Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 59–80

MAPPINGS OF FINITE DISTORTION: GLOBAL HOMEOMORPHISM THEOREM

Ilkka Holopainen and Pekka Pankka

University of Helsinki, Department of Mathematics P.O. Box 4, FI-00014 Helsinki, Finland ilkka.holopainen@helsinki.fi; pekka.pankka@helsinki.fi

Abstract. We study mappings of finite distortion between Riemannian manifolds. We establish Väisälä’s inequality for moduli of path families and as an application we prove a version of the so-called global homeomorphism theorem for mappings of finite distortion under a subexponential integrability condition on the distortion.

1. Introduction

Zorich’s theorem for quasiregular mappings states that every quasiregular local homeomorphism from R^{n} into R^{n} is a homeomorphism if n ≥ 3; see [18]. A geometric version of this theorem, known as the Global homeomorphism theorem, states that every quasiregular local homeomorphism from an n-parabolic Riemannian n-manifold into a simply connected Riemannian n-manifold is an embedding if n ≥ 3; see [19] and [1, 6.30].

Proofs of these results are based on geometrical properties of quasiregular mappings and especially on path family techniques like Poletsky’s inequality. In [10] Koskela and Onninen proved that substitutes of the Poletsky inequality and its generalization, Väisälä’s inequality, hold for mappings of (sub)exponentially integrable distortion. Thus these mappings have similar geometrical properties as quasiregular mappings in this sense. Using the Euclidean results of Koskela and Onninen we first generalize these inequalities for mappings of finite distortion between Riemannian manifolds. Then we combine Poletsky’s inequality with methods introduced by Zorich to extend the Global homeomorphism theorem for mappings of finite distortion.

Let Γ_{M}^{∞} be the family of all paths γ in a Riemannian manifold M whose locus |γ| is not contained in any compact subset of M. We say that M is ω -parabolic, where ω: M → [0, ∞] is measurable, if the ω -weighted n-modulus of Γ_{M}^{∞} is zero; see Section 3 for details.

2000 Mathematics Subject Classification: Primary 30C65.

Both authors were supported in part by the Academy of Finland, project 53292.

Theorem 1 (Global homeomorphism theorem). Let M be a K^{n−1}( ·)- parabolic Riemannian n-manifold, N a simply connected Riemannian n-manifold, both of dimension n ≥ 3, and let f: M → N be a local homeomorphism of finite distortion K( ·) satisfying the condition (A). Then f is a homeomorphism onto its image and the set N \ fM has zero n-capacity.

See Section 2 for the definition of a mapping of finite distortion between Riemannian manifolds.

The proof of the global homeomorphism theorem depends on the following two propositions.

We say that a point q ∈ N is an asymptotic limit of f if there exists a path γ: [a, b[→ M belonging to Γ_{M}^{∞} such that (f ∘ γ)(t) → q as t → b. We denote by E(f) the set of all asymptotic limits of f.

Proposition 2. Let M be a K( ·)^{n−1} -parabolic Riemannian n-manifold, N a Riemannian n-manifold, with n ≥ 3, and let f: M → N be a local homeomorphism of finite distortion K( ·) satisfying (A). Then E(f) is σ -compact, cap_{n}E(f) = 0, and N \ fM ⊂ E(f).

The following proposition extends the well-known simply connectedness result [13, Lemma 3.3] and its generalization [12, Theorem 6.13] to the case of σ -compact sets on Riemannian manifolds.

Proposition 3. Let N be a connected Riemannian manifold of dimension n ≥ 2 and let E be a σ -compact subset of N such that ℋ ^{n−k}(E) = 0 for some integer k ∈ {1, …, n}. Then the inclusion N \ E ↪ N induces an isomorphism π_{m}(N \ E) → π_{m}(N) for all 0 ≤ m ≤ k − 1.

2. Mappings of finite distortion on Riemannian manifolds

We assume throughout the paper that M and N are C^{∞}, oriented Riemannian n-manifolds without boundary. We denote by Γ(X) the set of measurable vector fields on M and by Γ^{k}(M) the set of C^{k} -smooth vector fields on M. The measure on M given by the Riemannian volume form will be denoted by m_{M}.

A locally integrable vector field X ∈ Γ(M) is called a weak gradient of a function u ∈ L_{loc}^{1}(M) if

∫_{M}⟨X, Y ⟩ dm_{M} = − ∫_{M} u divY dm_{M}

holds for every compactly supported vector field Y ∈ Γ^{1}(M). We denote by ∇u the weak gradient of u. The Sobolev space W_{loc}^{1,p}(M), 1 ≤ p < ∞, consists of all functions u ∈ L_{loc}^{p}(M) whose weak gradient ∇u belongs to L_{loc}^{p}(M).

Let us recall the definition of a mapping of finite distortion of an open set G ⊂ R^{n}. We say that f: G → R^{n} is a mapping of finite distortion if the following conditions are satisfied:

(a) f ∈ W_{loc}^{1,1}(G, R^{n}).

(b) The Jacobian determinant J( · , f) of f is locally integrable.

(c) There exists a measurable function K: G → [1, ∞], finite almost everywhere,

such that

∥Df(x)∥^{n} ≤ K(x)J(x, f) a.e.

Here Df(x) is the formal derivative of f at x, that is, the linear map R^{n} → R^{n} defined by the partial derivatives D_{i}f(x) as Df(x)e_{i} = D_{i}f(x). Furthermore, ∥Df(x)∥ = max{|Df(x)v|: v ∈ R^{n}, |v| = 1} and J(x, f) = det Df(x). If K ∈ L^{∞}(G), the condition (b) implies that f ∈ W_{loc}^{1,n}(G) and we recover the class of mappings of bounded distortion, also called quasiregular mappings. It is a deep result of Reshetnyak that a non-constant quasiregular mapping is locally Hölder continuous, discrete, and open; see [14]. Recently these topological properties have been established for mappings of finite distortion under minimal integrability assumptions on the distortion function K; see e.g. [4], [6], and [8].

In this paper we are mainly interested in locally homeomorphic mappings of finite distortion between Riemannian manifolds. In particular, we assume that mappings in question are a priori continuous. Therefore we may use local representations to define mappings of finite distortion on Riemannian manifolds.

Definition 4. We say that a continuous mapping f: M → N has finite distortion if, for each x ∈ M, there exist orientation-preserving charts (U, φ) at x and (V, ψ) at f(x) such that fU ⊂ V and that ψ ∘ f ∘ φ^{−1}: φU → R^{n} is a mapping of finite distortion in φU ⊂ R^{n}.

If f: M → N is a mapping of finite distortion, we define for a.e. x ∈ M a linear map Df(x): T_{x}M → T_{f(x)}N by

Df(x) = Dψ^{−1}(f(x)) ∘ D(ψ ∘ f ∘ φ^{−1})(φ(x)) ∘ Dφ(x),

where φ and ψ are any orientation-preserving chart mappings at x and at f(x), respectively. Observe that the definition of Df(x) is independent of the choice of φ and ψ. Now there exists a measurable (distortion) function K: M → [1, +∞], finite almost everywhere, such that

(1)  ∥Df(x)∥^{n} ≤ K(x)J(x, f)  a.e.,

where ∥Df(x)∥ = max{|Df(x)v|: v ∈ T_{x}M, |v| = 1} and J(x, f) = det Df(x). If we want to emphasize the role of the function K in (1), we call f a mapping of finite distortion K.

Our main objective in this section is to extend the basic topological and metric properties of mappings of finite distortion from the Euclidean case to the Riemannian setting. The results we consider are valid only under some additional assumptions on the distortion function K. To describe these assumptions, we consider infinitely differentiable and strictly increasing functions, called Orliczfunctions, 𝒜: [0, ∞[→ [0, ∞[, with 𝒜 (0) = 0 and lim_{s→∞} 𝒜 (s) = ∞. We say that a mapping f of finite distortion K satisfies the condition (A) if (A-0) exp(𝒜 (K)) ∈ L_{loc}^{1}(M), where 𝒜 is an Orlicz-function such that

(A-1) ∫_{1}^{∞} \frac{𝒜′(t)}{t} dt = ∞, and

(A-2) t𝒜′(t) increases to ∞ for sufficiently large t.

For the next lemma we recall that, for each L > 1 and x ∈ M, the Riemannian normal coordinates at x when restricted to a sufficient small neighborhood U of x provide an L-bilipschitz chart φ: U → R^{n}.

