Foliations and Global Inversion

Transcription

1 Foiations and Goba Inversion E. Cabra Bareira Department of Mathematics Trinity University San Antonio, TX January 2008 Abstract We consider topoogica conditions under which a ocay invertibe map admits a goba inverse. Our main theorem states that a oca diffeomorphism f : M R n is bijective if and ony if H n 1 (M) = 0 and the pre-image of every affine hyperpane is non-empty and acycic. The proof is based on some geometric constructions invoving foiations and toos from intersection theory. This topoogica resut generaizes in finite dimensions the cassica anaytic theorem of Hadamard-Pastock, incuding its recent improvement by Noet-Xavier. The main theorem aso reates to a conjecture of the aforementioned authors, invoving the we known Jacobian Conjecture in agebraic geometry. 1 Introduction In this paper we are concerned with the probem of finding topoogica conditions ensuring that a oca diffeomorphism is bijective. A cassica resut in this direction is the we-known Hadamard-Pastock Theorem (see [4] and [10]). It states that a Banach space oca diffeomorphism f : X X is bijective provided inf x X Df(x) 1 1 > 0. (1.1) 1

2 The proof of the Hadamard-Pastock theorem foows from simpe arguments invoving covering spaces. In recent years new topoogica and geometric ideas have been introduced in the subject of goba invertibiity, pushing the fied in different directions (see, for instance, [5], [6], [7], [8], [11], [14], [17], and [18]). The emerging picture reveas that goba invertibiity is aso infuenced by more subte topoogica phenomena. In [7], Noet and Xavier estabished a substantia improvement to the Hadamard-Pastock theorem when dim X <. Using degree theory, they showed in [7] that a oca diffeomorphism f : R n R n is bijective if there exists a compete Riemannian metric g on R n such that, v S n 1, inf x R n Df(x) v g > 0. (1.2) Notice that (1.2) is an improvement over (1.1) since Df(x) 1 1 = Df(x) 1 1 = inf v =1 Df(x) v. Furthermore, it is easy to produce exampes that satisfy (1.2) but not (1.1). Arguments from eementary Morse theory (see [9, p.112]) show that if (1.2) hods, then the pre-images of affine hyperpanes H must satisfy f 1 (H) R = R n (note that Df(x) v = f(x), v ). In particuar, by the Künneth formua, f 1 (H) is acycic (reca that a topoogica space is caed acycic if it has the homoogy of a point). In this paper we show that the above mentioned anaytica resuts are but a manifestation of a topoogica phenomenon. Theorem 1.1. A oca diffeomorphism f : R n R n is bijective if and ony if the pre-image of every affine hyperpane is non-empty and acycic. In Section 3 we wi point out a connection between the above theorem and the Jacobian Conjecture in agebraic geometry. The non-trivia haf of Theorem 1.1 consists in estabishing injectivity and surjectivity. Its proof is based on some geometric constructions invoving foiations, and the computation of intersection numbers of certain chain compexes. Theorem 1.1 aso aows for an anaytic coroary that is stronger than the resuts in [7], in the sense that one can choose the metric to suit the unit vector v. 2

3 2 Preiminaries Given a compact smooth manifod M n and a finite cover, we woud ike to have a systematic way to describe the intersections of the sets in the cover. Likewise, once a point is given we want to describe exacty a the sets in the cover that contain the given point. To this end, we wi consider a trianguation of M and view the top dimensiona ces as the sets of the covering. We set our notation as foows. Denote by T(M) a trianguation on M (whose existence is guaranteed by [16]) and et e(k) j be the j th k-ce of T(M). The set of indexes of k-ces wi be denoted by E(k) N. Aso, given a trianguation T(M), et T k (M) be the k-skeeton of M. Whenever the context is cear, we wi refer to the trianguated space simpy as M. This combinatoria approach aow us to easiy address the properties we mentioned above. For instance, given a simpex e(k) j, the star of e(k) j describes a the simpexes that contain e(k) j. In our resuts, we wi be interested in finding a the (k + 1)-simpexes that contain e(k) j. This is easiy accompished by ooking at the vertices of the ink of e(k) j, denoted by Lk (e(k) j ). We now review the basic definitions from intersection theory. We define in M n the intersection number (mod 2) between A p a p-cyce and B q a q-cyce, where p + q = n by #(A p, B q ). We note that when A p, B q represent transverse submanifods, then #(A k, B n k ) represents the number of geometric intersections mod 2. The property that we highight is that intersection number depends ony on the homoogy cass. For detais and forma definitions we refer the reader to [13]. Finay, we can aso define inking numbers between cyces. Let X p and Y q 1 be two nonintersecting cyces in R n with p+q = n. For Z p+1 a bounding chain of X p, i.e. Z p+1 = X p, we define the inking number between X p and Y q 1 as Lk(X p, Y q 1 ) = #(Z p+1, Y q 1 ), (2.1) which is independent of the choice of the bounding chain of X p. 3

