Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 411–437

UNIFORM DOMAINS OF HIGHER ORDER II

Pekka Alestalo and Jussi Väisälä

Helsingin yliopisto, Matematiikan laitos PL 4, Yliopistonkatu 5, FIN-00014 Helsinki, Finland

Abstract. We continue the investigation of (p, c)-uniform domains in a normed space. Special emphasis is on the basic properties of homological (p, c)-uniformity.

1. Introduction

1.1. This paper is a continuation to [Al], in which the first author introduced the class of (p, c)-uniform domains, where p ≥ 0 is an integer and c ≥ 1. We recall the definition. Suppose that E is a real normed space with dim E ≥ 2. A domain G ⊂ E is homotopically (p, c)-uniform if every continuous map f: S^{p} → G has a continuous extension g: \overline{B}^{p+1} → G satisfying the lens condition

(1.2)  d(x, |f|) ≤ cd(x, ∂G)  for all x ∈ |g|,

and the turning condition

(1.3)  d(|g|) ≤ cd(|f|).

Here |f| = im f, and d(A) is the diameter of a set A. Together these conditions are called the uniformity conditions. For p = 0, we obtain the c-uniform domains in the ordinary distance sense. The class defined above has been independently considered by J. Heinonen and S. Yang [HY].

A homological version of the definition was also given in [Al]. The domain G is homologically (p, c)-uniform if each reduced singular p-cycle f in G bounds a chain g such that (1.2) and (1.3) are true; now |f| is the carrier of f.

To abbreviate terminology we shall often replace the word ‘homotopically’ and ‘homologically’ by ‘htop’ and ‘hlog’, respectively.

The main emphasis in this paper will be on the basic properties of homological uniformity. Several results on hlog (p, c)-uniformity were given in [Al] only for p = 1. Our first task is to extend these for arbitrary p. This is done in Section 3. Before that, we develop in Section 2 a technique to construct singular homologies, called the prismatic method. In the rest of the paper we shall consider relations between htop and hlog uniformity, some related topological properties, and pdimensional plumpness, which was introduced in [Al] in the homotopical case; a homological version will be given in Section 6.

1991 Mathematics Subject Classification: Primary 30C65.

Part III of this investigation is under preparation. It will contain, for example, results on cartesian products and planar sections of null-sets for htop and hlog (p, c)-uniform domains.

We thank J. Partanen, who read the manuscript and made valuable remarks. Part of this article was written while the first author was visiting the University of Bielefeld in 1995–96. The hospitality of SFB 343 is hereby acknowledged.

1.4. Terminology and notation. The basic notation will be the same as in the first part; see [Al, p. 6]. Throughout the paper, E will denote a real normed space, and we shall usually assume that dim E ≥ 2. However, results dealing with porosity are relevant also in the one-dimensional case. We let |a − b| denote the distance between points a, b in any metric space X. Open and closed balls in X are written as B(a, r) and \overline{B}(a, r). More generally, if A ⊂ X, we let B(A, r) and \overline{B}(A, r) denote the open and closed r-inflations of A. By a map we mean a continuous function. The image set of a map f: A → B will be written as |f| = fA. If f, g: A → X are two maps, we set ∥f − g∥ = sup{|fa − ga| : a ∈ A}.

Our main reference on homology theory is the book [Ro] of J.J. Rotman. We shall use reduced singular homology with integral coefficients. Let X be a topological space and let p be an integer. The group S_{p}(X) of singular p-chains is the free abelian group generated by all singular p-simplexes, that is, maps σ: Δ^{p} → X of the standard p-simplex Δ^{p}. Thus each γ ∈ S_{p}(X) has a normal representation γ = ∑_{j∈J} n_{j}σ_{j}, where J is a finite set, the singular simplexes σ_{j} are all distinct, and the integers n_{j} nonzero. We write σ < γ if σ is a singular simplex appearing in the normal representation of the chain γ. The normal representation can then be written in the form γ = ∑_{σ<γ} n_{σ}σ. The carrier of γ is the set

|γ| = ∪{|σ| : σ < γ}.