Lemma 5. Let f: M → N be a continuous mapping of finite distortion K. Suppose that (U, φ) and (V, ψ) are orientation-preserving L-bilipschitz charts on M and on N, respectively, such that fU ⊂ V . Then h = ψ ∘ f ∘ φ^{−1}: φU → R^{n} is a mapping of finite distortion L^{4n}K ∘φ^{−1} . Furthermore, if f satisfies (A), then so does h.

Proof. The mapping h has finite distortion by our definition. Since φ and ψ are L-bilipschitz, they satisfy ∥Dψ( ·)∥ ≤ L, ∥Dφ^{−1}( ·)∥ ≤ L, J( · , ψ) ≥ L^{−n}, and J( · , φ^{−1}) ≥ L^{−n}. For x ∈ φU, we write y = ψ(f(φ^{−1}(x))). Then by the chain rule, h satisfies a distortion inequality

∥Dh(x)∥^{n} = ∥Dψ(y) ∘ Df(φ^{−1}(x)) ∘ Dφ^{−1}(x)∥^{n}
≤ ∥Dψ(y)∥^{n}∥Df(φ^{−1}(x))∥^{n}∥Dφ^{−1}(x)∥^{n}
≤ L^{4n}K(φ^{−1}(x))J(y, ψ)J(φ^{−1}(x), f)J(x, φ^{−1})
= L^{4n}K(φ^{−1}(x))J(x, h)

for a.e. x ∈ φU.

Suppose that f satisfies (A) with an Orlicz-function 𝒜 . Then h satisfies the condition (A) with an Orlicz-function 𝒜̃(t) = 𝒜 (t/L^{4n}) since J( · , φ^{−1}) ≥ L^{−n} in φU.

Using Lemma 5 we obtain the results of [8, Theorem 1.3] and [9, Theorem 1.1] in our setting.

Theorem 6. Let f: M → N be a continuous mapping of finite distortion satisfying (A). Then f is either constant or both open and discrete. Moreover, if f is non-constant, J(x, f) > 0 for a.e. x ∈ M, m_{M}(B_{f}) = 0, and f satisfies Lusin conditions (N) and (N^{−1}), that is, for every measurable E ⊂ M,

m_{M}(E) = 0 if and only if m_{N}(fE) = 0. Here B_{f} is the branch set of f, that is, the set of points x ∈ M where f

fails to be a local homeomorphism.

3. Väisälä’s inequality

For the statement and proof of Väisälä’s inequality we fix some terminology. Let I and J be intervals on R. If α: J → M is a subpath of β: I → M, that is, J ⊂ I and α = β | J, we denote α ⊂ β.

Let f: M → N be a continuous, discrete, and open mapping, and let β: [a, b[→ N be a path, where b ∈ [a, ∞]. A maximal f -lifting of β starting at a point x ∈ f^{−1}(β(a)) is a path α: [a, c[→ M, a ≤ c ≤ b, such that

(i) α(a) = x, (ii) f ∘ α = β | [a, c[, and (iii) if c < c′ ≤ b, then there does not exists a path α′: [a, c′[→ M such that

α = α′ | [a, c[ and f ∘ α′ = β | [a, c′[. Moreover, a maximal f -lifting is total, if f ∘ α = β. We use the same terminology for paths defined on compact intervals.

Let Γ be a path family in M, 1 ≤ p < ∞, and let ω: M → [0, +∞] be measurable. The weighted p-modulus of Γ, with the weight ω, is defined by

(2)  M_{p,ω}(Γ) = inf_{ϱ} ∫_{M} ϱ^{p}ω dm_{M},

where the infimum is taken over all admissible functions ϱ for the path-family Γ, that is, over all Borel functions ϱ: M → [0, ∞] such that

∫_{γ} ϱ ds ≥ 1

for every locally rectifiable γ ∈ Γ, see [10]. If ω ≡ 1, (2) defines the usual pmodulus of Γ which we denote by M_{p}(Γ).

Theorem 7 (Väisälä’s inequality). Let f: M → N be a continuous, nonconstant mapping of finite distortion K satisfying the condition (A). Let Γ be a path family in M, Γ′ be a path family in N, and m be a positive integer such that the following is true: For every path β: I → N in Γ′ there are paths α_{1}, …, α_{m} in Γ such that f ∘ α_{j} ⊂ β for all j and such that for every x ∈ M and t ∈ I the equality α_{j}(t) = x holds for at most i(x, f) indices j. Then

(3)  M_{n}(Γ′) ≤ \frac{M_{n,K_{I}( · ,f)}(Γ)}{m}.

Here i(x, f) is the local topological index of f at x; see, for example, [16] or [15]. The weight K_{I}( · , f) on the right-hand side of (3) is the inner distortion function of f defined by

K_{I}(x, f) = \{ \begin{matrix}\frac{∥D^{#}f(x)∥^{n}}{J(x, f)^{n−1}} , & if J(x, f) ≠ 0, \\ 1, & if J(x, f) = 0 and ∥D^{#}f(x)∥ = 0, \\ ∞, & if J(x, f) = 0 and ∥D^{#}f(x)∥ ≠ 0, \end{matrix}

where D^{#}f(x) is the matrix of cofactors of Df(x). The function K_{I}( · , f) satisfies a point-wise inequality K_{I}(x, f) ≤ K^{n−1}(x), [5, Section 6].

We need the following version of the change of variable formula; see [2] for a more general Euclidean version.

Lemma 8. Let f: M → N be a continuous mapping of finite distortion satisfying the condition (N). Let E ⊂ M be a measurable set and u: fE → [0, ∞] a measurable function. Then

∫_{E} u(f(x))J(x, f) dm_{M} = ∫_{N} u(y)N(y, f, E) dm_{N}.

Proof. Let (U, φ) and (V, ψ) be orientation-preserving charts on M and N, respectively, such that fU ⊂ V . Let F ⊂ E ∩ U be a measurable set, v: ψ(fF) → R, v(y) = u(ψ^{−1}(y)), and h = ψ ∘ f ∘ φ^{−1}: φU → ψV . Then

\begin{aligned}∫_{F} u(f(x))J(x, f) dm_{M} & = ∫_{φF} u(f(φ^{−1}(x)))J(φ^{−1}(x), f)J(x, φ^{−1}) dx \\ & = ∫_{φF} v(h(x))J(h(x), ψ^{−1})J(x, h) dx. \end{aligned}

Furthermore, by the (Euclidean) change of variables formula [2], we obtain

∫_{φF} v(h(x))J(h(x), ψ^{−1})J(x, h) dx = ∫_{ψV} v(y)J(y, ψ^{−1})N(y, h, φF) dy
= ∫_{V} v(ψ(y))N(ψ(y), h, φF ) dm_{N}
= ∫_{V} u(y)N(y, f, F) dm_{N}.

Let {(U_{i}, φ_{i})} and {(V_{j}, ψ_{j})} be atlases of M and N, respectively, consisting of orientation-preserving mappings such that for every i there exists j_{i}, with fU_{i} ⊂ V_{j_{i}}. Let E_{1} = E ∩ U_{1} and E_{i} = (E ∩ U_{i}) \ E_{i−1} for i ≥ 2. Then the sets E_{i} are disjoint and E = ∪E_{i}. The claim follows by applying the equations above to the sets E_{i}.

Let us recall the definition of the absolutely precontinuity of a continuous, discrete, and open mapping f: M → N on a path (see [10, p. 18] and [15, p. 40]). Let β: I_{0} → N be a closed rectifiable path, and let α: I → M be a path such that f ∘ α ⊂ β. Let s_{β}: I_{0} → [0, l(β)] be the length function of β. If s_{β} is constant on some interval J ⊂ I, β is also constant on J, and the discreteness of f implies that also α is constant on J. It follows that there exists a unique path α^{∗}: s_{β}(I) → M such that α = α^{∗} ∘ (s_{β} | I). We say that f is absolutely precontinuous on α if α^{∗} is absolutely continuous.

We have the following version of Poletsky’s lemma.

Lemma 9. Let f: M → N be a continuous mapping of finite distortion satisfying (A). Let Γ be a family of paths γ in M such that f ∘ γ is locally rectifiable and there exists a closed subpath of γ on which f is not absolutely precontinuous. Then M_{n}(fΓ) = 0.