4 3 Injectivity Let us consider a oca diffeomorphism f : M R n, where M is a smooth connected manifod. Our goa is to understand under which topoogica conditions the map f is injective. There is a conceptua ink between injectivity and connectedness. For instance, it is cear that a ocay invertibe map is injective if and ony if the pre-image of every 0-dimensiona affine subspace (i.e., a point) is connected (possiby empty). An anaogous statement can be made if one goes one dimension higher and considers ines instead of points, that is, a ocay invertibe map is injective if the pre-image of every ine is connected. In view of these observations, Noet and Xavier [7] made the foowing conjecture. Conjecture 3.1. A oca diffeomorphism f : R n R n is injective if the pre-images of every affine hyperpane is connected (possiby empty). At the present time this conjecture remains open and its significance is better seen in Agebraic Geometry where it woud provide a positive answer for the Jacobian Conjecture (reca that the Jacobian Conjecture states that a poynomia oca bihoomorphism F : C n C n is invertibe, see [2], [15]). Indeed, if F : C n C n is a poynomia oca bihoomorphism, and H C n is a rea hypersurface foiated by compex hyperpanes V, then by a Bertini type theorem F 1 (V ) is connected for a generic V (see [12], Cor. 1 of Theorem 3.7). From this one can easiy check that F 1 (H) is connected and hence one woud estabish the Jacobian Conjecture. The resut beow estabishes a weaker version of the Noet-Xavier conjecture, where connectedness is repaced by acycicity. Theorem 3.2. A oca diffeomorphism f : R n R n is injective if the preimage of every affine hyperpane is either empty or acycic. In fact, we observe that we may weaken the hypotheses of Theorem 3.2 to obtain the foowing stronger resut. We say that an affine hyperpane H R n is parae to a ine in R n provided that H = or H. 4

5 Theorem 3.3. For n 3, et f : M R n be a oca diffeomorphism where M is a (necessariy non-compact) connected manifod with H n 1 (M) = 0. If there exists a ine in R n such that the pre-image of every affine hyperpane parae to is either empty or acycic, then f is injective. The proof of Theorem 3.3 is based on geometric constructions of chain compexes, the computation of the intersection number between these objects, and the maxima ift of ines. Since the computation of intersection numbers is done with objects beonging to the domain of f, we need to require the extra assumption on the homoogy of M. Observe that the cases in Theorem 3.3 when n = 1, 2 are triviay true without any extra assumptions on M. We stress that in our arguments we wi ony require the existence of oca ifts. In fact, by Hadamard-Pastock Theorem [10], if a ines admit goba ifts the map is aready bijective. We refer to a oca ift of a ine = {tw w R n, t R} with respect to f as a path α : ( ε, ε) M, ε > 0 such that f (α(t)) = tw. Observe that by the Inverse function theorem, if f is a diffeomorphism a oca ift of a ine aways exists in the above sense. We say that α : ( δ, δ) M is the maxima ift of if δ = sup{ε α admits a oca ift for ε > 0}. Furthermore, has a goba ift if its maxima ift satisfies δ =. Finay, what is important for us is the fact that the maxima ift of a ine is propery embedded in the domain. In our notation, α is propery embedded if it eaves every compact set of M as t increases to δ. 3.1 Beginning of the proof of Theorem 3.3 Assume that there is a point p in the image of f with at east two distinct points q 0 and q 1 in its pre-image. Since transations do not change any of the hypotheses, we assume for simpicity that p = 0. Our goa is to construct a (n 1)-cyce Γ n 1 so that the intersection number of Γ n 1 with the maxima ift of the ine passing through the origin wi necessariy be zero, as the maxima ift is propery embedded. A simpe argument wi then show that f must have a critica point aong the ift, thus estabishing the desired contradiction. 5

6 First, we give an outine of the proof. Consider ε > 0 so that the ba V = B(0; ε) f(m) has diffeomorphic pre-images U 0 and U 1 around q 0 and q 1, respectivey. Next, et Y be the (n 1)-equatoria disk of V determined by, that is, the intersection of the orthogona hyperpane to and V. The cyce Γ n 1 we seek wi be constructed to resembe a topoogica cyinder that connects the induced equatoria disks of U 0 to U 1, denoted by X 0 and X 1, respectivey. We construct Γ n 1 as foows. Take a hyperpane parae to which intersects V tangentiay at v and is denoted by H v. As we change the hyperpane H v by moving it around V, the pre-images u i U i of v, for i = 0, 1, can be continuousy connected by paths in f 1 (H v ), at east for nearby hyperpanes. In this way we construct sma atera pieces of Γ n 1. One then tries to put together a those oca data. In so doing, one is forced to consider the situation where, for a fixed hyperpane H v, there are mutipy-defined paths joining the same pre-images of points in f 1 (H v ). Whenever this occurs, the topoogica hypotheses that f 1 (H v ) is acycic wi be used to fi in the gaps. See Fig. 3.1 for a depiction of this process when n = 3. X 1 q 1 u 1 f 1 (H v ) W n 1 Y p v q 0 X0 u 0 H v Figure 3.1: Construction of paths connecting X 0 to X 1 by the revoution of affine hyperpanes. 6