For p = −1 there is a unique (empty) map σ: Δ^{−1} = ∅ → X, and we identify S_{−1}(X) = Z. For p ≤ −2 we have S_{p}(X) = 0. The boundary homomorphism ∂ = ∂_{p}: S_{p}(X) → S_{p−1}(X) is defined in the well-known way. For a 0-chain γ = ∑_{σ<γ} n_{σ}σ we have ∂γ = ∑_{σ<γ} n_{σ}. The kernel of ∂_{p} is the group Z_{p}(X) of p-cycles, and the p^{th} homology group of X is H_{p}(X) = Z_{p}(X)/ im ∂_{p+1} . In the literature, the group H_{0}(X) is often written as H̃_{0}(X). It is a free abelian group of rank m − 1 where m is the number of the path components of X. The group H_{−1}(X) is trivial unless X = ∅, in which case H_{−1}(X) = Z. We write z ∼ z′ if the cycle z is homologous to z′.

Suppose that Q(p) is a property of a set involving an integer p ≥ 0. We say that a set A has the property Q(p) completely if A has the property Q(k) for all 0 ≤ k ≤ p. For example, a domain G is completely hlog (p, c)-uniform if it is hlog (k, c)-uniform for all 0 ≤ k ≤ p.

A domain is hlog (p)-uniform if it is hlog (p, c)-uniform for some c ≥ 1. A similar convention is applied to any property involving a pair (p, c) with p ∈ Z and c ≥ 1.

To illustrate the difference between complete and noncomplete uniformity, we consider the infinite cylinder Z = B^{n−1} × R ⊂ R^{n}, n ≥ 3. It is not difficult to show that Z is htop and hlog (p)-uniform for all 1 ≤ p ≤ n − 2, in fact, (p, \sqrt{2} )- uniform. However, Z is not (0)-uniform and hence not completely htop or hlog (p)-uniform for any p.

It is possible to consider (p, c)-uniform domains G in the extended space Ė = E ∪ {∞} with ∞ ∈ G. However, this essentially means that G \ {∞} is (p, c)-uniform; see [Vä_{3} , 5.4]. In this paper we consider only the case G ⊂ E. All closures and boundaries are taken in E, except when we are using the compactness method in Ṙ^{n}.

2. Prisms

2.1. Summary of Section 2. We develop the prismatic method, which produces homologies of a given singular cycle. It will be applied several times in later sections. The method can be regarded as an elaboration of the proof of the homotopy axiom in singular homology.

2.2. Basic concepts. Let Δ = [v_{0}, …, v_{p}] be a p-simplex in R^{n} with the given order of vertices. We let Δ̃ denote the simplicial complex consisting of all faces of Δ, including Δ. For I = [0, 1] we consider the prism Δ × I ⊂ R^{n+1} . We identify Δ = Δ × {0} and write v′_{j} = (v_{j}, 1) for 0 ≤ j ≤ p. The vertices of Δ × I are ordered as (v_{0}, …, v_{p}, v′_{0}, …, v′_{p}). Let P Δ be the standard triangulation of Δ × I. Its (p + 1)-simplexes are

s_{j} = [v_{0}, …, v_{j}, v′_{j}, …, v′_{p}],

0 ≤ j ≤ p. The ordering of the vertices of P Δ defines an orientation of P Δ.

For any oriented simplicial complex K, we let C_{∗}(K) = (C_{p}(K))_{p∈Z} denote the associated augmented chain complex with integral coefficients; see [Ro, p. 147].