Proof. Let {(U_{i}, φ_{i})} and {(V_{j}, ψ_{j})} be smooth atlases of M and N, respectively, such that each φ_{i} and ψ_{j} is an orientation-preserving L-bilipschitz mapping, and that for every i there exists j_{i} satisfying fU_{i} ⊂ V_{j_{i}}. Since the modulus of a path family depends only on locally rectifiable paths, we may assume that there are only locally rectifiable paths in fΓ. Let Γ̃ be the set of paths α in M such that α is a closed subpath of some γ ∈ Γ and f is not absolutely precontinuous on α. Every path α ∈ Γ̃ has a subpath β such that f is not absolutely precontinuous on β and |β| ⊂ U_{i} for some i. Let Γ̃_{i} be a set of those subpaths β of paths α ∈ Γ̃. Then the path family Γ is minorized by ∪Γ̃_{i}, that is, every path in Γ has a subpath in ∪Γ̃_{i}.

Since mappings φ_{i} and ψ_{j} are smooth L-bilipschitz mappings, we have that f ∘ φ_{i}^{−1} is not absolutely precontinuous on paths φ_{i} ∘ γ ∈ φ_{i}Γ̃_{i}, and furthermore ψ_{j_{i}} ∘ f ∘ φ_{i}^{−1} is not absolutely precontinuous on paths φ_{i}Γ̃_{i}. Thus, by Lemma 5 and [10, Lemma 4.3], we have

M_{n}((ψ_{j_{i}} ∘ f ∘ φ_{i}^{−1})φ_{i}Γ̃_{i}) = 0.

Since mappings ψ_{j_{i}} are L-bilipschitz, we have

M_{n}(fΓ) ≤ ∑_{i} M_{n}(fΓ̃_{i}) ≤ ∑_{i} L^{2n}M_{n}(ψ_{j_{i}}fΓ̃_{i}) = 0.

Now Väisälä’s inequality can be proved exactly as in the Euclidean case, see [15, Theorem II.9.1]. For the reader’s convenience we sketch the main steps.

Proof of Väisälä’s inequality. Let C ⊂ M be such that m_{M}(C) = 0, B_{f} ⊂ C and f is differentiable in M \ C. The differentiability of f a.e. in M follows from [15, VI Lemma 4.4], Condition (A), and Lemma 5. As J(x, f) > 0 a.e. in M, we may assume that J(x, f) > 0 for every x ∈ M \ C. Since f satisfies the condition (N), we find a Borel set B of measure zero containing f(C). Thus B_{f} ⊂ f^{−1}(B) and f is differentiable at every x ∈ M \ f^{−1}(B) with J(x, f) > 0.

We may assume that every β ∈ Γ′ is locally rectifiable and since m_{N}(B) = 0, we may also assume that

∫_{β} χ_{B} ds = 0

for every β ∈ Γ′. By Lemma 9, we may assume that if α is a path in Γ such that f ∘ α ⊂ β ∈ Γ′ then f is locally absolutely precontinuous on α.

Let ϱ be a non-negative Borel function M → R such that

∫_{γ} ϱ ≥ 1

for every γ ∈ Γ. Define σ: M → R by

σ(x) = \{ \begin{matrix}\frac{ϱ(x)}{min_{|v|=1} |Df(x)v|}, & if x ∈ M \ f^{−1}(B), \\ 0, & if x ∈ f^{−1}(B). \end{matrix}

Let ϱ′: N → R be the function

ϱ′(y) = \frac{1}{m}χ_{fM}(y) sup_{A} ∑_{x∈A} σ(x),

where A runs through over all subsets A ⊂ f^{−1}(y) such that #A ≤ m.

To see that ϱ′ is a non-negative Borel function, we apply the argument in the proof of [15, Theorem II.9.1]. Moreover, by the same argument as in the proofs of [15, Theorem II.9.1, Theorem II.8.1], we have that

∫_{β} ϱ′ ds ≥ 1

for all β ∈ Γ′. Poletsky’s Lemma 9 is employed also at this point.

Let (Ω_{i}) be an exhaustion of M by relatively compact sets Ω_{i} such that \overline{Ω}_{i} ⊂ Ω_{i+1} . We set

\begin{matrix}ϱ_{i} = & ϱχ_{\overline{Ω}_{i}}: M → R, \\ σ_{i} = & σχ_{\overline{Ω}_{i}}: M → R, \end{matrix}

and

(4)  ϱ′_{i}: M → R,  ϱ′_{i}(y) = \frac{1}{m}χ_{f \overline{Ω}_{i}}(y) sup_{A} ∑_{x∈A} σ_{i}(x).

Suppose y_{0} ∈ f \overline{Ω}_{i} \ f( \overline{Ω}_{i} ∩ B_{f}) and let k = #(f^{−1}(y_{0}) ∩ \overline{Ω}_{i}). Then there is a connected neighborhood V of y_{0} and k inverse mappings g_{μ}: V → D_{μ} with

\overline{Ω}_{i} ∩ f^{−1}V = ⋃\{ \overline{Ω}_{i} ∩ D_{μ}: 1 ≤ μ ≤ k\}.

For each y ∈ V we define L_{y} ⊂ J := {1, …, k} as follows. If k ≤ m, then L_{y} = J. If k > m, then #L_{y} = m, and for each μ ∈ L_{y} and ν ∈ J \ L_{y} either σ_{i}(g_{μ}(y)) > σ_{i}(g_{ν}(y)) or σ_{i}(g_{μ}(y)) = σ_{i}(g_{ν}(y)) and μ > ν. Thus the sum over the set L_{y} gives the supremum in the formula (4). Then

\begin{aligned}ϱ′_{i}(y) & = \frac{1}{m}χ_{f \overline{Ω}_{i}}(y) sup_{A} ∑_{x∈A} σ_{i}(x) = \frac{1}{m}χ_{f \overline{Ω}_{i}}(y) ∑_{μ∈L_{y}} σ_{i}(g_{μ}(y)) \\ & = \frac{1}{m} ∑_{μ∈L_{y}} σ_{i}(g_{μ}(y)) \end{aligned}

for y ∈ V . Furthermore, for L ⊂ J, the sets V_{L} = {y ∈ V : L_{y} = L} are pairwise disjoint Borel sets. By Hölder’s inequality,

ϱ′_{i}(y)^{n} ≤ \frac{1}{m} ∑_{μ∈L_{y}} σ_{i}(g_{μ}(y))^{n}.

Now

∫_{V_{L}} ϱ′_{i}(y)^{n} dm_{N}(y) ≤ \frac{1}{m} ∑_{μ∈L} ∫_{V_{L}}(σ_{i} ∘ g_{μ})^{n} dm_{N}.

By Lemma 8 we have the inequality

∫_{V_{L}} ϱ′_{i}(y)^{n} dm_{N}(y) ≤ \frac{1}{m} ∑_{μ∈V_{L}} ∫_{g_{μ}V_{L}} σ_{i}(x)^{n}J(x, f) dm_{M}(x).

Since J(x, f) > 0 for a.e. x ∈ g_{μ}(V_{L}),

K_{I}(x, f) = \frac{∥D^{#}f(x)∥^{n}}{J(x, f)^{n−1}} = \frac{J(x, f)}{min_{|v|=1} |Df(x)v|^{n}} ,

for a.e. x ∈ g_{μ}(V_{L}). Thus

∫_{V_{L}} ϱ′_{i}(y)^{n} dm_{N}(y) ≤ \frac{1}{m} ∑_{μ∈L} ∫_{g_{μ}V_{L}} ϱ_{i}(x)^{n}K_{I}(x, f) dm_{M}(x).

As in [15, pp. 51–52], we conclude that

∫_{N} ϱ′_{i}(y)^{n} dm_{N}(y) ≤ \frac{1}{m} ∫_{M} ϱ_{i}(x)^{n}K_{I}(x, f) dm_{M}(x).

Since (ϱ_{i}) and (ϱ′_{i}) are increasing sequences tending to ϱ and ϱ′, respectively, we have

∫_{N} ϱ′(y)^{n} dm_{N}(y) ≤ \frac{1}{m} ∫_{M} ϱ(x)^{n}K_{I}(x, f) dm_{M}(x).

Thus M_{n}(Γ′) ≤ M_{n,K_{I}( · ,f)}(Γ)/m.

As a direct consequence of Väisälä’s inequality we obtain a version of Poletsky’s inequality.