7 In order to determine how the atera pieces wi fit together and how such gaps shoud be fied, we consider a combinatoria decomposition of Y in terms of a trianguation. Here we observe that Y = S n 2, so such trianguation aways exist. The process of putting together the pieces of Γ n 1 wi be done in steps according to the dimension of the carrier of each point. More precisey, first we consider a point and the (n 2)-ces it may possiby beong and construct chain compexes that correspond to the atera pieces indicated above. Next, points that beong to the ower dimensiona skeeton of Y wi be consider more than once in the initia step. Hence, in the foowing step we consider the (n 3)-skeeton of Y and determine the bounding chains according to the higher dimensiona ces that contain it. We repeat this process unti we consider the 0-skeeton of Y. The existence and properties of Γ n 1 are estabished in the foowing emma. Lemma 3.4. There exists a geometric (singuar) chain compex Γ n 1 S n 1 (M) that may be represented as Γ n 1 = X 0 + W n 1 + X 1 such that W n 1 is a chain compex with W = X 0 + X 1 and for a q supp W n 1 (in its image in M), there exists v Y so that q f 1 (H v ). The proof of Lemma 3.4 foows the outine above where we wi construct a the singuar chain compexes of W n 1 and the attaching maps. This argument uses ideas from combinatoria topoogy and we postpone it unti next section. We proceed to estabish Theorem 3.3, but first we remark that we are interested in the existence of a geometric intersection (i.e., number of points in the set theoretica intersection) between Γ n 1 and the maxima ift of. Therefore we consider intersection numbers and homoogy with Z 2 coefficients, thus avoiding heavier notationa concerns regarding orientation and eaving the proof simper and more geometric. Assuming Γ n 1 is constructed as in Lemma 3.4, we compute the intersection number of Γ n 1 and the maxima ift of starting at q 0 which we denote by γ. We caim that γ must intersect Γ n 1 in another point besides q 0 and we wi show that it is q 1. First, we see that γ is propery embedded in M. Indeed, if we decompose γ as γ γ + as the maxima ift of in the negative and positive direction, respectivey, starting at q 0. It is then cear that γ and γ + are not entirey contained in any compact subset of M, otherwise the ift woud not be maxima. 7

8 Now, the fact that Γ n 1 S n 1 (M), i.e., a cyce and H n 1 (M) = 0 impies that there exists a bounding singuar chain Σ n, with Σ n = Γ n 1 and a compact set K so that Σ n K. Thus Γ n 1 is a representative of the trivia eement in H n 1 (M, M K). We aso have that γ H 1 (M, M K) and from the fact that intersection numbers depend ony on the homoogy cass, we have #(Γ n 1, γ) = 0. (3.1) Indeed, H n 1 (M, M K) = H n 1 (M/M K) and since Σ n K, we have Γ n 1 0 in H n 1 (M/M K) as we. From Lemma 3.4 and by definition of intersection numbers, we can write (3.1) as: 0 = #(Γ n 1, γ) = #(X 0, γ) + #(W n 1, γ) + #(X 1, γ) = #(X 1, γ), (3.2) where the first term is 1 since f is a oca diffeomorphism and the images of γ and X i are orthogona and the second term is zero since γ W n 1 =. Therefore, it must be that #(X 1, γ) = 1. In particuar, γ X 1 and by the choice of ε, it must be that γ X 1 = {q 1 }. For a geometric depiction see Fig Finay et α f 1 () be the path segment from q 0 to q 1. The image of α is a oop in that has a point p f(m) that is furthest from p. Now it is cear that f fais to be ocay invertibe at the corresponding pre-image of p, giving us the desired contradiction. Therefore f must be injective. 3.2 Reassembage of Hyperpanes and a Chain Compex Construction We now estabish Lemma 3.4, needed to compete the proof of Theorem 3.3. Whie outining the construction of Γ n 1 earier, we encountered a key probem which simpy put is attributed to the ack of uniqueness on the choice of the path used to connect the pre-images of a point in Y. In our construction this is refected as foows: athough each path may be defined continuousy within a neighborhood of a fixed point, as we consider the intersection of 8

9 U 1 q 1 Γ n 1 f 1 () V p q 0 U 0 Figure 3.2: Construction of a cosed chain compex Γ n 1 by revoving affine hyperpanes. two neighborhoods there wi possiby be two choices of paths. We caim that whenever ambiguity occurs, we may use the hypotheses of acycicity of the pre-images of hyperpanes to define chain compexes to circumvent this probem. We do this by considering a trianguation of Y with sufficienty sma mesh to be determined during the proof. Heuristicay, we view a neighborhood of a generic point as the top dimensiona ce containing it and the trianguation wi provide a way to keep track of the intersection of the mutipe neighborhoods. Let e(k) j be the ces of such trianguation, where k = 0,...,n 2 denotes the dimension of each ce and j E(k) N is the indexing set of the k-ces. From the initia choice of ε > 0, we may aso define an induced trianguation via the oca diffeomorphism on X 0 and X 1 with ces e(k) 0 j and e(k)1 j, respectivey. We construct W n 1 in n 1 steps which we enumerate from 0 to n 2. In step k we consider points in the (n 2 k)-skeeton of Y, denoted by Y (n 2 k), and show that the possiby mutipy defined chains are obtained by ooking at a the higher dimensiona ces containing such points and that 9

