math.GT 2010-02-24 0

Homeomorphism and diffeomorphism groups of non-compact manifolds with the Whitney topology

For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and H_c(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group H_c(M) is locally contract…

Authors: Taras Banakh, Kotaro Mine, Katsuro Sakai

HOMEOM O RPHISM AND DIFFEOMO RPHISM GROUPS OF N ON-COMP A CT MANIF OLDS WITH THE WHITNEY TOPOLOGY T ARAS BANAKH, K OT ARO MINE, KA TSURO SAKAI ∗ , AND T A TSUHIK O Y AGASAKI ∗∗ Abstract. F or a non-compac t n -manifol d M let H ( M ) b e the group of homeomorphisms of M endow ed with the Whitne y top ology and H c ( M ) the subgroup o f H ( M ) consisting of homeomorphisms with compa ct support. It is sho wn that the group H c ( M ) is locall y con tractible and the identit y comp onen t H 0 ( M ) of H ( M ) is an op en normal subgroup in H c ( M ). This induces the topological fact orization H c ( M ) ≈ H 0 ( M ) × M c ( M ) for the mapping class group M c ( M ) = H c ( M ) / H 0 ( M ) wi th the discrete topol ogy . F urthermore, for an y non-compact surface M , the pair ( H ( M ) , H c ( M )) is l ocally homeomorphic to (  ω l 2 , ⊡ ω l 2 ) at the identit y id M of M . Thus the gr oup H c ( M ) is an ( l 2 × R ∞ )-manifold. W e also study top ological prop erties of the group D ( M ) of diffeomorphisms of a non-compac t smo oth n -manifold M endo we d with the Whitne y C ∞ - topology and the subgroup D c ( M ) of D ( M ) consisting of all diffeomorphisms with compact support. It is sho wn that the pair ( D ( M ) , D c ( M )) is lo cally homeomorphic to (  ω l 2 , ⊡ ω l 2 ) at the identit y id M of M . Hence the gr oup D c ( M ) is a topological ( l 2 × R ∞ )-manifold f or any dimension n . 1. I ntr oduction In this pa p er we study topo logical prop erties of groups of homeo morphisms and diffeomor phisms of non- compact manifolds endow ed with the Whitney top ology and rec ognize their loca l topolo gical t yp e. F or a σ - compact n -manifold M po ssibly with b ounda ry le t H ( M ) denote the gr oup of homeomo rphisms of M endow ed with the Whitney top olo g y and H c ( M ) the subgroup of H ( M ) consisting of ho meomorphisms with compac t supp ort. In this top ology , each f ∈ H ( M ) has the fundamental neighborho o d system U ( f ) =  g ∈ H ( M ) : ( f , g ) ≺ U  ( U ∈ cov( M )) , where cov( M ) is the set of op en covers o f M and the no ta tion ( f , g ) ≺ U means that f and g a r e U -ne ar (i.e., every po int x ∈ M admits a set U ∈ U with f ( x ) , g ( x ) ∈ U ) (Prop ositio n 3.1). The supp ort of h ∈ H ( M ) is defined by supp( h ) = cl  x ∈ M : h ( x ) 6 = x  . The group H ( M ) is a top ological group a nd the identit y comp onent H 0 ( M ) o f H ( M ) lies in the subgr oup H c ( M ) (Prop osition 3.3 (1)). In case M is a compact n -manifold, the Whitney top ology on the gr oup H ( M ) coincides with the compact- op en top ology . Hence, H ( M ) is co mpletely metrizable and lo ca lly contractible ([10], [11]). When M is a compact surface (or a finite graph), the group H ( M ) is an l 2 -manifold (the ca se o f finite g raphs is the result of [2]; the case of compact surfaces is a combination of the results of [2 3], [14] and [30]). F or n ≥ 3, it is still an op en problem if the homeomo rphism group H ( M ) of a compact n -manifold M is an l 2 -manifold [3], which is called the Homeomor phism Gr o up Problem [33]. 