Corollary 10 (Poletsky’s inequality). Let f: M → N be a continuous mapping of finite distortion K satisfying the condition (A). Let Γ be a path family on M. Then

M_{n}(fΓ) ≤ M_{n,K^{n−1}( ·)}(Γ).

4. Parabolic manifolds and Global homeomorphism theorem

Let X and Y be topological spaces. As in the introduction we denote by

Γ_{X}^{∞} = {γ : |γ| ⊄ C for any compact C ⊂ X}

the family of all paths γ in X tending to infinity. If A ⊂ X and f: A → Y , we say that a point y ∈ Y is an asymptotic limit of f at the infinity of X if there exists a path γ: [a, b[→ A belonging to the family Γ_{X}^{∞} such that lim_{t→b} f ∘ γ(t) = y. We denote by E(f; X) the set of asymptotic limits of f at the infinity of X. If A = X, we omit the phrase “at the infinity of X ”, and write E(f) = E(f; X).

As an application of Poletsky’s inequality we prove the Global homeomorphism theorem (Theorem 1). In the proof we employ Propositions 2 and 3 that were announced in the introduction. The proofs of these propositions are presented in the following Subsections 4.1 and 4.2.

4.1. The proof of Proposition 2. The proof of Proposition 2 is based on the Euclidean localization method presented in [19]. We formulate this localization as follows.

Proposition 11. Let M, N, and f: M → N be as in Proposition 2. Suppose that q_{0} ∈ fM and (U, φ) is a chart at q_{0} such that φU contains the closed unit ball \overline{B}^{n} and φ(q_{0}) = 0. Then for every p_{0} ∈ f^{−1}(q_{0}) the set E(φ∘f | V; M)∩B^{n} has zero n-capacity, where V is the p_{0} -component of the set (φ ∘ f)^{−1}B^{n}.

The proof of Proposition 11 is divided into the following lemmata. For any sets C_{1} and C_{2} in a topological space D we denote by Δ(C_{1}, C_{2}; D) the family of all paths γ in D such that |γ| ∩ C_{i} ≠ ∅, i = 1, 2.

Lemma 12 ([19, Lemma 2]). Let n ≥ 3, T ⊂ R be measurable, r: T → R be a positive bounded measurable function, and x_{t} ∈ S^{n−2}(0, r(t))×{t} for every t ∈ T. Then, for every R > 0,

M_{n}( ⋃_{t∈T} Δ(\{(0, t)\}, \{x_{t}\}; B^{n−1}(0, r(t)) × \{t\}) ≥ c_{n} \frac{ℋ ^{1}(T_{R})}{R} ,

where T_{R} = r^{−1}(]0, R]) and c_{n} is a constant depending only on n.

Proof. Let us denote Γ_{t} = Δ({(0, t)}, {x_{t}}; B^{n−1}(0, r(t)) × {t}) for every t ∈ E and Γ′ = Δ({0}, {e_{1}}; B^{n−1} × {0}). It is well known that M_{n}(Γ′; R^{n−1} × {0}) > 0, where the n-modulus is taken with respect to the (n − 1)-dimensional Lebesgue measure of R^{n−1} × {0} ⊂ R^{n}. Using the rotational symmetry and the dilatation

(x_{1}, …, x_{n−1}) ↦ (\frac{x_{1}}{r(t)}, …, \frac{x_{n−1}}{r(t)} ), (x_{1}, …, x_{n−1}) ∈ R^{n−1},

we obtain

M_{n}(Γ_{t}; R^{n−1} × \{t\}) = \frac{M_{n}(Γ′; R^{n−1} × \{0\})}{r(t)}

for every t ∈ T. We set c_{n} = M_{n}(Γ′; R^{n−1} × {0}).

Let ϱ be an admissible function for the path family ⋃_{t∈E} Γ_{t}. Then

∫_{R^{n}} ϱ^{n} dx ≥ ∫_{T} (∫_{R^{n−1}} ϱ^{n} dy) dt ≥ ∫_{T} \frac{c_{n}}{r(t)} dt ≥ c_{n} \frac{ℋ ^{1}(T_{R})}{R} .

For the proof of Proposition 11 let us introduce some notation. We denote h = φ ∘ f | V: V → φU. For every y ∈ S^{n−1} , define γ_{y}: [0, 1] → \overline{B}^{n} by γ_{y}(t) = ty, and let γ̃_{y} be the maximal h-lifting of γ_{y} starting at p_{0} . For each y ∈ S^{n−1} , let λ(y) be the length of the maximal interval where γ̃_{y} is defined, that is, the maximal interval is either [0, λ(y)[ or [0, λ(y)]. Furthermore, we denote

E = {y ∈ S^{n−1} : λ(y) < 1},
C = {λ(y)y ∈ B^{n} : y ∈ E},
F = \{ty ∈ \overline{B}^{n} : λ(y) ≤ t ≤ 1, y ∈ E\},
G = B^{n} \ F,

and

Γ = {γ_{y} : y ∈ E}.

Finally, we denote by G′ the p_{0} -component of h^{−1}G.

Lemma 13. The set G is star-shaped and open, and E ⊂ S^{n−1} is closed.

Proof. Clearly G is star-shaped since

G = {ty ∈ B^{n} : 0 ≤ t < λ(y), y ∈ S^{n−1}}.

The openness of G follows from the fact that h is a local homeomorphism and therefore homeomorphic in a neighborhood of γ̃_{y}([0, c]), where c < λ(y) for every y; see [18, Remark 1]. The same reasoning shows that S^{n−1} \ E is open in S^{n−1} , and hence E is closed.

The following lemmata are discussed in [19, Section 3]; see also [13]. We say that a subset A of a topological space X is relatively locally connected at x ∈ Ā if every neighborhood U of x contains a neighborhood W of x such that W ∩ A is connected.

Lemma 14. (a) The mapping h | G′: G′ → G is a homeomorphism.

(b) Let x ∈ (∂G′) ∩ V be such that G is relatively locally connected at h(x) ∈ ∂G. Then there are arbitrary small neighborhoods W of x such that h(W ∩ G′) = hW ∩ G and h | W: W → hW is a homeomorphism.

(c) If G is relatively locally connected at q ∈ B^{n} ∩ ∂G, then for every pair of paths α, β: [a, b] → G ∪ {q} such that α([a, b[) ⊂ G, β([a, b[) ⊂ G, α(a) = β(a) = 0, and that α(b) = β(b) = q, the path α has a total h-lifting starting at p_{0} if and only if the path β has a total h-lifting starting at p_{0} . Moreover, if α̃ and β̃ are the total h-liftings of α and β, respectively, then α̃(b) = β̃(b).

(d) If q ∈ C and γ: [a, b] → G ∪ {q} is a path such that γ([a, b[) ⊂ G, γ(a) = 0, and γ(b) = q, then the maximal h-lifting of γ starting at p_{0} belongs to the path family Γ_{M}^{∞}.

Proof. (a) The formula x ↦ γ̃_{y}(t), where x = ty ∈ G, with t ≥ 0 and y ∈ S^{n−1} , gives the inverse mapping. Since h is a local homeomorphism, the continuity of the inverse mapping follows.

(b) It follows from the assumptions that there exists a neighborhood W of x such that h | W: W → hW is a homeomorphism and hW ∩ G is connected. Furthermore, W can be chosen to be arbitrarily small. Since x ∈ ∂G′, the set W ∩ G′ is non-empty, and therefore G′ ∩ (h | W)^{−1}(hW ∩ G) ≠ ∅. Now (h | W)^{−1}(hW ∩ G) ⊂ G′ since G′ is the p_{0} -component of h^{−1}G and (h | W)^{−1}(hW ∩ G) is connected. Thus hW ∩ G = h((h | W)^{−1}(hW ∩ G) ∩ W ) ⊂ h(W ∩ G′). The inclusion h(W ∩ G′) ⊂ hW ∩ G holds trivially.

(c) It is sufficient to show that if α has the total h-lifting α̃ starting at p_{0} , then the maximal h-lifting of β, say β̃, starting at p_{0} is total and β̃(b) = α̃(b).