We consider the case where Δ is the standard p-simplex Δ^{p} = [e_{0}, …, e_{p}]. Let α_{0}: C_{∗}(Δ̃^{p}) → C_{∗}(P Δ^{p}) be the chain map induced by the inclusion Δ̃^{p} ⊂ P Δ^{p}. Let X be a topological space and let σ: Δ^{p} → X be a singular p-simplex in X. By a prismoid of σ we mean a chain map

φ: C_{∗}(P Δ^{p}) → S_{∗}(X)

satisfying the condition

(2.3)  σ = φα_{0}Δ^{p};

here Δ^{p} is considered as an element (generator) of C_{p}(Δ̃^{p}). Figure 1 illustrates two prismoids φ_{1}, φ_{2} of a singular 1-simplex σ. The chain map φ_{1} maps all elementary chains to elementary chains; the prismoid φ_{2} is somewhat more complicated.

Figure 1

Let 0 ≤ q ≤ p − 1 and let ε: Δ^{q} → Δ^{p} be an order-preserving isometry onto a q -face of Δ^{p}. Then ε induces a chain map ε_{#}: C_{∗}(Δ̃^{q}) → C_{∗}(Δ̃^{p}). Moreover, the map ε × id: Δ^{q} × I → Δ^{p} × I defines a simplicial map π = π_{ε}: P Δ^{q} → P Δ^{p} and a chain map π_{#}: C_{∗}(P Δ^{q}) → C_{∗}(P Δ^{p}). The following functorial property is

easily verified: If Δ^{r} ⟶^{ζ} Δ^{q} ⟶^{ε} Δ^{p} are order-preserving isometries onto faces, then

(2.4)  π_{εζ} = π_{ε}π_{ζ}, (π_{εζ})_{#} = π_{ε#}π_{ζ#}.

Suppose that φ and ψ are prismoids of singular p-simplexes σ and τ, respectively. We say that these prismoids are compatible if φπ_{ε#} = ψπ_{ζ#} whenever 0 ≤ q ≤ p − 1 and ε, ζ: Δ^{q} → Δ^{p} are order-preserving isometries onto faces such that σε = τζ. Observe that this is a relevant property also in the case σ = τ, φ = ψ; we may then say that φ is self-compatible. See Figure 2.

Let γ be a singular p-chain in X with normal representation γ = ∑_{σ<γ} n_{σ}σ. A prism of γ is a chain map

φ = ∑_{σ<γ} n_{σ}φ_{σ}: C_{∗}(P Δ^{p}) → S_{∗}(X)

such that

(1) φ_{σ} is a prismoid of σ for each σ < γ,

(2) φ_{σ} and φ_{τ} are compatible for all pairs σ, τ < γ, including the case σ = τ.

By a prism of the zero chain we mean the zero map. We see that a prismoid of a singular simplex is a prism if and only if it is self-compatible.

Figure 2

2.5. Construction of prisms. We shall make use of prisms several times in this paper to obtain homologies of given cycles and sometimes also prisms of more general chains. The prisms will be constructed by a process where we start with the 0-chains of P Δ^{p} and proceed by induction to higher-dimensional skeletons of P Δ^{p}. We shall next explain this process.

Let X be a topological space and let

γ = ∑_{σ<γ} n_{σ}σ

be a p-chain in X. We want to construct a prism φ = ∑_{σ<γ} n_{σ}φ_{σ} of γ. The aim of the construction usually is to get a chain γ_{1} = φα_{1}Δ^{p}, where the chain map α_{1}: C_{∗}(Δ̃^{p}) → C_{∗}(P Δ^{p}) is induced by the natural embedding of Δ^{p} onto Δ^{p} × {1}, such that γ_{1} is in some sense better situated than γ.

Starting with 0-chains, let v be a vertex of P Δ^{p}. If v = e_{j} for some 0 ≤ j ≤ p, we let φ_{σ}v be the elementary 0-chain σv ∈ X. If v = e′_{j} = (e_{j}, 1), we can choose φ_{σ}v to be a rather arbitrary point in X. However, to ensure the compatibility of the various φ_{σ}, we choose these points in a compatible way, which means that φ_{σ}e′_{j} = φ_{τ} e′_{k} whenever σe_{j} = τe_{k}. The maps φ_{σ} are then extended linearly to homomorphisms φ_{σ}: C_{0}(P Δ^{p}) → S_{0}(X).