2000 Mathematics Subje ct Classific ation. 22A05, 46A13, 46T05, 46T10, 54H11, 57N05, 57N17, 57N20, 57S05, 58D05, 58D15. Key wor ds and phr ases. Homeomorphism group, diffeomorphism group, transformation group, the Whitney top ology , σ - compact manifold, non-compact surface, ( l 2 × R ∞ )-manifold, F r ´ ec het space, LF-space, top ological group, b ox pro duct, small box pro duct. This work is supp orted by Gran t-in-Ai d f or Scientific Research (No.17540061 ∗ , No.19540078 ∗∗ ). 1 2 T. BANAKH, K. MINE , K. SAKAI, AND T. Y A GASAKI In this pap er w e ar e concer ned with the ca se that M is a noncompact σ -co mpact n -manifold. In this cas e the Whitney to po lo gy on H ( M ) is still very impo rtant in Geo metric T op olo gy , but it has rather bad lo c a l prop erties. Our obser v ations in this pap er mean that the subgro up H c ( M ) of H ( M ) is q uite nice from the top ological viewp oint. First we reca ll basic fa c ts on the lo cal to po logical mo dels of the g roups H ( M ) and H c ( M ). A F r´ echet sp ac e is a completely metrizable lo cally co n vex top olo gical linear space and an LF-space is the direct (o r inductive) limit o f increa sing sequence of F r´ echet spa ces in the categ o ry of lo cally conv ex top ologica l linear spac es. A top olo gical characterization of LF-spaces is given in [7]. The simplest non- trivial example of an LF-space is R ∞ , the direct limit of the tow er R 1 ⊂ R 2 ⊂ R 3 ⊂ · · · , where each space R n is identified with the hyperplane R n × { 0 } ⊂ R n +1 . In [24] P . Mankiewicz studied LF-spaces and prov ed that an infinite-dimensional separable LF-space is homeomorphic to ( ≈ ) either R ∞ , l 2 or l 2 × R ∞ . The space l 2 × R ∞ is homeomo rphic to the count able small b ox p ow er ⊡ ω l 2 of the Hilbert space l 2 , whic h is a subspace of the b ox pow er  ω l 2 . One s ho uld notice that the b ox p ow er  ω l 2 is neither loc a lly connected, nor sequential, nor normal (see [17], [34]). In the pap ers [4] and [5] we ha ve alrea dy shown tha t ( H ( R ) , H c ( R )) ≈ ( H ([0 , ∞ )) , H c ([0 , ∞ ))) ≈ (  ω l 2 , ⊡ ω l 2 ) . Moreov er, it is proved in [5] that if M is a non-c o mpact se pa rable graph then H c ( M ) is an ( l 2 × R ∞ )-manifold. Thu s, a s a non-compact version o f the Homeomo rphism Group Problem, w e can exp ect the fo llowing: Conjecture. F or any non-c omp act σ -c omp act n -manifold M p ossibly with b oundary, the p air ( H ( M ) , H c ( M )) is lo c al ly home omo rphic t o (  ω l 2 , ⊡ ω l 2 ) at the identity id M of M . In p articular, the gr oup H c ( M ) is an ( l 2 × R ∞ ) -manifold. 