First we observe that β̃ is given on [a, b[ by β̃ | [a, b[= (h | G′)^{−1} ∘ β | [a, b[ since h | G′: G′ → G is a homeomorphism. By (b), there exists a neighborhood W of α̃(b) such that h(W ∩ G′) = hW ∩ G. Thus there exists c ∈]a, b[ such that β([c, b[) ⊂ hW ∩ G = h(W ∩ G′) since hW is a neighborhood of q. It follows that β̃([c, b[) ⊂ W ∩ G′. Since W can be chosen to be arbitrarily small, β̃(t) → α̃(b) as t → b. Hence β̃ is total and β̃(b) = α̃(b).

(d) Let γ: [a, b] → G∪{q} be as in the claim and let γ̃ be the maximal h-lifting γ̃ of γ starting at p_{0} . Suppose that γ̃ ∉ Γ_{M}^{∞}. Then γ̃ is total and γ̃(b) ∈ ∂G′. Let z = q/|q|. We may assume that a = 0 and b = λ(z). Let W be a neighborhood of γ̃(b) such that h | W: W → hW is a homeomorphism and hW is an open ball around q. Then there exists c ∈]0, b[ such that γ_{z}([c, b[) ⊂ hW ∩ G. Since G is open, there exists, for every t ∈ [c, b[, an open ball B(γ_{z}(t), r_{t}) ⊂ hW ∩ G. Furthermore, since G is star-shaped with respect to 0, there exists b_{t} ∈]0, b[ such that the ray between 0 and γ(t′) intersects the ball B(γ_{z}(t), r_{t}) for all t′ ∈ [b_{t}, b[. Thus we can connect the points γ(t′) and γ_{z}(t) by a path in hW ∩ G. Since W can be chosen to be an arbitrarily small neighborhood of γ̃(b), we conclude that lim_{t→b} γ̃_{z}(t) = γ̃(b). This leads to a contradiction since γ̃_{z} ∈ Γ_{M}^{∞}. Hence γ̃ ∈ Γ_{M}^{∞}.

Lemma 15. The topological dimension of E is zero and ℋ ^{n−1}(E) = 0. Furthermore,

(5)  G ⊂ hV ⊂ \overline{G} = \overline{B}^{n}

and

(6)  G′ ⊂ V ⊂ \overline{G}′.

Proof. Suppose that E contains a non-degenerate continuum K. By applying an additional rotation, we may assume that there exists a ∈]0, 1[ such that K ∩ (R^{n−1} × {t}) ≠ ∅ for every t ∈ [−a, a]. Let δ ∈]0, 1[ be such that B^{n}(0, δ) ⊂ G, let

D = {(z_{1}, …, z_{n}) ∈ B^{n} : |z_{n}|/|z| < a or |z_{n}| < δa}

and let g: D → B^{n} be the mapping that is the identity in D ∩ B^{n}(δ) and g(z_{1}, …, z_{n}) = (z_{1}, …, z_{n−1}, δz_{n}/|z|) in D \ B^{n}(δ). Then g is L-bilipschitz and for every z ∈ K it maps the ray between λ(z)z and z to the ray between (λ(z)z_{1}, …, λ(z)z_{n−1}, δz_{n}/|z|) and (z_{1}, …, z_{n−1}, δz_{n}/|z|).

Define r: [−δa, δa] → [δ, 1] by

r(t) = sup{s: B^{n−1}(0, s) × {t} ⊂ g(G ∩ D)}.

By the openness of G, r is lower semi-continuous. Thus r is a measurable function and for every t ∈ [δa, δa] there exists x_{t} ∈ S^{n−2}(0, r(t)) × {t} such that x_{t} ∈ g(D ∩ C).

We set

Γ = ⋃_{t∈[−δa,δa]} Δ({(0, t)}, {x_{t}}; B^{n−1}(0, r(t)) × {t}).

By Lemma 12,

M_{n}(Γ) ≥ 2c_{n}δa,

where c_{n} is a positive constant depending only on n. On the other hand,

M_{n}(Γ) ≤ M_{n}((g ∘ h)Γ_{M}^{∞})

since Γ ⊂ (g ∘h)Γ_{M}^{∞} by Lemma 14. Furthermore, since g is L-bilipschitz, we have

M_{n}((g ∘ h)Γ_{M}^{∞}) ≤ L^{2n}M_{n}(hΓ_{M}^{∞}).

Finally, applying Poletsky’s inequality (Corollary 10) we obtain

M_{n}(hΓ_{M}^{∞}) ≤ M_{n,K^{n−1}( ·)}(Γ_{M}^{∞}) = 0

since M is K^{n−1}( ·)-parabolic. Hence M_{n}(Γ) = 0 which is a contradiction. Thus the components of E are singletons.

To prove that ℋ ^{n−1}(E) = 0, consider the path family Γ = {γ_{z}|[δ, 1] : z ∈ E}. Then by [17, 7.5] and Poletsky’s inequality,

ℋ ^{n−1}(E) = (log \frac{1}{δ})^{n−1}M_{n}(Γ) ≤ (log \frac{1}{δ})^{n−1}M_{n,K^{n−1}}(Γ_{M}^{∞}) = 0.

Clearly G ⊂ hV ⊂ \overline{B}^{n} and \overline{G} ⊂ \overline{B}^{n}. Thus (5) follows if we show that \overline{B}^{n} ⊂ \overline{G}. Suppose that there exists a point q ∈ \overline{B}^{n} \ \overline{G}. Then there is an open ball B^{n}(q, r) ⊂ R^{n} \ \overline{G}. In particular, B^{n}(q/|q|, r) ∩ S^{n−1} ⊂ E, which is a contradiction since E is totally disconnected.

The inclusion G′ ⊂ V in (6) is clear, so it remains to show that V ⊂ \overline{G}′. Suppose that there is a point x ∈ V \\overline{G}′. Then x has a neighborhood U ⊂ V \\overline{G}′. Since h is an open mapping, hU ⊂ B^{n} is open. Therefore hU ∩ G ≠ ∅ since \overline{G} = \overline{B}^{n}. This is a contradiction.

Proof of Proposition 11. Here we use the notation introduced before Lemma 13. By Lemma 15, the topological dimension of E is zero, and therefore E does not locally separate the sphere S^{n−1} , see [3, Corollary IV.4]. Hence every point x ∈ S^{n−1} has arbitrary small neighborhoods U ⊂ S^{n−1} of x such that each (S^{n−1} \ E) ∩ U is path-connected. Recall that E is closed. Thus G is relatively locally connected at every point q ∈ \overline{B}^{n}. Furthermore, for every q ∈ \overline{B}^{n}, there exists a path γ: [0, 1] → G ∪ {q} such that γ([0, 1[) ⊂ G, γ(0) = 0, and γ(1) = q. Such a path can be constructed piece-wise by connecting points q_{i} ∈ W_{i} ∩ G and q_{i+1} ∈ W_{i+1} ∩ G by a path in W_{i} ∩ G, where (W_{i}) is a nested sequence of neighborhoods of q shrinking to q such that each W_{i} ∩ G is connected.

Let Ω ⊂ B^{n} be the set of all points y ∈ B^{n} such that, for every path γ: [0, 1] → G ∪ {y}, with γ([0, 1[) ⊂ G, γ(0) = 0, and γ(1) = y, the unique maximal h-lifting γ̃ starting at p_{0} is total. In particular, γ̃(1) ∈ V is defined, and therefore y = h(γ̃(1)) ∈ hV . We claim that Ω = hV . It suffices to prove that hV \ G ⊂ Ω. Fix y ∈ hV \ G and let γ: [0, 1] → G ∪ {y} be a path such that γ([0, 1[) ⊂ G, γ(0) = 0, and γ(1) = y. As in the proof of Lemma 14(c), the maximal h-lifting of γ, say γ̃, starting at p_{0} is given on [0, 1[ by γ̃ | [0, 1[= (h | G′)^{−1} ∘ γ | [0, 1[. Let x ∈ h^{−1}(y). Then x ∈ ∂G′ since h is an open mapping. Let W be a neighborhood of x given by Lemma 14(b). Then γ([c, 1[) ⊂ hW ∩G = h(W ∩ G′) for some c ∈]0, 1[. As in the proof of Lemma 14(c), we conclude that γ̃ is a total h-lifting of γ starting at p_{0} . Hence y ∈ Ω and therefore Ω = hV .