Assume that 0 ≤ q ≤ p and that we have defined the chain maps φ_{σ}: C_{q}(P Δ^{p}) → S_{q}(X) for all σ < γ. Let s be a (q + 1)-simplex of P Δ^{p} and let σ < γ. If s ∈ Δ̃^{p}, we set φ_{σ}s = σε_{s} ∈ S_{q+1}(X), where ε_{s}: Δ^{q+1} → Δ^{p} is the orderpreserving isometry onto s. Assume that s is not in Δ̃^{p}. Then ∂s ∈ C_{q}(P Δ^{p}), and thus φ_{σ}∂s is defined. The chain φ_{σ}s ∈ C_{q+1}(X) must satisfy the condition ∂φ_{σ}s = φ_{σ}∂s. In the applications this will be possible by some conditions which are valid in the situation. Otherwise we can choose φ_{σ}s rather arbitrarily, but to ensure compatibility, we choose φ_{σ}s = φ_{τ} t whenever φ_{σ}∂s = φ_{τ} ∂t. We again express this by saying that the chains φ_{σ}s are chosen in a compatible way. The maps φ_{σ} are extended linearly to homomorphisms φ_{σ}: C_{q+1}(P Δ^{p}) → S_{q+1}(X).

After the case q = p we clearly have chain maps φ_{σ}: C_{∗}(P Δ^{p}) → S_{∗}(X), which are prismoids of σ. It is a routine exercise to show that φ = ∑_{σ<γ} n_{σ}φ_{σ} is a prism of γ.

In the applications, at the step q = 0 of the induction, the 1-chains φ_{σ}s can usually be chosen to be singular 1-simplexes, but for q ≥ 1 we must in general choose a more general chain.

2.6. Simple prisms. Every continuous map h: Δ^{p} × I → X induces a chain map \overline{h}: C_{∗}(P Δ^{p}) → S_{∗}(X) in the following way: Let s be a q -simplex of P Δ^{p}, and let ε_{s}: Δ^{q} → s be the order-preserving affine homeomorphism. Then \overline{h}s is the singular simplex hε_{s}: Δ^{q} → X.

Suppose that σ: Δ^{p} → X is a singular p-simplex and that h: Δ^{p} × I is a continuous extension of σ. Then clearly \overline{h} is a prismoid of σ, but \overline{h} need not be self-compatible. For example, if p = 1, then \overline{h} is self-compatible if and only if either σ(e_{0}) ≠ σ(e_{1}) or h(e_{0}, t) = h(e_{1}, t) for all t ∈ I.

We say that a prism φ = ∑_{σ<γ} n_{σ}φ_{σ} of a p-chain γ is simple if each φ_{σ} is of the form \overline{h}_{σ} for some h_{σ}: Δ^{p} × I → X.

To construct a simple prism of a chain γ = ∑_{σ<γ} n_{σ}σ we can apply the method of 2.5. Now the maps h_{σ} are defined by skeletonwise induction. At the first step, the points h_{σ}e′_{i} are chosen in a compatible way, by which we mean that h_{σ}e′_{i} = h_{τ} e′_{j} whenever σe_{i} = τe_{j}. Assume that the maps h_{σ} are defined in the q -skeleton of P Δ^{p} and that s is a (q + 1)-simplex in P Δ^{p} \ Δ̃^{p}. Then h_{σ}|∂s is defined, and we must extend this map to s. These extensions are chosen in a compatible way, that is, h_{σ}|s = h_{τ} |t whenever h_{σ}|∂s = h_{τ} |∂t.