1 Here, w e say that a pair ( X ′ , X ) of top olog ic al spaces X ⊂ X ′ is lo c al ly home omorp hic to a pair ( Y ′ , Y ) if each point x ∈ X has an op en neighborho o d U ⊂ X ′ such that the pair ( U, U ∩ X ) is ho meomorphic to the pair ( V , V ∩ Y ) for so me open s et V ⊂ Y ′ . In this pap er we fir st show that the gr oup H c ( M ) is lo cally contractible in an y dimension n . Prop ositio n 1 . F or any σ -c omp act n - manifold M p ossibly with b oundary, t he gr oup H c ( M ) is lo c al ly c on- tr actible. Thu s the identit y comp onent H 0 ( M ) is an op en norma l subgro up o f H c ( M ) a nd it induces the to po logical factorization H c ( M ) ≈ H 0 ( M ) × M c ( M ) , where M c ( M ) = H c ( M ) / H 0 ( M ) is the mapping class gr oup of M with the discrete topo logy . As mentioned ab ov e, the conjecture ab ove has b een pr oved in the case n = 1, see [5]. Here we solve the conjecture affir ma tively in the case n = 2. Theorem 2. Supp ose M is a non-c omp act σ -c omp act 2 -manifold p ossibly with b oundary. Then t he p air ( H ( M ) , H c ( M )) is lo c al ly home omorphic t o (  ω l 2 , ⊡ ω l 2 ) at the identity id M of M . In p articular, the gr oup H c ( M ) is an ( l 2 × R ∞ ) -manifold. 1 Because of the non-paracom pactness of  ω l 2 , we av oid to say “  ω l 2 -manifold” or “(  ω l 2 , ⊡ ω l 2 )-manifold pair ”. HOMEOMORPHISM AND DIFFEOM ORPHISM GR OUPS 3 An y σ -compact 2-ma nifo ld M p ossibly with b oundary admits a P L-structure [2 6]. When M is equipp ed with a PL-structure, the sy m b ol H P L ( M ) denotes the subgroup o f H ( M ) consisting of PL-homeomorphisms with res p ect to this P L-structure. A subspace A of a space X is said to b e homotopy dense (abbrev. HD) if there exists a homotopy φ t : X → X such that φ 0 = id X and φ t ( X ) ⊂ A ( t ∈ (0 , 1]). Prop ositio n 3. S u pp ose M is a n on- c omp act σ - c omp act PL 2 -manifold p ossibly with b oundary. Then the sub gr oup H P L c ( M ) is homotopy dense in H c ( M ) . W e a lso study the lo cal top ologica l type of groups of diffeomorphisms of non-compa ct smo oth manifolds endow ed with the Whitney C ∞ -top ology . F or a smo oth n - manifold M , let D ( M ) denote the gro up of a ll diffeomorphisms of M endowed with the Whitney C ∞ -top ology . Let D 0 ( M ) denote the identit y comp o nent of the diffeomo rphism gr o up D ( M ) a nd D c ( M ) deno te the subgr oup of D ( M ) consisting of all diffeomorphisms of M with compact supp ort. Theorem 4. F or a non-c omp act σ -c omp act smo oth n -manifold M without b oundary, the p air ( D ( M ) , D c ( M )) is lo c al ly home omorph ic to (  ω l 2 , ⊡ ω l 2 ) at the identity id M of M . In p articular, the gr oup D c ( M ) is a top olo gic al ( l 2 × R ∞ ) -manifold. This implies that the identit y co mpo nent D 0 ( M ) is an op en normal subg r oup of D c ( M ) and it induces the top ologica l factor ization D c ( M ) ≈ D 0 ( M ) × M ∞ c ( M ) , M ∞ c ( M ) = D c ( M ) / D 0 ( M ) . In [8] we have shown that ( D ( R ) , D c ( R )) ≈ (  ω l 2 , ⊡ ω l 2 ) . In the succeeding pap er [6], we determine the global top ologica l types of the gro ups H c ( M ) for non-compact s urfaces M and the gro ups D c ( M ) for some kind of non-co mpact s mo o th n -manifolds M . This pap er is org anized as follows: Section 2 co n tains the ba sic facts on the b ox pro ducts and the s mall box pro ducts, and Section 3 contains g eneralities o n ho meomorphism groups with the Whitney top ology . In Section 4 we introduce some fundamen tal notations on transformatio n gro ups and in Section 5 w e fo rmulate the notion of strong top olo gy on transforma tion gr oups and study some basic pro per ties. In Section 6 we apply thes e results to gr o ups of homeomorphisms and diffeo mo rphisms of noncompact manifolds and prov e Theorems 2 and 4 toge ther with P rop ositions 1 and 3. 