeomorphic. On the other hand, these spaces are not strongly homeomorphic b ecause the connecte d comp onen ts of those spaces endo w ed with the strong top ology are homeomorphic to the (non-homeomorphic) spaces ⊡ ω R and ⊡ ω l 2 , r esp ectiv ely . The top ological c haracteriza tion of the spaces Π ω R and Π ω l 2 w as giv en b y T oru ´ nczyk [14]. An analogous problem for the resp ectiv e b ox-prod u cts is op en. Problem 8.7. Give a top ological c haracterization of th e sp aces  ω R and  ω l 2 . 9. A cknowledgment The authors thank the referee wh ose constructiv e criticism resulted in considerable impro v emen t of the present ation of this pap er. Referen ces [1] R .D. Anderson, Hilb ert sp ac e is home omorphic to the c ountable infini te pr o duct of li nes , Bull. Amer. Math. So c. 72 (1966), 515–519. [2] T. Banakh, On hyp ersp ac es and home omorphism gr oups home omorphic to pr o ducts of absorbing set s and R ∞ , Tsukuba J. Math. 23 (1999) 495–504 . [3] T. Banakh, K. Mine and K . Saka i, Classifying home omorphism gr oups of infinite gr aphs , T op ology App l. 157 (2009), 108–122 . [4] T. Banakh, K. Sak ai, Char acterizations of ( R ∞ , σ ) - or ( Q ∞ , Σ) -manifol ds and their appli c ations , T op ology Ap pl. 106 (2000), 115–134. [5] T. Banakh, D. Rep o v ˇ s, A top olo gic al char acterization of LF-sp ac es , p reprint (arXiv : 0911.0609). [6] M. Bestvina, J. Mogilski, Char acter izing c ertain inc omplete infinite-dimensional absolute r etr acts , Michig an Math. 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T oru´ nczyk, Char act erizing Hilb ert sp ac e top ol o gy , F und. Math. 111 (1981), 247–262 . (T. Banakh) Dep ar tment of M a them a tics, Iv an Franko N a tional Uni versity of L viv, Ukraine, and Instytut Ma tema tyki, Un iwersytet Humanistyczn o-Przyr odniczy Jana Kocano wskiego w Kie lca ch, Poland E-mail addr ess : t.o.tbanakh @gmail.com (T. Y agas aki) Division of Ma thema tics, Gradua te School of Science and Technology, Kyoto Institute of Technology, Kyoto, 606-8585, Jap an E-mail addr ess : yagasaki@ki t.ac.jp