We remark that the standard direct proof of the homotopy axiom of singular homology makes use of simple prisms. Indeed, suppose that h: X × I → Y is a homotopy connecting the maps f_{0}, f_{1}: X → Y . Let z = ∑_{σ<z} n_{σ}σ be a p-cycle in X. Setting h_{σ} = h ∘ (σ × id): Δ^{p} × I → Y we obtain a simple prism of f_{0#}z. It follows from Theorem 2.10 below that f_{0#}z is homologous to f_{1#}z.

2.7. Preparations. We introduce some concepts and notation in order to state and prove the main result 2.10 on prisms. Let Δ = [v_{0}, …, v_{p}] be an ordered psimplex in R^{n}. As above, we let α_{0}, α_{1}: C_{∗}(Δ̃) → C_{∗}(P Δ) be the chain maps induced by the vertex maps v_{j} ↦ v_{j} and v_{j} ↦ v′_{j} = (v_{j}, 1), respectively. It is well known that α_{0} and α_{1} are connected by a chain homotopy D = D_{Δ} defined as follows: For each q -face s = [u_{0}, …, u_{q}] of Δ, considered as an element of C_{q}(Δ̃), Ds is the (q + 1)-chain

Ds = ∑_{j=0}^{q}(−1)^{j}[u_{0}, …, u_{j}, u′_{j}, …, u′_{q}]

in C_{q+1}(P Δ). This map is extended linearly to a homomorphism D: C_{q}(Δ̃) → C_{q+1}(P Δ) for each 0 ≤ q ≤ p. The chain homotopy law states that

(2.8)  ∂D + D∂ = α_{1} − α_{0}.

The chain homotopy D has the following functorial property: Suppose that Γ = [u_{0}, …, u_{q}] is another ordered simplex with q ≤ p and that ε: Γ → Δ is an affine order-preserving embedding onto a q -face of Δ. Then ε induces a chain map ε_{#}: C_{∗}(Γ̃) → C_{∗}(Δ̃). Moreover, the affine embedding ε×id: Γ×I → Δ×I induces a simplicial map π_{ε}: P Γ → P Δ and a chain map π_{ε#}: C_{∗}(P Γ) → C_{∗}(P Δ). Then for each j ∈ Z we have the commutative diagram

[figure: C_{j}(Γ̃); D_{Γ}; ε_{#}; C_{j}(Δ̃); D_{Δ}; C_{j+1}(P Γ); π_{ε#}; C_{j+1}(P Δ).]

Furthermore, if Γ_{1} ⟶^{ζ} Γ_{2} ⟶^{ε} Δ are order-preserving affine embeddings, then obviously

(2.9)  π_{(εζ)#} = π_{ε#}π_{ζ#}.

Suppose that γ is a singular p-chain in a topological space X and that φ is a prism of γ. Let D^{p}: C_{∗}(Δ̃^{p}) → C_{∗}(P Δ^{p}) be the above chain homotopy connecting the chain maps α_{0}, α_{1}: C_{∗}(Δ̃^{p}) → C_{∗}(P Δ^{p}). We again consider Δ^{p} as an element of C_{p}(Δ̃^{p}). The chain

g(φ) = φD^{p}Δ^{p} ∈ S_{p+1}(X)

will play a central role in the theory.

For 0 ≤ j ≤ p we let ε_{j}: Δ^{p−1} → Δ^{p} denote the standard face map, which embeds Δ^{p−1} onto the face of Δ^{p} opposite to e_{j}. Then ε_{j} ×id induces a simplicial map π_{j}: P Δ^{p−1} → P Δ^{p} as above, and we can define the chain maps

\begin{aligned}ε & = ∑_{j=0}^{p}(−1)^{j}ε_{j#}: C_{∗}(Δ̃^{p−1}) → C_{∗}(Δ̃^{p}), \\ π & = ∑_{j=0}^{p}(−1)^{j}π_{j#}: C_{∗}(P Δ^{p−1}) → C_{∗}(P Δ^{p}). \end{aligned}

Observe that εΔ^{p−1} = ∂Δ^{p}.