2. Box and small box products In this section we r ecall some basic prop erties o n b ox pro ducts and small b ox pro ducts. Let ω a nd N denote the sets of non- neg ative integers a nd po sitive integers, respe c tiv ely . Definition 2. 1. (1) The b ox pr o duct  n ∈ ω X n of a sequence of topolog ical spa c es ( X n ) n ∈ ω is the countable pro duct Q n ∈ ω X n endow ed with the b ox top ology g e nerated by the base consis ting of b oxes Q n ∈ ω U n , where U n is an op en subset of X n . (2) The smal l b ox pr o duct ⊡ n ∈ ω X n of a sequence of p o int ed spaces ( X n , ∗ n ) n ∈ ω is the subspa ce of  n ∈ ω X n defined by ⊡ n ∈ ω X n =  ( x n ) n ∈ ω ∈  n ∈ ω X n : ∃ m ∈ ω ∀ n ≥ m, x n = ∗ n  . (3) The pair (  n ∈ ω X n , ⊡ n ∈ ω X n ) is denoted by the sym b ol (  , ⊡ ) n ∈ ω X n . 4 T. BANAKH, K. MINE , K. SAKAI, AND T. Y A GASAKI The small b ox pr o duct ⊡ n ∈ ω X n has a canonical distinguished p oint ( ∗ n ) n ∈ ω . F or a sequence of subsets A n ⊂ X n ( n ∈ ω ), let ⊡ n ∈ ω A n = ⊡ n ∈ ω X n ∩  n ∈ ω A n , where it is no t a ssumed that ∗ n ∈ A n . If ∗ n 6∈ A n for infinitely many n ∈ ω then ⊡ n ∈ ω A n = ∅ . Identifying Q i ≤ n X i with the closed s ubs pa ce  ( x i ) i ∈ ω ∈ ⊡ i ∈ ω X i : x i = ∗ i ( i > n )  , we can regar d ⊡ n ∈ ω X n = S n ∈ ω Q i ≤ n X i . When X n = X for all n ∈ ω , we write  ω X and ⊡ ω X instead of  n ∈ ω X n and ⊡ n ∈ ω X n , which are called the b ox p ower and the smal l b ox p ower of X , resp ectively . Then w e ca n reg ard ⊡ ω X = S n ∈ ω X n , where X ⊂ X 2 ⊂ X 3 ⊂ · · · . Prop ositio n 2.2. If e ach finite pr o duct Q i ≤ n X i is p ar ac omp act, t hen the smal l b ox pr o duct ⊡ i ∈ ω X i is also p ar ac omp act. Pr o of. By the characteriz a tion of para compactness (Theor em 5.1 .11 in [1 2]), it suffices to show that every op en cover U of ⊡ i ∈ ω X i has a σ -lo cally finite o pen refinement. F or e a ch n ∈ N , we shall construct a lo cally finite o p en collection V n in ⊡ i ∈ ω X i which covers Q i ≤ n X i and refines U . Each x ∈ Q i ≤ n X i has a ba sic op en neighborho o d ⊡ i ∈ ω U x i which is contained some member of U . Then { Q i ≤ n U x i : x ∈ Q i ≤ n X i } is a n op en cov er o f Q i ≤ n X i , which has a lo cally finite op en refinement U n . F or each U ∈ U n , choose x ∈ Q i ≤ n X i so that U ⊂ Q i ≤ n U x i and define V U = U × ⊡ i>n U x i . Then V n = { V U : U ∈ U n } ( n ∈ ω ) s atisfy the require d conditions. Consequently , S n ∈ N V n is a σ -lo cally finite op en r efinement of U .  A sequence of maps φ n : ( X n , ∗ n ) → ( Y n , ∗ n ) ( n ∈ ω ) induces a contin uous map ⊡ n ∈ ω φ n : ⊡ n ∈ ω X n → ⊡ n ∈ ω Y n , ( ⊡ n ∈ ω φ n )(( x n ) n ∈ ω ) = ( φ n ( x n )) n ∈ ω . Lemma 2.3. F or a c omp act sp ac e K and a se quenc e of maps φ n : ( X n × K , {∗ n } × K ) → ( Y n , ∗ n ) ( n ∈ ω ) , the map Φ :  ⊡ n ∈ ω X n  × K → ⊡ n ∈ ω Y n , Φ  ( x n ) n ∈ ω , y ) = ( φ n ( x n , y )) n ∈ ω . is c ontinuous. Pr o of. The pr o of is str aightforw ard. T ake any p oint (( x n ) n ∈ ω , y ) o f ⊡ n ∈ ω X n × K a nd any op en neighbor ho o d V of Φ(( x n ) n ∈ ω , y ) in ⊡ n ∈ ω Y n . W e may assume that V is of the for m V = ⊡ n ∈ ω V n , where V n is an o pe n neighborho o d of φ n ( x n , y ) in X n . There exists m ∈ ω such that x n = ∗ n for n > m . F or n = 0 , 1 , · · · , m , choose op en neighborho o ds U n of x n in X n and W n of y in K such that φ n ( U n × W n ) ⊂ V n . F or n > m , since φ n ( {∗ n } × K ) = { ∗ n } ⊂ V n and K is co mpact, there exists an op en neighborho o d U n of x n = ∗ n in X n such that φ n ( U n × K ) ⊂ V n . Then U = ⊡ n ∈ ω U n and W = T m n =0 W n are o pe n neighborho o ds of ( x n ) n in ⊡ n ∈ ω X n and y in K , respe c tively . Now, it is easy to see that Φ( U × W ) ⊂ V . This completes the pro of.  F or example, a sequence of homotopies φ n t : ( X n , ∗ n ) → ( Y n , ∗ n ) ( n ∈ ω ) induces a homo to p y ⊡ n ∈ N φ n t : ⊡ n ∈ N X n → ⊡ n ∈ N Y n . This simple obser v ation leads to some useful cons equences. Definition 2 .4. A subs pace A o f a spa ce X is ca lled homotopy dense (abbrev. HD) in X re l. a s ubs e t A 0 of A if ther e ex ists a homoto p y φ t : X → X ( t ∈ [0 , 1]) such that φ 0 = id X , φ t | A 0 = id a nd φ t ( X ) ⊂ A ( t ∈ (0 , 1]). HOMEOMORPHISM AND DIFFEOM ORPHISM GR OUPS 5 Prop ositio n 2.5. L et ( X n , A n , ∗ n ) n ∈ ω b e a se quenc e of p ointe d p air of sp ac es. If e ach A n is HD in X n r el. the p oint ∗ n , t hen ⊡ n ∈ ω A n is HD in ⊡ n ∈ ω X n . Remark 2.6. (1) Suppose X is a metrizable s pa ce, A 0 ⊂ A ⊂ X and A 0 is a closed s ubs et of X . If A is HD in X , then A is HD in X rel. A 0 . (2) Suppose G is a topolo g ical g r oup and H is a subgr oup o f G . If H is HD in G , then H is HD in G rel. the identit y elemen t e of G . Definition 2.7. (1) A subspace A of a space X is called c ontr actible in X (r el. a point a ∈ A ) if there exists a homotopy φ t : A → X ( t ∈ [0 , 1]) such that φ 0 = id A and φ 1 ( A ) is a singleton ( φ t ( a ) = a for every t ∈ [0 , 1], whence φ 1 ( A ) = { a } ). (2) A space X is called ( st r ongly ) lo c al ly c ontr actible at x ∈ X if every neighbor ho o d U o f x co n tains a neighborho o d V of x which is co n tractible in U (rel. x ). (3) A p ointed spa ce ( X , ∗ ) is said to b e (i) lo c al ly c ontr actible if X is lo cally contractible at any p oint of X and strongly lo cally contractible at the point ∗ , and (ii) c ontr actible if X is contractible in X r el. ∗ . Prop ositio n 2. 8 . If p ointe d sp ac es ( X n , ∗ n ) ( n ∈ ω ) ar e (lo c al ly) c ontr actible , then the smal l b ox pr o duct ⊡ n ∈ ω X n is also (lo c al ly) c ontr actible as a p ointe d sp ac e. Remark 2. 9. A spa ce X is ca lled semi-lo c al ly c ontr actible at a point x ∈ X if x has a neigh b orho o d V in X which contracts in X . It is easy to s ee that if a top o logical gro up G is se mi-lo cally contractible at the identit y element e ∈ G then G is s tr ongly lo cally co nt rac tible at every x ∈ G , henc e the pointed spa c e ( G, e ) is lo cally contractible. Indeed, if h : V 0 × [0 , 1] → G is a contraction of a neighbor ho o d V 0 of e ∈ G , then we can define a contraction h ′ : V 0 × [0 , 1] → G by h ′ ( x, t ) = h ( e, t ) − 1 h ( x, t ). Since h ′ ( { e } × [0 , 1]) = { e } , every neighborho o d U of e con tains a neig hborho