We improved a version of Global Homeomorphism Theorem due to Katriel such that it can be applied to more general geometric objects: Hilbert Riemannian manifolds, graphs, and polyhedra.Mathematics, AppliedMathematicsSCI(E)0ARTICLE97-10610

Chang, Kung-Ching

Liu, Jia-Quan

Rendiconti del Seminario Matematico della Università di Padova

Numérisation de Documents Anciens Mathématiques

REND. SEM. MAT. UNIV. PADOVA, Vol. 108 (2002)Global Homeomorphism Theoremfor Manifolds and Polyhedra (*).KUNG-CHING CHANG (**) - JIA-QUAN LIU (**)ABSTRACT - We improved a version of Global Homeomorphism Theorem due toKatriel such that it can be applied to more general geometric objects: HilbertRiemannian manifolds, graphs, and polyhedra.1. Introduction.Hadamard’s Global Homeomorphism Theorem (GHT) is concernedwith a differential mapping F between Banach spaces X and Y , in which,the speed of decay of the function VF 8 (x)21 V21 plays an important role.In fact, he (1906) only studied finite dimensional case. Extensions weremade by many authors, e.g., P. Levy (1920), Caccioppoli (1932), Banach-Mazur (1934), etc. It would be natural to ask whether this may be ex-tended to other geometric objects, or the differentiability of the map Fmay be removed. We should mention the pioneer works of F. Browder[Br], Prodi-Ambrosetti [PA], R. Plastock [Pl], and A. D. Ioffe [Io]. In[Ka], Katriel generalized the result to certain continuous mappings be-tween metric spaces. The main point in his approach is to use the surjec-tion constant due to Ioffe in [Io]. However, in [Ka], the metric space Y isassumed to be «nice». Although this notion includes Banach spaces, andthe unit sphere of a Hilbert space as special examples, it is very restric-tive. The purpose of this paper is to improve the version of GHT in [Ka],so that it can be applied to more general geometric objects. Following(*) Supported by NSFC, MCME.(**) Indirizzo degli AA.: LMAM, School of Mathematical Sciences, PekingUniversity, Beijing, 100871, China.Kung-Ching Chang - Jia-Quan Liu98Browder and Plastock, our idea returns to show that F is a covering map.The crucial point is to verify a kind of weak properness of F . The new in-gredient in the verification relies on a recent deformation lemma for con-tinuous functions on metric spaces due to Corvellec [Co]. Accordingly,the restriction on Y is considerably reduced. Not only all Hilbert-Rie-mannian manifolds but also infinite graphs as well as the abstract poly-hedra are included.Before going to state our main result, we introduce the necessary no-tations and terminologies. Let (X , r) be a metric space, g : XKR 1 be acontinuous function. (xX , (dD0, a continuous mapH : Bd (x)3 [0 , d) KX is called admissible, if it satisfiesr(H(y , t), y) G t , ((y , t) Bd (x)3 [0 , d).One (cf. J. N. Corvellec, M. DeGiovanni, and M. Marzocchi [CDM])defines the weak slope of g at x to beNdgN(x) 4 sup ]s [0 , 1Q)N )dD0, )H : Bd (x)3 [0 , d) KX , s.t.g(H(y , t) ) Gg(y)2st ((y , t) Bd (x)3 [0 , d)(,where H is admissible.A point x0 is called critical if NdgN(x0 ) 40, otherwise, it is called regu-lar. For a map F : XKY between two metric spaces, we setSur (F , x)(t) 4 sup ]rF0NBr (F(x) ) %F(Bt (x) )( (xX , (tD0,and setsur (F , x) 4 lim inftK101tSur (F , x)(t), (xX .The later is called the surjection constant of F at x , see [Io]. Our mainresults, which includes the versions of GHT for manifolds and polyhedraas special cases reads as follow:THEOREM 1.1. Let (X , r), (Y , d) be two complete and path connect-ed metric spaces. Assume(H1) (y0Y , there is a neighborhood U4U(y0 ), there is a continuousfunction