With this notation we give the basic result of prism theory:

2.10. Theorem. Suppose that φ is a prism of a singular chain γ ∈ S_{p}(X). Then ψ = φπ is a prism of ∂γ. Setting γ_{1} = φα_{1}Δ^{p} we have

∂g(φ) + g(ψ) = γ_{1} − γ.

If γ is a cycle, then γ_{1} is also a cycle, and ∂g(φ) = γ_{1} − γ.

Proof. Let γ = ∑_{i=1}^{m} n_{i}σ_{i} be the normal representation. We define an equivalence relation in the set {1, …, m} × {0, …, p} by calling (i, j) equivalent to (k, l) if σ_{i}ε_{j} = σ_{k}ε_{l}. Let G be the set of all equivalence classes and let H ∈ G. Set

m_{H} = ∑_{(i,j)∈H}(−1)^{j}n_{i},

and let τ_{H} denote the map σ_{i}ε_{j}: Δ^{p−1} → X for (i, j) ∈ H. Then

∂γ = ∑_{i=1}^{m} ∑_{j=0}^{p}(−1)^{j}n_{i}σ_{i}ε_{j} = ∑_{H∈G} ∑_{(i,j)∈H}(−1)^{j}n_{i}σ_{i}ε_{j} = ∑_{H∈G} m_{H}τ_{H},

where the singular (p − 1)-simplexes τ_{H} are all distinct. However, this is usually not a normal representation, since some numbers m_{H} may be zero. In particular, if ∂γ = 0, then m_{H} = 0 for all H ∈ G.

By the compatibility condition of the prism φ = ∑_{i=1}^{m} n_{i}φ_{i}, the chain map φ_{i}π_{j#}: C_{∗}(P Δ^{p−1}) → S_{∗}(X) does not change if (i, j) is replaced by an equivalent pair. Given H ∈ G, we can therefore write ψ_{H} = φ_{i}π_{j#} for all (i, j) ∈ H. Clearly ψ_{H} is a prismoid of τ_{H}. Moreover,

ψ = φπ = ∑_{H∈G} ∑_{(i,j)∈H}(−1)^{j}n_{i}φ_{i}π_{j#} = ∑_{H∈G} m_{H}ψ_{H}.

To prove that ψ is a prism of ∂γ it suffices to show that if H, K ∈ G, then the prismoids ψ_{H} and ψ_{K} are compatible. Assume that 0 ≤ q ≤ p − 1 and that ε_{H}, ε_{K}: Δ^{q} → Δ^{p−1} are order-preserving isometries onto faces such that τ_{H}ε_{H} = τ_{K}ε_{K}. Choose (i, j) ∈ H and (k, l) ∈ K. Writing π_{H} = π_{ε_{H}} and π_{K} = π_{ε_{K}} we must show that ψ_{H}π_{H#} = ψ_{K}π_{K#} , that is,

(2.11)  φ_{i}π_{j#}π_{H#} = φ_{k}π_{l#}π_{K#}.

Setting ζ = ε_{j}ε_{H} and η = ε_{l}ε_{K} we have σ_{i}ζ = τ_{H}ε_{H} = τ_{K}ε_{K} = σ_{k}η. The maps ζ, η = Δ^{q} → Δ^{p} are order-preserving isometries. By the compatibility of the prismoids φ_{i} and φ_{k} we have φ_{i}π_{ζ#} = φ_{k}π_{η#} . This implies (2.11) by (2.9), and we have proved that ψ is a prism of ∂γ.

The chain homotopy law (2.8) and the chain map law ∂g = g∂ give

∂g(φ) = ∂φD^{p}Δ^{p} = φ∂D^{p}Δ^{p} = φα_{1}Δ^{p} − φα_{0}Δ^{p} − φD^{p}∂Δ^{p} = γ_{1} − γ − φD^{p}∂Δ^{p}.