o d V of e such that h ′ ( V × [0 , 1]) ⊂ U . Then the r e striction h ′ | V × [0 , 1] is a contraction of V in U fix ing the identit y element e . Since the topolo g ical g r oup G is homogeneo us, it follows that G is stro ngly locally co n tractible at every x ∈ G . Finally we discuss the box pr o ducts o f top ologica l gro ups. As usual, w e rega rd a top olo gical g roup as a po int ed spa ce by distinguishing the iden tit y element. F or top ologica l g roups ( G n ) n ∈ ω , the b ox pro duct  n ∈ ω G n is a top ological group under the co or dinatewise multip lication, and the sma ll b ox pr o duct ⊡ n ∈ ω G n is a top ologica l s ubg roup of  n ∈ ω G n . Suppo se G is a topolo gical g roup with the identit y element e ∈ G . An y sequence ( G n ) n ∈ ω of subgr oups of G induces the natura l m ultiplication map p : ⊡ n ∈ ω G n → G, p ( x 0 , . . . , x k , e, e , . . . ) = x 0 · x 1 · · · x k . Lemma 2.10. The map p is c ontinuous. Pr o of. Fix any p oint x = ( x 0 , . . . , x k ) ∈ ⊡ n ∈ ω G n and take any neighbor ho o d V of its image p ( x ) = x 0 · · · x k in G . Replacing x by a lo nger sequence if necess ary , we ca n ass ume that x k = e . By the contin uity of the group multiplication, find a sequence of neighborho o ds U n of x n in G , n ≤ k , such that U 0 · U 1 · · · U k − 1 · U k · U k ⊂ V . Now, inductively co nstruct a decrea sing sequence ( U n ) n>k of o pen neighbo r ho o ds of e in G suc h that U n · U n ⊂ U n − 1 for all n > k . Such a choice will guara nt ee that U 0 · · · U k − 1 · U k · U k +1 · · · U n ⊂ U 0 · · · U k − 1 · U k · U k ⊂ V ( n > k ) . 6 T. BANAKH, K. MINE , K. SAKAI, AND T. Y A GASAKI Then the set U =  n ∈ ω U n ∩ ⊡ n ∈ ω G n is an op en neighbor ho o d o f x in ⊡ n ∈ ω G n and p ( U ) ⊂ [ n>k ( U 0 · · · U n ) ⊂ V , which pr ov es the con tinuit y of p at x .  Let p : X → Y be a c o nt inuous ma p. A lo c al s e ction o f p a t y ∈ Y is a map s : V → X defined o n a neighborho o d V of y in Y such tha t ps = id. When V = Y , the map s is ca lled a se ct ion o f p . Lemma 2.11. Supp ose ( G n ) n ∈ ω is a se quenc e of su b gr oups of G such t hat G n ⊂ G n +1 for e ach n ∈ ω and G = S n ∈ ω G n . If the multiplic ation map p : ⊡ n ∈ ω G n → G has a lo c al se ction at e , then t he fol lowing hold: (1) The map p has a lo c al se ction at any p oint of G and a lo c al se ction s at e with s ( e ) = ( e , e, · · · ) . (2) If e ach G n is lo c al ly c ontr actible, then so is G . (3) Su pp ose G is p ar ac omp act and H is a sub gr oup of G . If H n = H ∩ G n is HD in G n for e ach n ∈ ω , then H is H D in G . Pr o of. (1) The verification is simple and omitted. (2) By Remar k 2.9 and Pr op osition 2.8, the s mall box pro duct ⊡ n ∈ ω G n is lo cally c o nt rac tible. Since the map p has a lo cal sectio n at any point, the group G is also lo cally contractible. (3) Since G is para c ompact, it suffices to show that each g ∈ G ha s an open neighborho o d U in G with a homotopy φ t : U → G s uch that φ 0 = id and φ t ( U ) ⊂ H ( t ∈ (0 , 1]). By (1) the map p admits a lo cal s e ction s : U → ⊡ n ∈ ω G n at the p oint g . By Rema rk 2.6 (2) and Prop ositio n 2.5, the sma ll b ox product ⊡ n ∈ ω H n is HD in ⊡ n ∈ ω G n by a n absorbing homotopy ψ t . Then the homotopy φ t is defined by φ t = pψ t s .  