10 these chains give rise to a cyce. Then by using the acycicity hypotheses, we have that such cyce can be reaized as the boundary of another chain compex which wi be the buiding bocks of W n 1. Step 0: The initia process is anaogous to what has been outined before, but to estabish our notation we provide the forma argument. Given the initia trianguation of Y, take v e(n 2) and et u i be the pre-image of v in X i for i = 0, 1. From the connectedness hypotheses of f 1 (H v ), there is a path W 1(v) f 1 (H v ) joining u 0 to u 1, that is, W 1 (v) is a 1-chain with W 1(v) = u 0 + u 1. Next, we can continuousy modify W 1 (v) for a points in a neighborhood of v Y. This foows because W 1 (v) is compact and f is a oca diffeomorphism. By repeating this construction for every point in Y, we obtain a cover of Y from which we extract a finite subcover as Y is compact. Then take finitey many barycentric subdivisions of Y unti its mesh is smaer than the minimum diameter of the subcover. Finay we redo the assignment of W 1(v) for each v e(n 2) using the newy obtained trianguation. This has the property that for each (n 2)-ce we may define a (n 1)-chain compex denoted by W 1 e(n 2) from the continuous famiy of paths for each E(n 3). Step 1: In this next step, we consider points in the (n 3)-skeeton of Y as these are the points which we possiby assigned two different 1-chain compexes in the previous step. For v e(n 3), we may identify a the (n 2)-ces that contain e(n 3) by ooking at the vertices of Lk(e(n 3) ). In this case, we have precisey two points as e(n 3) beongs to exacty two top dimensiona ces say, e(n 2) 1 and e(n 2) 2. From the previous step, we constructed two possiby distinct chain compexes W1 1(v) and W 2 1(v) contained f 1 (H v ) joining u 0 to u 1. If it is the case they are aready the same, we are done. Otherwise, consider the 1-chain U 1(v) = W 1 1(v) + W 2 1 (v). We caim U 1(v) is a cyce. Indeed, U1 (v) = W 1 1 (v) + W2 1 (v) = u 0 + u 1 + u 0 + u 1 = 0, since we are using Z 2 -coefficients. From the hypotheses that f 1 (H v ) is acycic, we have that U 1 (v) is the boundary of a 2-chain denoted by W 2(v). Now, using the fact that f is a oca diffeomorphism and W 2 (v) is compact, we can continuousy define W 2 (u) for a u in a neighborhood of v in 10

11 Y (n 3). Note that in this step we are ony considering points in the (n 3)- skeeton. Therefore we obtain a cover of Y (n 3) which by compactness we extract a finite subcover. Next, we iterate finitey many barycentric subdivisions of the trianguation on Y unti its mesh is smaer than the minimum diameter of the subcover. We then redo the construction of the chain compexes up to this point in step 0 and 1 using the new trianguation. We do this so the 2-chain compex defined above can be continuousy assigned for each point within a (n 3)-ce and we obtain a (n 1)-ce denoted by W 2 e(n 3) for each E(n 3). Step k: For a generic step k (1 < k n 2), we consider points in the (n 2 k)-skeeton of Y. For v e(n 2 k), we ook at the (n 1 k)-ces that contain e(n 2 k). This is the case because in the previous step k 1, we have defined k-chains Wi k (v) over v these ces that v beong, for some i. A systematic way to consider these ces is to ook at the vertices of Lk(e(n 2 k) ). Let us assume that those are e(n 1 k) 1, e(n 1 k) 2,...,e(n 1 k) j. We now define U k(v) = W1 k (v) + + Wj k (v) f 1 (H v ) and we caim that U k (v) is a cyce. Indeed, ( j ) j U k (v) = Wi k (v) = Wi k (v) i=1 i=1 j = Ui k 1 (v) = (3.3) W k 1 (v) i=1 where the chains U k 1 i (v) were constructed in the previous step in a simiar manner and corresponds to the index of a (n k)-ces that contains v. The chain U k 1 i (v) is formed by ooking at a the (n k)-ces that contain e(n 1 k) i and hence wi contain e(n 2 k). Therefore, these (n k)-ces can aso be determined by ooking at the edges of Lk(e(n 2 k) ). Observe that for a fixed i, as we ook at the chains of type W k 1 (v) that comprise U k 1 i (v) we can aternativey ook at the coection of edges in Lk(e(n 2 k) ) that make up U k 1 i (v) and the chains W k 1 (v) wi be the vertices of such edges. However, because each edge contains exacty two vertices, as we do this for a i each term in the ast summation in (3.3) appears twice. Since our computation uses Z 2 coefficients, we have (3.3) is zero estabishing that U k (v) is a cyce. 11

12 Again, from the hypotheses that f 1 (H v ) is acycic, we find a bounding (k + 1)-chain W k+1 (v) f 1 (H v ) of U k k+1 (v), that is, W (v) = U k(v). Next, an anaogous argument as in step 1 is used to find a neighborhood of v in Y (n 2 k) where the assignment of W k+1 (v) is continuous for a points within it. This foows from the oca diffeomorphism of f and compactness if W k+1 (v). This induces a cover of Y (n 2 k) and by compactness we extract a finite subcover. Finay, we take finitey many barycentric subdivisions of Y unti the mesh is smaer than the minimum diameter of the subcover. Using the new trianguation, we repeat the assignment of the chain compexes in each of the previous steps 0 through k. In particuar, this defines W k+1 (v) for a points in e(n 2 k) and by the modification argument indicated above, we obtain a (n 1)-chain denoted by W k+1 e(n 2 k) for each E(n 2 k). Once we have competed a the n 1 steps, we put together the singuar compexes constructed from the (n 1)-chains in each step by means of their attaching maps aong their common boundary which wi be expicity computed. In order to finish the proof, we show that W n 1 = X 0 + X 1 and thus once we attach X 0 and X 1 to the boundary we wi have the cyce Γ n 1 we seek. We consider the decomposition of W n 1 from the (n 1)-chains in each step, that is, W n 1 = n 2 k=0 For simpicity, et S k = E(n 2 k) E(n 2 k) W k+1 e(n 2 k). (3.4) W k+1 e(n 2 k), then W n 1 = 12