Next we show that E(h; M) ∩ B^{n} = B^{n} \ hV which then implies that E(h; M) ∩ B^{n} is σ -compact. Since hV = Ω, the inclusion E(h; M) ∩ B^{n} ⊂ B^{n} \ hV follows immediately from the definition of Ω. On the other hand, for every point y ∈ B^{n} \Ω, there exists a path γ: [0, 1] → G∪{y}, with γ([0, 1[) ⊂ G, γ(0) = 0, and γ(1) = y, whose maximal h-lifting γ̃ starting at p_{0} is not total. If γ̃ ∉ Γ_{M}^{∞}, there is an increasing sequence of positive numbers t_{i} such that t_{i} → 1 and γ̃(t_{i}) → x ∈ M. Since f is a local homeomorphism at x and γ(t_{i}) → y ∈ B^{n}, there is a connected neighborhood U of x such that (φ∘f)U ⊂ B^{n}. Hence x ∈ V and, furthermore, y ∈ hV = Ω. We conclude that γ̃ ∈ Γ_{M}^{∞}, and so y ∈ E(h; M). We have proved that E(h; M) ∩ B^{n} is σ -compact.

To show that E(h; M) ∩ B^{n} has zero n-capacity, we study the family Γ_{0} of all paths γ such that |γ| ∩ E(h; M) ∩ B^{n} ≠ ∅. Since E(h; M) ∩ B^{n} ⊂ F and the set F has zero measure by Fubini’s theorem and Lemma 15, it is sufficient to consider only paths γ satisfying |γ| ∩ G ≠ ∅. Let γ ∈ Γ_{0} . We may assume that γ is defined on the unit interval and γ(1) ∈ E(h; M). We show that γ ∈ hΓ_{M}^{∞}. If |γ| ∩ F = {γ(1)}, we have by Lemma 14(c) that γ ∈ hΓ_{M}^{∞}. Assume that |γ| ∩ F ≠ {γ(1)}. If γ ∉ hΓ_{M}^{∞}, it has a total h-lifting, say γ̃, starting at h^{−1}(γ(0)). Let {W_{i}} be a finite open cover of |γ̃| such that h | W_{i} is a homeomorphism, and hW_{i} ∩ G is connected. Then there is a path β: [0, 1] → B^{n} such that β(0) = γ(0), β([0, 1[) ⊂ (⋃_{i} hW_{i}) ∩ G, and β(1) = γ(1). Then β has a total h-lifting starting at h^{−1}(γ(0)). This contradicts Lemma 14(c), since β(1) = γ(1) ∈ E(h; M) ∩ B^{n}. Thus the maximal h-lifting of γ is not total and γ ∈ hΓ_{M}^{∞}. By Poletsky’s inequality, M_{n}(Γ_{0}) = 0, and so E(h; M) ∩ B^{n} has zero n-capacity.

In the proof of Proposition 2 we apply the following lemma.

Lemma 16 ([11, 2.3]). Let M be a Riemannian manifold, C ⊂ M compact and L > 1. Then there exists R = R(C, L) > 0 such that for every x ∈ C there exist a neighborhood V of x and an L-bilipschitz diffeomorphism ψ: V → B^{n}(R) with ψ(x) = 0.

Proof of Proposition 2. Let (Ω_{i}) be an increasing sequence of relatively compact open subsets of N and, for every i, let {q_{i,j}} be a countable dense subset of Ω_{i}. Since f is a local homeomorphism, f^{−1}({q_{i,j}}) is a countable dense subset of M.

For every i, let R_{i} > 0 be such that there exists a 2-bilipschitz diffeomorphism ψ_{i,j}: V_{i,j} → B^{n}(R_{i}), where V_{i,j} is a neighborhood of q_{i,j} as in Lemma 16.

Suppose that γ̃: [a, b[→ M belongs to the family Γ_{M}^{∞} and that the limit lim_{t→b}(f ∘ γ̃)(t) = q ∈ N exists. Let γ = f ∘ γ̃. Then, by the density of {q_{i,j}}, there exists c ∈]a, b[ such that γ([c, b[) ⊂ V_{i,j} for some i and j. Thus, for some i, j, and p ∈ f^{−1}(q_{i,j}), we have γ̃([c, b[) ⊂ Ṽ_{i,j}, where Ṽ_{i,j} is the p-component of f^{−1}(V_{i,j}). It follows that q ∈ E(f | Ṽ_{i,j}; M).

Since ψ_{i,j}^{−1}(E(ψ_{i,j} ∘ f | Ṽ_{i,j}; M) ∩ B^{n}(R_{i})) ⊂ E(f), we have

E(f) = ⋃_{i,j} ψ_{i,j}^{−1}(E(ψ_{i,j} ∘ f | Ṽ_{i,j}; M) ∩ B^{n}(R_{i})).

Therefore E(f) is σ -compact and has zero n-capacity since it is a countable union of σ -compact sets of zero n-capacity by Proposition 11.

It remains to prove that N \ fM ⊂ E(f). We first show that ∂(N \ fM) ⊂ E(f). Let q ∈ ∂(N \ fM). Since f is an open mapping, q ∉ fM. Choose i and j such that q ∈ V_{i,j}. Fix p ∈ f^{−1}(q_{i,j}) and let Ṽ_{i,j} be the p-component of f^{−1}V_{i,j} and h_{i,j} = (ψ_{i,j} ∘ f) | Ṽ_{i,j}. Let γ: [0, 1] → B^{n}(R_{i}) be a path such that γ(0) = 0 and γ(1) = ψ_{i,j}(q). Then the maximal h_{i,j} -lifting of γ is not total since ψ_{i,j}(q) ∉ h_{i,j}Ṽ_{i,j}. Thus ψ_{i,j}(q) ∈ E(h_{i,j}; M), and so q ∈ E(f). Since E(f) has zero n-capacity, the topological dimension of ∂(N \ fM) is zero. Thus N \ fM has no interior points and N \ fM = ∂(N \ fM) ⊂ E(f).

4.2. The proof of Proposition 3. Let X be a compact topological space. We define the distance of continuous mappings F: X → R^{n} and H: X → R^{n} by

d(F, H) = max_{x∈X} |F (x) − H(x)|.

In the proof of Proposition 3 we use the method introduced in [13]. In order to obtain the necessary approximation results, we prove a quantitative version of [13, Lemma 3.3] for all homotopy groups in question. The closed unit interval is denoted by I in this section.

Proposition 17. Let n, k, and m be integers such that 1 ≤ m ≤ k ≤ n. Let Ω ⊂ R^{n} be a domain, E ⊂ Ω be a compact set such that ℋ ^{n−k}(E) = 0 and let F: I^{m} → Ω and f: ∂I^{m} → Ω \ E be continuous mappings satisfying

(7)  tf(x) + (1 − t)F (x) ∈ Ω

for every x ∈ ∂I^{m}, and every t ∈ I.

Then, for every ε > 0, there exists a continuous mapping H: I^{m} → Ω \ E satisfying H | ∂I^{m} = f and

d(F, H) < d(f, F | ∂I^{m}) + ε.

The proof of Proposition 17 is based on the following lemma.

Lemma 18. Let A ⊂ R^{n} be such that ℋ ^{n−k}(A) = 0 and let P ⊂ R^{n} be a set that has a countable covering by k -dimensional hyperplanes. Then (P + y) ∩ A = ∅ for a.e. y ∈ R^{n}.

Proof. The proof is almost verbatim to the proof in the 2-dimensional case presented in [13, Lemma 3.2] and is omitted.

Proof of Proposition 17. We proceed using the proof of [13, Lemma 3.3] as our guideline. To avoid technical difficulties we prove the claim for continuous mappings F: J^{m} → Ω and f: ∂J^{m} → Ω \ E, satisfying the condition (7) for every y ∈ ∂J^{m} and t ∈ I, where J = [−1, 1].

Let us simplify the notation by introducing following auxiliary functions. Let ϱ be a metric in J^{m} given by the l_{∞} -norm in R^{m}, that is,

ϱ(x, y) = max{|x_{1} − y_{1}|, …, |x_{m} − y_{m}|}

for every x = (x_{1}, …, x_{m}) and y = (y_{1}, …, y_{m}) in J^{m}.

For every r ∈]0, 1[, we define λ_{r}: J^{m} → J^{m} by

λ_{r}(x) = \{ \begin{matrix}\frac{x}{r}, & ϱ(x, 0) ≤ r, \\ \frac{x}{ϱ(x, 0)}, & ϱ(x, 0) > r. \end{matrix}

Then λ_{r} maps the set rJ^{m} homeomorphically onto J^{m} and the set J^{m} \ rJ^{m} onto the boundary ∂J^{m} keeping the boundary fixed.