g4gy0 : UKR1 , and there is a constant bD0 such thatg(y0 ) 40, g(y) D0, NdgN(y) Db , (yU0]y0(.And assume that F : XKY is a local homeomorphism satisfyingGlobal homeomorphism theorem for manifolds and polyhedra 99(H2) (yY , ) a neighborhood U4U(y) and ) a constant kD0 suchthatsur (F , x) Fk , (xF 21 (U) .Then F is a covering map. Furthermore, if p 1 (Y) %F * p 1 (X), then F is aglobal homeomorphism.Let us recall the weak properness of a map F between path connect-ed topological spaces X and Y:(H3 )( path s : [0 , 1 ] KY with s (0) 4y0 , (x0F 21 (y0 ), and ( half openpath g : [0 , d) with dG1, satisfying F(g(t) ) 4s (t), (t [0 , d), the limitlimtKd20g(t) exists.It is known ([D] or [Ch]) that if a local homeomorphism F is weaklyproper, then it possesses the Uniqueness Path-Lifting Property. Accord-ing to F. E. Browder [Br] and R. Plastock [Pl], a map between path con-nected, locally path-connected and locally simply connected topologicalspaces is a covering map if and only if it is a local homeomorphism andpossesses the Uniqueness Path-Lifting Property.The following deformation lemma is due to J. N. Corvellec [Co]: LetX be a metric space, f : XKR 1 be continuous, and let aEc , and sD0 beconstants. Assume that f 21 [a , c] is complete and that NdfN(x) Ds , (x f 21 ( (a , c] ). Then there is a deformation h : [0 , 1 ]3 fcK fc , where fc44]xXNf (x) Gc(, such thatf (h(t , x) ) G f (x), x fa ¨ h(t , x) 4x , and h(1 , fc ) % fa .i.e., fa is a strong deformation retract of fc . More precisely, in fact, by thesame proof, there is a continuous function t on fc such that 0 E t(x) GGs21 (f (x)2a),(x f 21 (a , c], there exists a pathg : [0 , t(x) ] K f 21 [a , c]such that g(0) 4x , g(t(x) )  f 21 (a), and r(g(t), x) G t , (t [0 , t(x) ].PROOF OF THEOREM 1. We are going to verify:(1) (y0Y , there exists a neighborhood V4V(y0 ) such that V is path-connected and simply connected.(2) the map F is weakly proper.Step 1. (The choice of V). Since there are U , g : UKR 1 and bD0 sat-isfying (H1 ), without loss of generality, we may assume that U is closed.(cD0, define V4]yUNg(y) Gc(. V is a closed neighborhood of y0 , sois complete. Now we apply the Deformation Lemma due to Corvellec, VKung-Ching Chang - Jia-Quan Liu100is contractible and path connected. Combining with the fact that F is alocal homeomorphism, X is also locally path-connected and locally simplyconnected.Step 2. (The weak properness of F) First we needLEMMA 1. Let V be defined above. Then under the assumptions ofTheorem 1 there are a continuous t : F 21 (V) KR 1 , and a patha : [0 , t(x) ] KF 21 (V) such that a(0) 4x , F(a(t(x) ) ) 4y0 , 0 G t(x) GG 1kbg(F(x) ), and r(a(t), x) G t , (xF 21 (V).PROOF. Define f4g i F : F 21 (V) KR 1 . Again f is continuous, andF 21 (V) is complete. Provided by a result due to Ioffe [Io], we haveNdfN(x) FNdgN(y)3sur (F , x) Fbk»4aD0,(xF 21 (V0]y0() 4 f 21 (0 , c] .Now, we apply the Deformation Lemma due to Corvellec, there exista continuous function t : fcKR 1 satisfying 0 E t(x) Ga21 f (x), and apath s : [0 , t(x) ] K fc such that s (0) 4x , and s (t(x) )  f 21 (0). Noticingthat fc4F 21 (V), and f04F 21 (y0 ), the lemma is proved.LEMMA 2. Under the assumptions of Theorem 1, if 0 EdG1 andg : [0 , d) KX is a path such that limtKd20F(g(t) ) 4y0Y , then the limitlimtKd20g(t) exists in X.PROOF. Suppose that g(t) has no limit as tKd20. Then there existeD0 and tnE tn8Ed with tnKd20, such that r(g(tn ), g(tn8 ) ) Fe , andF(g(tn ) ), F(g(tn8 ) ) Ky0 . Therefore we may assume xn4g(tn ), and x 8n 44g(tn8 ) F 21 (V), where V4V(y0 ) is defined in step 1. Let g and f4g i Fbe