13 n 2 S k. We now anayze each term separatey. For k = 0; k=0 S 0 = = E(n 2) = E(n 2) E(n 2) W 1 e(n 2) W 1 e(n 2) + E(n 2) (e(0) 0 + e(0)1 ) e(n 2) + = X 0 + X 1 + E(n 2) W 1 e(n 2) E(n 2) j n 3 W 1 e(n 3) j n 3, j n 3 W 1 e(n 3) j n 3 where j n 3 denotes the index of a (n 3)-ces that beong to the boundary of e(n 2). In genera, for 0 < k < n 2; S k = e(n 2 k) = E(n 2 k) = E(n 2 k) E(n 2 k) W k+1 W k+1 e(n 2 k) + E(n 2 k) W k+1 Wj k n 1 k e(n 1 k) + j n 1 k + W k+1 e(n 3 k) jn 3 k, j n 3 k E(n 2 k) e(n 2 k) where j n 1 k denotes the index of a (n 1 k)-ces that contain e(n 2 k) and j n 3 k denotes the index of a (n 3 k)-ces that beong to the boundary of e(n 3 k). Finay, for k = n 2; S n 2 = = E(0) = E(0) E(0) W n 1 e(0) W n 1 e(0) + E(0) W n 2 j 1 e(0) + 0, j 1 13 W n 1 e(0)

14 where j 1 denotes the index of a 1-ces that contain e(0) in its boundary. Observe that the second summation term of S k is the same as the first summation term of S k+1 as we count each chain twice. Since we are using Z 2 -coefficients, (3.4) simpifies to W n 1 = X 0 + X 1. Finay, from the construction of W n 1 we see that for each q supp W n 1, q f 1 (H v ) for some v Y. This concudes the proof of Lemma Surjectivity In this section we consider the question of when a oca diffeomorphism f : M R n is surjective, based on the topoogy of the pre-images of hyperpanes. The trivia exampe of an incusion map of the region between two panes satisfies Theorem 3.3 but it is not surjective. This indicates that further assumptions must be added. On the other hand, we are abe to eiminate the homoogica assumption on the domain. Theorem 4.1. Let f : M R n be a oca diffeomorphism where M is a connected manifod. If the pre-image of every affine hyperpane is non-empty and acycic, then f is surjective. The proof is based on geometric constructions invoving foiation theory and the computation of inking numbers between certain singuar chain compexes in the range R n. We remark that since the computation of inking numbers wi occur in R n, it is not necessary to make any further assumptions on the homoogy groups of M. This is unike the situation in Theorem 3.3, where we assumed H n 1 (M) = 0. Combining Theorem 4.1 and Theorem 3.3, we obtain the foowing characterization of R n, for n 2. Theorem 4.2. A smooth connected manifod M is diffeomorphic to R n if and ony if H n 1 (M) = 0 and there exists a oca diffeomorphism f : M R n such that the pre-image of every affine hyperpane is non-empty and acycic. 14

15 4.1 Proof of Theorem 4.1 We estabish surjectivity by showing that for each R > 0 the ba of radius R is fuy contained in f(m), that is, B(0; R) f(m). Since transations do not change any of our hypotheses, et us assume that 0 f(m) and singe out o f 1 (0) M. Next, from the oca diffeomorphism assumption, there exists ε > 0 such that B(0; ε) f(m) and f 1 (B(0; ε)) has a diffeomorphic component W n 1 (ε) which contains o M. Observe that for R ε, we triviay have B(0; R) f(m), so we restrict ourseves to the case R > ε. We argue that we can find a way to expand B(0; ε) within the image of f so that it wi contain a ba of radius any R. To this end, we sha choose directions for this expansion as foows. For v S n 1, et H v be the canonica codimension one foiation of R n by hyperpanes orthogona to v. Since the eaf space of H v is homeomorphic to R, we parameterize the eaves of H v by H v (t) where t is the distance of the hyperpane H v (t) to the origin. Because H v (t) = H v ( t), we wi ony consider t 0. Let N v = f H v be the puback foiation of M which, by definition, has the connected components of f 1 (H v (t)) as eaves. Since our hypotheses states that the pre-images of hyperpanes are non-empty and connected, the eaf space of N v is homeomorphic to R and we then write N v (t) = f 1 (H v (t)). Next, we caim that for each v S n 1, we may find a goba transversa γ v to the foiation N v that may be used to expand the image of f. More precisey, we have the foowing resut. Lemma 4.3. For each u W n 1 (ε) with f(u) = εv, v S n 1, there exists a smooth path γ v : [0, ) M with γ v N v such that f ( γ v (t) ) H v (t) for t [0, R]. The proof foows directy from transverse modification arguments in foiation theory (see [1]) and the fact that the eaves of N v are non-empty and connected. As we proceed with the proof of Theorem 4.1, et us fix a canonica identification of S n 1 to B(0; ε) and W n 1 (ε). By appying Lemma 4.3, we obtain directions γ v from which to expand W n 1 (ε) up to γ v N v (R). Then for a fixed v, we can ocay modify γ v so that we can carry an entire 15