Let ε > 0. By continuity, there exists r > 0 such that the mapping F_{0}: J^{m} → R^{n},

F_{0}(x) = \{ \begin{matrix}F (λ_{r}(x)), & ϱ(x, 0) ≤ r, \\ (\frac{ϱ(x, 0) − r}{1 − r} )f(λ_{r}(x)) + (\frac{1 − ϱ(x, 0)}{1 − r})F (λ_{r}(x)), & ϱ(x, 0) > r, \end{matrix}

is well-defined and satisfies conditions F_{0} | ∂J^{m} = f and

(8)  d(F_{0}, F) < d(f, F | ∂J^{m}) + \frac{1}{4}ε.

Moreover, F_{0}(J^{m}) ⊂ Ω by (7).

Let δ = dist(f(∂J^{m}), E ∪ ∂Ω) > 0. Then there exist a triangulation of J^{m} and a simplicial approximation of F_{0} , say F_{1} , with respect to the triangulation such that

(9)  d(F_{1}, F_{0}) < min\{δ, \frac{1}{4}ε\}.
By Lemma 18, there exists y ∈ R^{n} such that F_{1}(J^{m}) + y ⊂ Ω \ E and
(10)  |y| < min\{δ − d(F_{1}, F_{0}), \frac{1}{4}ε\}.

Define F_{2}: J^{m} → Ω \ E by F_{2}(x) = F_{1}(x) + y. Then

tf(x) + (1 − t)F_{2}(x) ∈ Ω \ E

for every x ∈ ∂J^{m} and t ∈ I. Thus there exists r′ ∈]0, 1[ such that the continuous mapping H: J^{m} → Ω \ E,

H(x) = \{ \begin{matrix}F_{2}(λ_{r′}(x)), & ϱ(x, 0) ≤ r′, \\ (\frac{ϱ(x, 0) − r}{1 − r} )f(λ_{r′}(x)) + (\frac{1 − ϱ(x, 0)}{1 − r})F_{2}(λ_{r′}(x)), & ϱ(x, 0) > r′, \end{matrix}

is well-defined and satisfies conditions H|∂J^{m} = f and

(11)  d(H, F_{2}) < \frac{1}{4}ε.

The claim follows now from estimates (8), (9), (10), and (11).

Proposition 19. Let n, k, and m be integers such that 1 ≤ m ≤ k ≤ n. Let Ω ⊂ R^{n} be a domain, E ⊂ Ω be a σ -compact set such that ℋ ^{n−k}(E) = 0, and let F: I^{m} → Ω and f: ∂I^{m} → Ω \ E be continuous mappings satisfying (7) for every x ∈ ∂I^{m} and t ∈ I.

Then for every ε > 0 there exists a continuous mapping H: I^{m} → Ω \ E such that H | ∂I^{m} = f and d(H, F) < d(f, F | ∂I^{m}) + ε.

Proof. Let ε > 0 and let E_{0} ⊂ E_{1} ⊂ … ⊂ E be an increasing sequence of compact subsets exhausting E. Thus, using Proposition 17 iteratively, there exist mappings F_{i}: I^{m} → Ω \ E_{i}, i = 0, 1, …, such that F_{i} | ∂I^{m} = f and

d(F_{i+1}, F_{i}) < min\{δ_{0}, …, δ_{i}, \frac{1}{2}ε\}/2^{i},

where δ_{i} = dist(F_{i}(I^{m}), E_{i} ∪ (R^{n} \ Ω)) > 0. Moreover, we may assume that d(F, F_{0}) < d(f, F | ∂I^{m}) + \frac{1}{2}ε.

Since (F_{i}) is a uniformly convergent sequence of continuous mappings, the limit mapping H: I^{m} → R^{n}, H(x) = lim_{i→∞} F_{i}(x), is continuous. Since H | ∂I^{m} = f and d(H, F) < d(f, F | ∂I^{m}) + ε, it suffices to show that H(I^{m}) ⊂ Ω \ E.

For this it is sufficient to show that dist(H(I^{m}), E_{k} ∪ ∂Ω)) > 0 for every k. Fix k ≥ 0. By the triangle inequality, we have

δ_{k} = dist(F_{k}(I^{m}), E_{k} ∪ ∂Ω) ≤ d(F_{k}, F_{i}) + dist(F_{i}(I^{m}), E_{k} ∪ ∂Ω)

for every i ≥ k. Since d(F_{k}, F_{i}) ≤ \frac{1}{2}δ_{k},

dist(F_{i}(I^{m}), E_{k} ∪ ∂Ω) ≥ \frac{1}{2}δ_{k} > 0,

for every i ≥ k. Thus dist(H(I^{m}), E_{k} ∪ ∂Ω) > 0 for every k.

Proposition 20. Let N be a Riemannian n-manifold and E ⊂ N a σ - compact subset of N such that ℋ ^{n−k}(E) = 0 for some integer k ∈ {1, …, n}. Let 1 ≤ m ≤ k and F: I^{m} → N be a continuous mapping such that F (∂I^{m}) ⊂ N \ E. Then there exists a continuous mapping H: I^{m} → N \ E such that H | ∂I^{m} = F | ∂I^{m}. Moreover, given ε > 0, the mapping H can be chosen such that d(H, F) < ε.

Proof. Let us introduce some notation. For given integer l ≥ 1, let

V_{l} = \{\frac{1}{l}(v_{1}, …, v_{m}) ∈ I^{m} : v_{i} ∈ \{0, …, l\} for all i\} ⊂ \frac{1}{l}Z^{m}.

We say that vertices v and w of V_{l} are neighbors in V_{l} if |v − w| = 1/l. If v and w are neighbors in V_{l}, we define

[v, w] = {tv + (1 − t)w : t ∈ [0, 1]}.

If w_{1}, …, w_{j} are neighbors of v in V_{l} for some j ≥ 2, we define

(12)  [v, w_{1}, …, w_{j}] = [v, w_{1}] × ⋯ × [v, w_{j}].

For completeness we define [v] = {v} for every v ∈ V_{l}. The sets [v, w_{1}, …, w_{j}] defined in (12) are called j -faces. We denote by C_{j}^{l} the set of all j -faces in I^{m}. The set S_{j}^{l} = ∪{v : v ∈ C_{j}^{l}} is called the j -skeleton of I^{m}. The boundary of an j -face v with respect to the j -skeleton S_{j}^{l} is denoted by ∂_{s}v.

We construct a sequence F_{0}, …, F_{m} of mappings such that F_{m}(I^{m}) ⊂ N \ E and F_{m} | ∂I^{m} = F | ∂I^{m}. By the compactness of F (I^{m}) there exists a finite collection {(U_{α}, φ_{α})} of L-bilipschitz charts covering F (I^{m}). Fix an integer l > 0 such that for every v ∈ C_{m}^{l} there is α such that F (v) ⊂ U_{α}. Let δ > 0 be such that dist(F (v), N \ U_{α}) > δ if v ∈ C_{j}^{l} is contained in U_{α}. Since the collection of all faces is finite such δ exists. If ε > 0 is given, we assume that δ < ε.

Let us construct the mapping F_{0}: ∂I^{m} ∪ S_{0}^{l} → N \ E. Since ℋ ^{n}(E) = 0, there exists x_{v} ∈ B(F (v), δ) \ E for every vertex v ∈ V_{l}∩]0, 1[^{m}. We define F_{0} using formulas F_{0} | ∂I^{m} = F | ∂I^{m} and F_{0}(v) = x_{v} for every v ∈ V_{l}∩]0, 1[^{m}.

Suppose that for some j ≤ m − 1 we have constructed continuous mappings F_{0}, …, F_{j} satisfying properties: F_{i}: ∂I^{m} ∪ S_{i}^{l} → N \ E, F_{i} | (∂I^{m} ∪ S_{i−1}^{l}) = F_{i−1} , and d(F | v, F_{i} | v) < δ for every i-face v.

Next we construct the mapping F_{j+1} . We define F_{j+1} | ∂I^{m} = F | ∂I^{m}. For every v ∈ C_{j+1}^{l} not contained in ∂I^{m} we do the following construction. Let Ω_{v} be the δ -neighborhood of F (v), that is,

Ω_{v} = {q ∈ N : dist(q, F (v)) < δ}.