defined as above w.r.t y0 , one has f (xn ) K0. For large n , we may as-sume f (xn ) E13ae . According to Lemma 1, there is a paths : [0 , t(xn ) ] KF 21 (V), with s (0) 4xn , F(s (t(xn ) ) ) 4y0 andr(s (t), xn ) G t , where 0 E t(xn ) Ga21 f (xn ) E13e .Denoting z4s (t(xn ) ), we have r(z , xn ) G13e .Similarly, we have a path s 8 : [0 , t(xn8 ) ] KF 21 (V), satisfyings 8 (0) 4x 8n , F(s 8 (t(x 8n ) ) ) 4y0and r(z 8 , x 8n ) G13e , where z 84s 8 (t(x 8n ) ).Global homeomorphism theorem for manifolds and polyhedra 101Thus zcz 8 , both z , z 8F 21 (y0 ).Now, we construct a path h4s 8 i g i s*, where s*(t) 4s (12 t); i.e.,h is a path starting from z , goes along s* to xn , then along g to x 8n , andthen along s 8 to z 8 .We have h : [0 , 1 ] KF 21 (V) with h(0) 4z and h(1) 4z 8 . Again, wewrite f4g i F on F 21 (V).Applying the Deformation Lemma due to Corvellec, we obtain a pathj(t) 4z(1 , h(t) ) % f 21 (0) 4F 21 (y0 ), (t [0 , 1 ], where z : [0 , 1 ]3 fcK fcis the deformation satisfying z(1 , fc ) % f0 .This contradicts with the assumption that F is a local homeomor-phism. The lemma is proved.As to the last assertion, it is a well known fact in elementary topolo-gy, which follows directly from the Uniqueness Path Lifting propertyand the Homotopy Path Lifting Theorem.THEOREM 2. Suppose that X and Y are complete, path-connectedmetric spaces, and that F : XKY is a local homeomorphism. Assume(H1) and (H2). If X is path-connected after the removal of any discretesubset, and if there exists a continuous function h : YKR 1, satisfyingthe (PS) condition, and possessing a unique minimizer and a discreteset of maximizers as the only critical points, Then F is a globalhomeomorphism.PROOF OF THEOREM 2. Let Y0 be the set of maximizers of h, andY14Y0Y0 .According to the critical point theory for continuous functionals, Y1 iscontractible, because the only critical point of h1 »4hNY1 is the uniqueminimizer. Let X04F 21 (Y0 ) and X14X0X0 , then X0 is a discrete subsetof X, and then X1 is again path-connected. Provided by Theorem 1, F is acovering map, so is F1 »4FNX1 . However, F1 (X1 ) %Y1 , the later is con-tractible, F1 is a global homeomorphism, according to Theorem 1. Thisimplies that (zY1 , JF 21 (z) 41. Therefore, JF 21 (y) 41 (yY . Thisproves that F is a global homeomorphism.REMARK 1. Comparing Theorem 2 with Theorem 6.1 in [Ka], theonly difference is that in the definition of «nice» space, all functions gy0defined in (H1) are assumed to have the same property as h in Theorem2, (y0Y; while in our case only one such h is needed.We present here few examples:Kung-Ching Chang - Jia-Quan Liu102EXAMPLE 1 (Hilbert Riemannian manifold). Let (M , g) be aHilbert Riemannian manifold, modelled on a Hilbert space E with Rie-mannian metric g . It is known that the topology derived by the distance,which is defined by the length of the geodesics between two points, coin-cides with the given topology on M. With this distance d, M is a metricspace (M , d).Claim: (M , d) satisfies (H1 ).Indeed, (pM , let V be the domain of normal coordinates around p ,and let exp be the exponential map. Br (u p ), Br (p) are the r-balls withcenters at u pTp (M), the tangent space at p , and at pM respectively.We assume that Br (p) %V . Now, (y0Br/2 (p)0]p(, we set d44 12min ]d(y0 , p), r( and U4Bd (y0 ). Thus d(y , p) Gr , (yU , and thenj4expp21 (y) is well defined, and d(y , p) 4VjV , where V i V4V i Vp denotesthe norm induced by g on Tp (M). Let j 0 denotes the unit vector of j .Define H(y , t) 4expp (j2 tj 0 ) and h(y) 4Vexpp21 (y)V ((y , t) U33 [0 , d). Since,tEdGd(y0 , p)2d(y , y0 ) Gd(y , p) 4VjV ,H(y , t) Br (p) is well defined. It is easy to verify: d(H(y , t), y) 4 t , andh(H(y , t) ) 4VjV2 t4h(y)2 t . Therefore, (H1 ) is verified.COROLLARY 1. Suppose that (M , g) and (N , h) are Hilbert-Rie-mannian manifolds, and that F : MKN is a local homeomorphismsatisfying the following condition: there exists kD0, (pM , thereexists r4r(p) D0, such thatkr(q , p) Gd(F(q), F(p) ) (qBr (p),In particular, if F is a C 1 mapping, VdF(p)21 VFk (pM. Then F isa covering map. Furthermore if p 1 (N) %F * p 1 (M), then F is a globalhomeomorphism.PROOF OF COROLLARY 1. We only need to prove the C 1 case. Let rand d be the induced distance on M and N resp. Then (M , r), (N , d) arecomplete, path connected, locally path-connected, locally simply con-nected metric spaces, satisfying (H1 ). According to the Implicit FunctionTheorem, F is a local homeomorphism. It remains to verify:sur (F , p) Fk (pMIndeed, provided by the assumption on the boundedness of VdF 21 V ,Global homeomorphism theorem for manifolds and polyhedra 103there are a neighborhood U of p and a constant mFk , such thatBmr (u F(q) ) %dF(q)(Br (u q ) ), (qU .There exists dD0, such that if tEd, then Btk (F(p) ) %F(U)OV,where V is the domain of normal coordinates around F(p) in N.Therefore, (yBtk (F(p) ), h4expF(p)21 (y) TF(p) (N) is well defined.Setting s (s) 4expF(p) (sh), l(s) 4F 21 (s (s) ), (s [0 , 1 ], and x44F 21 (y), then l(0) 4p , l(1) 4x , and l 8 (0) 4 (dF(p) )21 hTp (M).We haver(x , p) GGGG01Vl 8 (s)Vl(s) ds01V(dF)21 (l(s) ) s 8 (s)Vl(s) ds1kd(y , F(p) )t .This proves the Corollary.REMARK 2. The conclusion in this Corollary could be extended tothe case of Finsler-Banach manifold, provided we are able to define theexponential map on such a manifold.LEMMA 3. Suppose that (Y , d) is a metric space. If (y0Y thereexist a neighborhood U4U(y0 ), a norm space (E , V .V), and a localhomeomorphism F : UKE such that(1) d(u , v) GVF(u)2F(v)V (u , vU .(2) F(y0 ) 4u and F(U) is a star-shaped with respect to u.Then T satisfies (H1).PROOF OF LEMMA 3. Now, we define g(y) 4VF(y)V and choose d (0 , VF(u)V), with Bd (y0 ) %U , and then defined on Bd (y0 )3 (0 , d) themapping:H(u , t) 4F21g12 tVF(u)Vh F(u)if uU0]y0(, and H(y0 , t) 4y0 .The verification is trivial.Kung-Ching Chang - Jia-Quan Liu104EXAMPLE 2. Let X and Y be subsets in Banach spaces E and F ,resp. in which each point has a star-shaped neighborhood, and let r andd be the induced metrics respectively.COROLLARY 2. Suppose that F : (X , r) K (Y , d) is a local homeo-morphism that d(F(x), F(y) ) Fkr(x , y), where k is a positive constant.Then F is a covering map.PROOF OF COROLLARY 2. It follows directly from Theorem 1 andLemma 3.One may consider other geometric objects.EXAMPLE 3 (Infinite graph). Let E be a Banach space with normV .V. Given a set of isolated points A4]a1 , a2 , R(, and a set of closedsegments connecting some pairs of these points L4]l1 , l2 , R(.If the intersection of any two segments is either empty or a point inA, then we denote them by a triple ]A , L , E(. The triple determines ametric space (Y , d), in whichY4 0i41Qliendowed with the reduced distance from E. It is called an infinitegraph.COROLLARY 3. Suppose that (X , r) and (Y , d) are two infinitegraphs determined by the triples ]A , L , E( and ]B , M , F( respect-ively. Assume that F : XKY is a mapping such that F maps (A , L) to(B , M), which is a local bijection, and is linear on each lL. If thereare positive constants m and k(a) such thatmGNF(l)NNlNGk(a) ,if a l , (lL , (aA , where N i N is the length of a segment. Then F is acovering map. Furthermore