16 neighborhood of v in W n 1 (ε) aong γ v using the compactness of γ v ( [0, R] ). Repeating this process for each v W n 1 (ε), we obtain a cover of W n 1 (ε). However, because there is no canonica choice for γ v, points beonging to the intersection of two neighborhoods may have mutipy defined paths. Our approach wi be simiar to the one in section 3. The key difference here is that we aso need to contro how each neighborhood is pushed aong the goba transversa γ v. Intuitivey, we wi push the ces of W n 1 (ε) aong γ v and possiby create broken pieces at each instant t. Using the hypotheses that N v (t) is acycic and f is a diffeomorphism we wi define bounding chains fiing the gaps in each eaf. Furthermore, we wi argue that these chain compexes constructed for s, t wi be homoogous, hence we say that they are homoogous reative to t. This process is depicted in Fig. 4.1 and it is stated precisey in the emma beow. W n 1 (R) W n 1 (t) W n 1 (ε) 0 γ v Figure 4.1: A oca assembage of chain compexes based on the trianguation of a sphere. Lemma 4.4. For each R > 0, there exists a famiy of geometric (singuar) chain compexes W n 1 (t) that are homoogous in M\ {o} for t (0, R] such 16

17 that: i) For t (0, ε], W n 1 (t) = f 1( B(0; t) ). ii) If q supp W n 1 (t), then q N v (t) for some v S n 1. The proof of Lemma 4.4 uses combinatoria topoogy and foiation theory to expicity construct such cyces. Since the process is engthy and rather technica, we postpone it and continue with the proof of Theorem 4.1. Our strategy to show that B(0; R) f(m) is by contradiction. Suppose there is p / f(m). We compute the inking number between p and f ( W n 1 (t) ) = Z n 1 (t) in two ways, yieding different vaues. This argument is simiar to standard reasoning in degree theory and is geometric in nature. As before, we work with Z 2 -coefficients. Notice that since f is continuous, we have that Z n 1 (t) is a famiy of homoogous cyces in R n \ {0}. Then for t (0, ε], Z n 1 (t) = B(0; t) and we have that the origin is contained in the inside of Z n 1 (ε), more precisey, the inking number between the origin and Z n 1 (ε) is equa to 1. From (ii) of Lemma 4.4, we have that 0 / Z n 1 (t) for each t (0, R] and as mentioned above, Z n 1 (t) Z n 1 (R) in R n \ {0}. Therefore as intersection numbers, thus inking numbers are invariant under the same homoogy cass, we have Lk(Z n 1 (t), 0) = 1 for each ε < t R, (4.1) where we consider the 0-norma cyce formed by the origin and a suitabe point in the compement of a compact set containing Z n 1 (t). We caim that p is inside Z n 1 (R), that is, Lk(Z n 1 (R), p) = 1. Indeed, consider the segment Y 1 from 0 to p. We have that Y 1 Z n 1 (R) =, otherwise it woud impy p f(m). By definition, #(Z n 1 (R) Y 1 ) = #(Y 1 Z n 1 (R)) = 0. (4.2) Computing the inking number between the cyce Z n 1 (R) and Y 1 using the fact that Y 1 is a norma 0-cyce, we have, 0 = #(Y 1 Z n 1 (R)) = Lk( Y 1, Z n 1 (R)) = Lk(Z n 1 (R), Y 1 ) = Lk(Z n 1 (R), 0 p) = Lk(Z n 1 (R), 0) Lk(Z n 1 (R), p). (4.3) 17

18 Combining (4.3) and Lk(Z n 1 (R), 0) = 1 we obtain Lk(Z n 1 (R), p) = 1. (4.4) Observe that Z n 1 (ε) = B(0; ε), hence Lk(Z n 1 (ε), p) = 0. Finay, from the assumption that p / f(m), we have Z n 1 (R) Z n 1 (ε) in R n {p} and again by the invariance of inking numbers on the homoogy cass we obtain, Lk(Z n 1 (R), p) = Lk(Z n 1 (ε), p) = 0. (4.5) This is a contradiction, therefore it must be the case that p f(m) and hence f is surjective. 4.2 The Construction of a Famiy of Homoogous Cyces We now compete the proof of Theorem 4.1 by estabishing the technica proof of Lemma 4.4. We empoy a simiar technique as in section 3, that is, we use trianguations as a too to keep track of intersections in the coverings. For simpicity, et W n 1 (ε) = W and consider a trianguation of W with ces e(k) ; k = 0,...,n 1 and E(k) N where E(k) is the set of indexes of a the k-ces in W. The idea of the construction of W n 1 (t) is in essence geometric, and can be outined as foows. Consider a trianguation with sufficienty sma mesh. For each top dimensiona ce of W, we push it aong a goba transversa γ v emanating from one of its points up to the eve R and use the oca modification of γ v for points within the ce to push these points. The key issue is that a point v beonging to the boundary of a top dimensiona ce may be pushed aong mutipe choices of γ v, one for each top dimensiona ce it beongs to. Hence the W n 1 (t) may not be we defined; geometricay, this wi create broken pieces at each eve. However, by considering the coections of ces that contain v, via the ink of v, we wi show that for each t, the mutipy defined chain compexes form a cyce in the pre-image of H v (t). Thus by acycicity, we can fi these gaps with bounding chains. Furthermore, the process wi be done so it is homoogous reative to t, that is, as we consider different chain compexes for each t. Now, as we begin to 18