3. Basic proper ties of h omeomorphism groups with the Whitney topology In this sec tio n, we lis t so me basic prop erties of the Whitney top olog y on homeomorphis m g roups. F or any topolo gical space M , let H ( M ) denote the gro up of homeomor phisms of M endow ed with the Whitney top ology . This top ology is gener a ted by the subsets U ( h ) =  g ∈ H ( M ) : ( h, g ) ≺ U  , ( h ∈ H ( M ) , U ∈ cov( M )) , and each h ∈ H ( M ) has the neig hborho o d ba sis U ( h ) ( U ∈ cov( M )). On the s pace C ( X , Y ) of all contin uous functions fro m X to Y , the Whitney top olo gy is usually defined as the gr aph top olo gy o r the WO 0 -top olo gy , that is, it is genera led b y Γ U = { f ∈ C ( X, Y ) : Γ f ⊂ U } , where U r uns through all o pen sets in X × Y a nd Γ f = { ( x, f ( x )) : x ∈ C } is the gr aph of f ∈ C ( X , Y ) (e.g., see [20, § 41]). The gr aph top olo gy or the WO 0 -top olo gy on H ( M ) is the subspace topolog y inherited from the space C ( M , M ) with this top olog y . In the space C ( X , Y ), the gr a ph topolo gy is not generaled by the s e ts U ( f ) = { g ∈ C ( X , Y ) : ( f , g ) ≺ U } ( f ∈ C ( X , Y ) , U ∈ cov( Y )) . F or completeness, we giv e a pro of of the following: Prop ositio n 3.1 . F or any top olo gic al sp ac e M , U ( h ) ( U ∈ cov( M )) is a neighb orho o d b asis of h ∈ H ( M ) in the gr aph top olo gy. HOMEOMORPHISM AND DIFFEOM ORPHISM GR OUPS 7 Pr o of. Fix h ∈ H ( M ). (1) Let W ⊂ M 2 be an op en set s uc h that Γ h ⊂ W . E ach x ∈ M has an open neighborho o d U x in M such that U x × h ( U x ) ⊂ W . Since h is a homeo morphism, U = { h ( U x ) : x ∈ M } ∈ cov( M ). T o see U ( h ) ⊂ Γ W , ta ke an y homeo morphism g ∈ U ( h ). F or every p oint z ∈ M , ther e exists y ∈ M such that { h ( z ) , g ( z ) } ⊂ h ( U y ). Since h is a bijection, we have z ∈ U y , hence ( z , g ( z )) ∈ U y × h ( U y ) ⊂ W . This means that Γ g ⊂ W and hence U ( h ) ⊂ Γ W . (2) T ake a ny cov er U ∈ cov( M ). F or each x ∈ M , choos e U x ∈ U with h ( x ) ∈ U x . Then W = [ x ∈ M h − 1 ( U x ) × U x ⊂ M × M is an op en neighbor ho o d of Γ h in M × M . T o see Γ W ⊂ U ( h ), take any g ∈ Γ W . Since Γ g ⊂ W , for any y ∈ M we can find x ∈ M such that ( y , g ( y )) ∈ h − 1 ( U x ) × U x and hence { h ( y ) , g ( y ) } ⊂ U x . This means that g ∈ U ( h ).  F or s ubspaces K ⊂ L ⊂ M , let E K ( L, M ) deno te the s pace o f embeddings f : L → M with f | K = id K endow ed with the compa ct-op en top olo gy . In compar ison with the Whitney top olog y (when M is Hausdorff ), each f ∈ E K ( L, M ) admits the fundamen tal neig h b orho o d system: U ( f , C ) = { g ∈ E K ( L, M ) : ( f | C , g | C ) ≺ U } ( C is a compact subset of L , U ∈ cov( M )). The g roup H ( M ) a cts on E ( L, M ) by the left comp osition. When M is paraco mpact, every op en c over of M admits a star-r efinement. This r e mark leads to the following basic fact. Prop ositio n 3.2. If M is p ar ac omp act, 2 then (i) H ( M ) is a top olo gic al gr oup and (ii) the natur al action of H ( M ) on E ( L, M ) is c ontinuous. Pr o of. F or the s ake of completeness we include the pro of. (i) It follows from the definition o f the