Then Ω_{v} ⊂ U_{α} for every U_{α} containing F (v). Fix one such α. Then, by Lemma 19, there exists a continuous mapping F̃_{v}: v → φ_{α}(Ω_{v} \ E) such that

F̃_{v} | ∂_{s}v = φ_{α} ∘ F_{j} | ∂_{s}v

and

d(F̃_{v}, φ_{α} ∘ F | v) < d(φ_{α} ∘ F_{j} | ∂_{s}v, φ_{α} ∘ F | ∂_{s}v)
+ (δ/L − d(φ_{α} ∘ F_{j} | ∂_{s}v, φ_{α} ∘ F | ∂_{s}v))
= δ/L.

We define F_{j+1} | v = φ_{α}^{−1} ∘ F̃_{v}. The mapping F_{j+1}: ∂I^{m} ∪ S_{j+1}^{l} → N \ E is welldefined and continuous. Moreover, F_{j+1} | ∂I^{m} ∪ S_{j}^{l} = F_{j}, and d(F | v, F_{j+1} | v) < δ for every v ∈ C_{j+1}^{l}. This concludes the induction step and the proof.

Proof of Proposition 3. Let us first consider the injectivity of the induced mapping π_{m}(N \E) → π_{m}(N). Let f and h be continuous mappings I^{m} → N \E such that f(∂I^{m}) = h(∂I^{m}) = {q} for some q ∈ N \ E and that there exists a homotopy F: I^{m} × I → N such that F : f ≃ h rel ∂I^{m}. Since m + 1 ≤ k, there exists by Proposition 20 a continuous mapping H: I^{m+1} → N \ E such that H | ∂I^{m+1} = F | ∂I^{m+1} . Thus H is a homotopy f ≃ h rel ∂I^{m} in N \ E. Hence the induced mapping π_{m}(N \ E) → π_{m}(N) is injective.

Let us show the surjectivity of the induced mapping. Let f: I^{m} → N be a representative of some homotopy class of π_{m}(N). We may assume that f(∂I^{m}) = {q}, where q ∈ N \ E. We have to show that there exists h: I^{m} → N \ E such that h ≃ f rel ∂I^{m} in N.

Let {(U_{α}, φ_{α})} be a finite collection of L-bilipschitz charts covering f(I^{m}). Let C_{j}^{l} and S_{j}^{l} be as in the proof of Proposition 20 for every l ≥ 1 and 0 ≤ j ≤ m. We fix l and choose δ > 0 as in the proof of Proposition 20 as follows. Let l ≥ 1 be such that for every v ∈ C_{m}^{l} there exists α such that f(v) ⊂ U_{α}. Let δ > 0 be such that for all 0 ≤ j ≤ m and v ∈ C_{j}^{l}, and for all β such that f(v) ⊂ U_{β}, we have d(f(v), N \ U_{β}) > δ/L^{2} . By Proposition 20, there exists a continuous mapping h: I^{m} → N \ E such that h | ∂I^{m} = f | ∂I^{m} and d(h, f) < δ/L^{2} .

For every 0 ≤ j ≤ m and every v ∈ C_{j}^{l}, we construct a mapping F_{v}: v×I → N as follows. Let α be such that f(v) ⊂ U_{α}. It follows from d(f, h) < δ/L^{2} that h(v) ⊂ U_{α} and the mapping F_{v}: v × I → N,

(x, t) ↦ φ_{α}^{−1}(tφ_{α}(h(x)) + (1 − t)φ_{α}(f(x))),

is well-defined and continuous. Moreover, it is a homotopy f | v ≃ h | v in U_{α}.

By construction, F_{v} | v × {0, 1} = F_{w} | w × {0, 1} for every v ∈ C_{j}^{l} and w ∈ C_{j−1}^{l} such that w ⊂ ∂_{s}v. Moreover, since d(F_{v} | w, F_{w}) < δ for such v and w, we may assume that F_{v} | w = F_{w}. Indeed, since F_{v}(v × I) and F_{w}(w × I) are contained in the same chart U_{α} for some α, we may use the same technique as in the proof of Proposition 17 to modify the boundary values of the mapping F_{v}. Since the mappings F_{v} and F_{v′} agree on the set v ∩ v′ for every v ∈ C_{j}^{l}, we may define a continuous mapping H: I^{m} × I → N by H | (v × I) = F_{v}. The mapping H is a homotopy f ≃ h rel ∂I^{m} in N.

4.3. The proof of the Global homeomorphism theorem. The last observation used in the proof of the Global homeomorphism theorem is the following homotopy lifting lemma.

Lemma 21. Let X, Y , and Z be topological spaces and let f: X → Y be a local homeomorphism. Then for every pair of continuous functions h: Z → X and F: Z × I → Y \ E(f) satisfying

(1) (f ∘ h)Z ⊂ Y \ E(f), and

(2) F | (Z × {0}) = f ∘ h there exists a continuous function H: Z × I → X such that f ∘ H = F and H | (Z × {0}) = h.

Proof. For every z ∈ Z there exists a unique total f -lifting of the path t ↦ F (z, t) starting at h(z), say γ_{z}. Thus we can define H(z, t) = γ_{z}(t) for every (z, t) ∈ Z ×I. Since f is a local homeomorphism, the mapping H is continuous.

Proof of the Global homeomorphism theorem. By Proposition 2, the set N \ fM has zero n-capacity. Thus it is sufficient to consider only the injectivity of f. We construct a certain mapping g: fM \ E(f) → M by using Lemma 21 and show that it is the inverse of f.

By Propositions 2 and 3, fM \ E(f) = N \ E(f) is non-empty and simply connected. Fix q ∈ fM \ E(f) and p ∈ f^{−1}(q). Let α: I → fM \ E(f) and β: I → fM \ E(f) be any paths such that α(0) = β(0) = q and α(1) = β(1). Both paths have total f -liftings, denoted by α̃ and β̃, starting at p. Since fM \ E(f) is simply connected, there exists a homotopy F: I^{2} → fM \ E(f) such that F: α ≃ β rel {0, 1}. By Lemma 21, there exists a homotopy H: I^{2} → M such that H | I × {0} = α̃ and f ∘ H = F. By the uniqueness of path lifting, H: α̃ ≃ β̃. The path I → M, t ↦ H(1, t), is a constant path since it is an f -lifting of the constant path I → N, t ↦ α(1), and f is a local homeomorphism. Thus β̃(1) = α̃(1). We define g: fM \ E(f) → M by g(y) = α̃(1), where α̃: I → M is an f -lifting starting at p of some path α: I → fM \ E(f), with α(0) = q and α(1) = y. By the argument above, the mapping g is well-defined. Let us denote X = g(fM \ E(f)). Clearly g is the inverse of f | X. Thus g is a local homeomorphism and f | X is injective.

We show that \overline{X} is both open and closed. Fix a point x ∈ \overline{X} . Let U be a connected neighborhood of x such that f | U is a homeomorphism. We show that U ⊂ \overline{X} . Since E(f) has zero n-capacity and f is an open mapping, the set f^{−1}E(f) has no interior points. Thus it is sufficient to show that U \ f^{−1}E(f) ⊂ X. By Proposition 3, fU \ E(f) is path-connected. Since U ∩ X ≠ ∅, there exists a point x′ ∈ U ∩ X. Let x′′ ∈ U \ f^{−1}E(f). Then there exists a path γ: I → fU \ E(f) such that γ(0) = f(x′) and γ(1) = f(x′′). Observe that f(x′) ∈ fU \ E(f), since x′ ∈ U ∩ X. Thus γ̃ = (f | U)^{−1} ∘ γ is an f -lifting of γ starting at x′. Since the f -lifting of γ is total, there exists a path connecting q and f(x′′) such that it has a total f -lifting from p to x′′. Therefore x′′ ∈ X. Thus U \ f^{−1}E(f) ⊂ X. Hence the set \overline{X} is both open and closed. Since M is connected, M = \overline{X} .

We show that f is injective on M. Suppose that there exist two points x and x′ in M such that f(x) = f(x′). Let U and U′ be disjoint neighborhoods of x and x′. Then there exist points z ∈ U ∩ X and z′ ∈ U′ ∩ X such that f(z) = f(z′). This is a contradiction with the injectivity of f | X. Thus the mapping f is injective.

80 Ilkka Holopainen and Pekka Pankka

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Received 17 December 2002