19 formay describe W n 1 (t), we wi do so in steps enumerated from 0 to n 1, outined beow. Step 0: For each point v W, suppose v e(n 1) for some E(n 1). From Lemma 4.3 we obtain a goba transversa γ v to the foiation N v. We then define the foowing 0-chains, that is, points where γ v intersect the eaves of N v ; Let W 0(v, t) = γ v(t) N v (t) for t [ε, R]. By compactness ( ) of γ v [0, R] and the fact that f is a oca diffeomorphism, there is a neighborhood V v M of γ v [0, R] such that we can continuousy modify γv to ( ) obtain a goba transversa γ v for a v in a neighborhood O v W of v, as depicted in Fig 4.2. γ v v o O v Figure 4.2: Loca modification of goba transversas. Then, by the compactness of W, we obtain a finite subcover of W from {O v }. Now with such subcover, we iterate finitey many barycentric subdivisions of W unti its mesh is smaer than the minimum diameter of the subcover. In this process, we obtain a new trianguation of W with the property that, for each v e(n 1), we may continuousy define γ v for a points in e(n 1). In fact, we define, for each t [ε, R] and E(n 1), a (n 1)-chain denoted by W 0 (t) e(n 1) which is topoogicay equivaent to W 0 (v, t) e(n 1) and varies continuousy on t. Step 1: Consider points v in the (n 2)-skeeton of W. Suppose v e(n 2) 19

20 for some E(n 2). Then v beongs to the intersection of two (n 1)-ces that can be determined by ooking at the vertices in Lk(e(n 2) ). Without oss of generaity et v e(n 1) 1 e(n 1) 2. Then for each t [ε, R], we have defined in the previous step the points W1 0(v, t), W 2 0(v, t) N v(t) aong path emanating from each top dimensiona ce. Now we can join such points by a path W 1(v, t) ying in N v(t) as it is acycic. Once we construct W 1 (v, t), we caim that it can be ocay modified for a points in a neighborhood of (v, t) in W (n 2) [ε, R]. Indeed, for each v W (n 2) and t [ε, R], the path W 1 (v, t) is compact and hence we may find a neighborhood U(v, t) M of W 1 (v, t) such that the (oca) gradient fow of the height function f v : M R n given by f v (x) = f(x), v can be used to continuousy define W 1 (v, t) for nearby t. Aso, the fact that f is a oca diffeomorphism continuousy defines W 1(v, t) for a nearby v in W (n 2). The process above provides a cover {U(v, t)} of W (n 2) [ε, R]. By compactness, we may find a finite subcover which induces a cover of W. Indeed, in step 0 each top dimensiona ce is pushed diffeomorphicay aong the goba transversa γ v. We can now iterate finitey many barycentric subdivisions of W so its mesh is smaer than the minimum diameter of the subcover above restricted to W. Next, we repeat a the constructions up to this point using the new trianguation. Observer that this guarantees that each ce e(n 2) is contained in a member of the finite subcover. Now et us consider a partition of [ε, R] induced by this subcover, that is, we have ε = t 0 < t 1 < < t N = R for some N N. From the choice of subdivision we can continuousy modify W 1(v, t) for t (t i, t i+1 ) and v e(n 2) by the argument above. The key probem is that for the endpoint we may possiby have two chains defined, each coming from the adjacent intervas. However, the fact that each eaf of N v is acycic yieds a simiar construction as the transverse modification method (see [1]) to ensure that whenever ambiguity occurs, the choice wi be homoogous reative to t. The detais are as foows; For each i = i,...,n 1, et the two choices for a bounding chain W 1(v, t i) be W 1(v, t i) and W 1(v, t i) +, where W 1(v, t i) is the chain defined continuousy from W 1(v, t), t (t i 1, t i ) and the second one, W 1(v, t i) +, is the chain defined continuousy from W 1(v, t), t (t i, t i+1 ). By defaut, we agree to aways choose W 1(v, t i) +. This wi not be ambiguous 20

21 because we can choose either chain compex. Indeed, W 1(v, t i) W 1(v, t i) + since from construction they have the same boundary and, by acycicity of N v (t i ), there is a bounding chain contained in N v (t i ). Finay, we must consider a new neighborhood Ũ(v, t i) of such bounding chain where the restriction of f is a diffeomorphism. Doing so for every v W (n 2) and i = 1,..., N we obtain a new cover of W by adding the coection of sets Ũ(v, t i) to the finite subcover considered up to this point. This is done to ensure that the process wi aways yied chains homoogous reative to t. Then iterate finitey many barycentric subdivisions of W to obtain a trianguation with mesh sufficienty sma to define chains W 1 (v, t) continuousy for a points v e(n 2) for each and by construction these chains are homoogous reative to t with continuousy varying bounding chains. This adaptation of the transverse modification argument produces for each t [ε, R] and E(n 2), a (n 1)-chain compex denoted by W 1(t) e(n 2) which is topoogicay equivaent to W 1(v, t) e(n 2), v e(n 2), and is homoogous reative to t. This concudes step 1. Now we give the genera procedure for 1 < k n 1. Step k: Consider points in the (n 1 k)-skeeton of W. Suppose v e(n 1 k) for some E(n 1 k). We are interested in identifying a the (n k)-ces that contain e(n 1 k) in its boundary, i.e., that contain v. This can be accompished by ooking at the vertices of Lk(e(n 1 k) ). for simpicity, suppose that those are e(n k) 1, e(n k) 2,...,e(n k) m for some m N. For each t [ε, R], consider the (k 1)-chain; W k 1 1 (v, t) + + Wm k 1 (v, t), (4.6) where the chains W k 1 j (v, t) were constructed in Step k 1. We caim that the (k 1)-chain in (4.6) is a cyce. Indeed, ( m ) ( ) m m Wj k 1 (v, t) = Wj k 1 (v, t) = W k 2 (v, t), (4.7) j=1 j=1 where corresponds to the index of a (n k + 1)-ces in W that contain e(n 1 k). Now the argument is competey anaogous to the one given in the injectivity case, i.e, it foows from the observation that an edge contains j=1 21