Whitney to p olo gy o n H ( M ) that the inv ersio n f 7→ f − 1 is contin u- ous. So, it remains to check that the c ompo sition is co n tinuous with resp ect to the Whitney to p olo gy . Given f , g ∈ H ( M ) and U ∈ c ov( M ), we should find V , W ∈ cov( M ) such that f ′ g ′ ∈ U ( f g ) for every f ′ ∈ V ( f ) and g ′ ∈ W ( g ), that is, ( f , f ′ ) ≺ V and ( g , g ′ ) ≺ W imply ( f g , f ′ g ′ ) ≺ U . By the paracompa ctness of M , there is a cov er V ∈ cov( M ) with S t ( V ) ≺ U . Let W = f − 1 ( V ) = { f − 1 ( V ) : V ∈ V } and as sume ( f , f ′ ) ≺ V and ( g , g ′ ) ≺ W . Since ( f g , f g ′ ) ≺ f ( W ) = V and ( f g ′ , f ′ g ′ ) ≺ V , it follows tha t ( f g , f ′ g ′ ) ≺ S t ( V ) ≺ U . The asser tion (ii) ca n be se en by the sa me argument.  The next prop os ition is the main result in this sec tio n. F or a subset L ⊂ M , let H ( M , L ) = { h ∈ H ( M ) : h | L = id } . Prop ositio n 3.3. If M is p ar ac omp act then (1) H 0 ( M ) ⊂ H c ( M ) and (2) every c omp act subsp ac e K ⊂ H c ( M ) is c ontaine d in H ( M , M \ K ) for some c omp act subset K ⊂ M . Pr o of. (1) It suffices to show that each f ∈ H ( M ) \ H c ( M ) can b e separa ted from id M by a clop en subset U of H ( M ). F or this pur po se, we compare the space H ( M ) with the additive gr oup  ω R . The la tter space contains the clop en subgro up c 0 = n ( a n ) n ∈ ω ∈  ω R : lim n →∞ a n = 0 o . F or any f ∈ H ( M ) \ H c ( M ), we can find a countable discrete subset X = { x n } n ∈ ω of M such tha t f ( X ) ∩ X = ∅ . Indeed, the set F = supp( f ) is no n-compact and clos e d in M , whe nc e F is paracompa ct. 2 It is prov ed in [13] that H ( X ) is a topological gr oup if X is metrizable. 8 T. BANAKH, K. MINE , K. SAKAI, AND T. Y A GASAKI Since the compactness coincides with the ps eudo compactness in the class o f pa racompact spa ces (see [12, 3.10.21 , 5 .1.20]), F is not pseudo co mpact and hence admits a contin uous unbounded function which extends to a co ntin uous function ξ : M → [0 , + ∞ ) b y the norma lit y of M . Since ξ | F is unbounded, we can choose a countable subse t X = { x n } n ∈ ω in F so that for each n ∈ ω , f ( x n ) 6 = x n , ξ ( x n ) > n and ξ ( x n ) > max  ξ ( x i ) , ξ ( f ( x i )) , ξ ( f − 1 ( x i )) : i < n  . Then f ( X ) ∩ X = ∅ and lim n →∞ ξ ( x n ) = ∞ . By the norma lit y o f M , ther e e x ists a Urysohn map λ : M → [0 , 1] with λ ( X ) = 0 and λ ( f ( X )) = 1. Since h ( X ) is discrete for ea ch h ∈ H ( M ), we hav e the map ϕ : H ( M ) →  ω R defined by ϕ ( h ) =  λ ( h ( x n ))  n ∈ ω . Since ϕ (id M ) = (0 , 0 , . . . ) a nd ϕ ( f ) = (1 , 1 , . . . ), it follows that U ≡ ϕ − 1 ( c 0 ) is a clop en neigh b orho o d o f id M with f 6∈ U . (2) Given a co mpact subset K ⊂ H c ( M ) we shall show that s upp( K ) = cl M  S h ∈K supp( h )  is compact. Assume conversely that the set s upp( K ) is not co mpa ct. The s ame argument as in (i) yields a contin uous function ξ : M → [0 , ∞ ) whos e res triction ξ | supp( K ) is unbounded. By induction we can choo se sequences of p oints x n ∈ supp( K ) and of homeomor phis ms h n ∈ K ( n ∈ ω ) such that h n ( x n ) 6 = x n , ξ ( x n ) > n and ξ ( x n ) > max ξ  S i

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