if every loop in Y is the image of a loop inX, then F is a global homeomorphism. In particular, if Y is a tree, thenF is a global homeomorphism.Global homeomorphism theorem for manifolds and polyhedra 105PROOF OF COROLLARY 3. The verification of (H1 ) is divided into twocases: (1) y0 int (l) for some lL , (2) y0A . In the former, it is a specialcase of Corollary 2. In the later, there is dD0 such that AOBd (y0 ) 4y0 .Define gy0 (y) 4Vy2y0 V , and H(y , t) 4y2tVy2y0 V(y2y0 ) (yV4YOOBd (y0 ).EXAMPLE 4 (Abstract Polyhedra). Let E be a Banach space withnorm V .V. Given a set of isolated points A%E . For any index set I44]i0 , i1 , R( finite or infinite, letD I4Cl]conv (ai0 , ai1 , R)(be a geometric simplex in E. A geometric complex G in E is defined to bea set of index sets satisfying:(1) If I 8%I , and IG; then I 8G .(2)D IOJ4D I1D J (I , JGThe subsetK4 0IGD Iendowed with the distance induced from E is called an infinitepolyhedron.Let (Z , d) be a metric space, and let t : KKZ be a surjectivesatisfyingC1 Vj2hVGd(t(j), t(h) ) GC2 Vj2hV , (j , hK .for some positive constants C1 , C2 . Let Y4t(K) endowed with the in-duced metric d, then (Y , d) is a metric space. We call it an abstractpolyhedron.Let (X , r), (Y , d) be two abstract polyhedra determined by]A , G , E , t(, and ]B , L , F , h(, respectively. F : XKY is called bi-Lip-schitzian, ifF : t(D I ) Kh(D J ), is a surjective, (IG , and for some J4J(I) L ,and is bi-Lipschitzian in each t(D I ), IG . i.e.,C1 (I) r(x , x 8 ) Gd(F(x), F(x 8 ) ) GC2 (I) r(x , x 8 ), (x , x 8t(D I ),where C1 (I), C2 (I) are positive constants depending on I.Kung-Ching Chang - Jia-Quan Liu106COROLLARY 4. Suppose that (X , r) and (Y , d) are two completepath connected polyhedra. If F : XKY is a bi-Lipschitzian map, andalso a local surjection, and if there is a positive constant k such thatC1 (I) Fk (IGThen F is a covering map. Moreover, let L and M be the graphs (1-skele-tons) determined by G and L respectively. If any loop in M is the imageof a loop in L; then F is a global homeomorphism.The proof is similar to that of Corollary 3 and that of Lemma 3.REMARK 3. An index set IG is called maximal, if (JG , JOIc¯ ,implies J%I . If we assume that the X is such an abstract polyhe-dron:(xX , t21 (x) is in at most finite many D I , where I is maximal in G .Then the assumption on the local homeomorphism of F in Corollary 4can be replaced by the assumption that the map: IKJ4J(I), is one toone.R E F E R E N C E S[Br] F. E. BROWDER, Covering spaces, Fibre Spaces and Local Homeomor-phisms, Duke Math. J., 21 (1954), pp. 329-336.[Ch] W. Y. CHEN, Nonlinear Functional Analysis, (in Chinese) Gansu Peo-ple’s Press (1982).[Co] J. N. CORVELLEC, On the Second Deformation Lemma, TMNA, 17(2001), pp. 55-66.[CDM] J. N. CORVELLEC - M. DEGIOVANNI - M. MARZOCCHI, Deformation prop-erties for continuous functionals and critical point theory, TMNA, 1(1993) pp. 151-171.[De] K. DEIMLING, Nonlinear Functional Analysis, Springer Verlag(1985).[Io] A. D. IOFFE, Global Surjection and Global Inverse Mapping Theorem inBanach Spaces, Ann. New York Acad. Sci., 491 (1987) pp. 181-188.[Ka] G. KATRIEL, Mountain Pass Theorems and Global HomeomorphismTheorems, Ann. Inst. H. Poincaré, Anal. Nonlinéaire, 11 (1994), pp. 189-210.[Pl] R. PLASTOCK, Homeomorphisms Between Banach Spaces, TAMS, 200(1974), pp. 169-183.[PA] G. PRODI - A. AMBROSETTI, Analisi Non Lineare, I Quaderno Scuola nor-male di Pisa (1973).Manoscritto pervenuto in redazione il 26 giugno 2002.

Global homeomorphism theorem for manifolds and polyhedra

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