22 exacty two vertices. Since N v (t) is acycic, there exists a k-chain W k (v, t) m that bounds W k 1 j (v, t). j=1 We now argue that the chain W k (v, t) may be continuousy modified with respect to v and within the same homoogy cass reative to t. This is simiar to the construction as in step 1, except that we repeat it for each interva of the partition obtained in step k 1, so we omit the detais. Finay, for each t [ε, R] and E(n 1 k), we obtain a (n 1)-chain denoted by W k(t) e(n 1 k) which for v e(n 1 k), is topoogicay equivaent to W k(v, t) e(n 1 k), and is homoogous reative to the parameter t. Remark 4.1. In the ast step n 1, we consider the 0-skeeton of W which is discrete, hence no further subdivisions are necessary. Once a n steps are competed, for each t [ε, R], we put together a the constructed chain compexes via the obvious attaching maps based on the intersection of the ces in W as indicated by their construction in each step. This defines W n 1 (t) as foows; W n 1 (t) = n 1 k=0 E(n 1 k) W k (t) e(n 1 k). We observe that W n 1 (t) W n 1 (s) in M\ {o} for t, s [ε, R]. Indeed, by construction if t, s (t i 1, t i ) for i = 1,...,N, we can use the oca gradient fow of the corresponding height functions to continuousy modify the chain W n 1 (t) to W n 1 (s), each chain W n 1 k (t) e(k) at a time. Otherwise, the ony probem is at the end points t i, where again by construction, the chains of W n 1 (t i ) are homoogous in N(t i ) = { p M p N v (t i ) for some v S n 1}, thus ensuring that W n 1 (t) are homoogous in M\ {o}. It now remains to show that W n 1 (t) is a cyce in M. The computation beow is quite simiar to the one done in section 3. For simpicity, et S k (t) = n 1 W k (t) e(n 1 k), so that W n 1 (t) = S k (t). E(n 1 k) k=0 22

23 Considering each term separatey, we have for k = 0; S 0 (t) = W 0 (t) e(n 1) = E(n 1) E(n 1) = 0 + W 0 (t) e(n 1) E(n 1) + E(n 1) j n 2 W 0 e(n 2) jn 2, W 0 (t) e(n 1) where j n 2 denotes the index of a (n 2)-ces that beong to the boundary of e(n 1). In genera, for 1 < k < n 1; S k (t) = W k (t) e(n 1 k) = = E(n 1 k) E(n 1 k) E(n 1 k) W k (t) e(n 1 k) + + E(n 1 k) W k (t) e(n 1 k) W k 1 j n k (t) e(n 1 k) + j n k + W k (t) e(n 2 k) jn 2 k, j n 2 k E(n 1 k) where j n k denotes the index of a (n k)-ces that contain e(n 1 k) and j n 2 k denotes the index of a (n 2 k)-ces that beong to the boundary of e(n 1 k). Finay, for k = n 1, S n 1 (t) = = E(0) = E(0) E(0) W n 1 (t) e(0) W n 1 (t) e(0) + E(0) W n 2 j 1 (t) e(0) + 0, j 1 23 W n 1 (t) e(0)

24 where j 1 denotes the index of a 1-ces that contain e(0) in its boundary. As we sum the terms in W n 1 (t), we see that the second summation term appearing in S k (t) coincided with the first summation in S k+1 (t). Indeed, these terms count the same objects twice, and hence we obtain zero (reca that we are working with Z 2 coefficients). Therefore W n 1 (t) is a cyce and this finishes the proof of Lemma Fina Remarks Having obtained independent resuts on injectivity and surjectivity, we note that our main resut foows from Theorem 3.3 and Theorem 4.1. Reca, Theorem 1.1. A oca diffeomorphism f : R n R n is bijective if and ony if the pre-image of every affine hyperpane is non-empty and acycic. We aso have the foowing anaytic condition that estabishes whether a oca diffeomorphism is bijective. Given a compete Riemannian metric g on R n and a smooth function h : R n R, the gradient of h reative to g, denoted by g h, satisfies g x ( g h, w) = dh x (w) for a w R n. Our anaytic resut is the foowing. Coroary 5.1. A oca diffeomorphism f : R n R n is bijective if for each v S n 1, there exists a compete metric g v on R n such that, inf f x R n gv v (x) gv > 0. (5.0) It is easy to see that such condition impies that the pre-images of hyperpanes are acycic, hence the resut foows. We compare this resut with the work in [7] where we can now choose the metric to suit the unit vector. Finay, We can aso state an anaytica resut impying ony injectivity. Coroary 5.2. A oca diffeomorphism f : R n R n is injective provided there exists w S n 1 with the property that for each unit vector v perpendicuar to w, there exists a compete Riemannian metric g v on R n such that, inf x R n Df(x) v gv > 0. (5.0) Observe that in our injectivity resuts we did not need topoogica hypotheses on the pre-image of every hyperpanes, hence the resut hods. 24

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