On the other hand, the functorial property of D implies

g(ψ) = φπD^{p−1}Δ^{p−1} = φD^{p}εΔ^{p−1} = φD^{p}∂Δ^{p},

and we obtain the formula of the theorem. The last statement is a direct consequence.

2.12. Metric conditions. In applications, the space X will always be metric, and we want that the chains γ_{1}, g(φ) and g(ψ) of 2.10 satisfy some metric conditions. In particular, we want that g(φ) and g(ψ) are not too far from γ and ∂γ, respectively. Let pr_{1}: Δ^{p} × I → Δ^{p} be the projection and let s be a simplex of P Δ^{p}. Let γ = ∑_{i=1}^{m} n_{i}σ_{i} be the normal representation. The construction of 2.5 will be carried out in such a way that

(2.13)  |φ_{i}s| ⊂ \overline{B}(σ_{i} pr_{1}s, r)

for some r > 0 and for all s and i. It is then clear that |g(φ)| ⊂ \overline{B}(|γ|, r). We show that also |g(ψ)| ⊂ \overline{B}(|∂γ|, r). Let x ∈ |g(ψ)|. With the notation of the proof of 2.10, we have x ∈ |ψ_{H}t| for some p-simplex t ∈ P Δ^{p−1} and for some H ∈ G with m_{H} ≠ 0. Choose (i, j) ∈ H. Then ψ_{H} = φ_{i}π_{j#} and τ_{H} = σ_{i}ε_{j}. Moreover, π_{j} = ε_{j} × id maps t onto a p-simplex s ∈ P Δ^{p} with pr_{1}s = ε_{j} pr_{1}t, and thus σ_{i} pr_{1}s = σ_{i}ε_{j} pr_{1}t ⊂ |τ_{H}|. By (2.13) this implies

|ψ_{H}t| = |φ_{i}s| ⊂ \overline{B}(|τ_{H}|, r).

Since m_{H} ≠ 0, we have |τ_{H}| ⊂ |∂γ|, and hence x ∈ \overline{B}(|∂γ|, r). We have proved the following result:

2.14. Theorem. Let X be a metric space and let γ ∈ S_{p}(X) be a chain with normal representation γ = ∑_{i=1}^{m} n_{i}σ_{i}. Suppose that r > 0 and that φ = ∑_{i=1}^{m} n_{i}φ_{i} is a prism of γ such that (2.13) holds for all simplexes s ∈ P Δ^{p} and for all 1 ≤ i ≤ m. Then, with the notation of 2.10, we have

|g(φ)| ⊂ \overline{B}(|γ|, r), |g(ψ)| ⊂ \overline{B}(|∂γ|, r).

3. Homological uniformity

3.1. Summary of Section 3. We extend the results of [Al] on hlog (1, c)- uniformity to hlog uniformity of all orders. The central result is Theorem 3.10, which states that complete hlog (p, c)-uniformity follows from lower-order uniformity and an apparently weaker condition, which involves only ‘nicely’ situated p-cycles. We apply this result to characterize the complements of completely hlog (p, c)-uniform domains in R^{n} in terms of compact families of sets.

3.2. Remark. It is possible to characterize the hlog (p, c)-uniform domains in terms of homology groups without mentioning cycles and chains. Let A be a compact subset of a domain G ⊂ E, and let c ≥ 1. Set

W (A, c, G) = {x ∈ G : d(x, A) ≤ c(d(A) ∧ d(x, ∂G))}.

Then the following conditions are quantitatively equivalent:

(1) G is a hlog (p, c)-uniform domain.

(2) The homomorphism H_{p}(A) → H_{p}(W (A, c, G)) is zero for all compact sets A ⊂ G.

More precisely, (1) implies (2) with the same constant c, and (2) implies (1) with c ↦ 2c + 1. The proof is straightforward.

We start with an elementary lemma, which is often useful when working with uniform domains.

3.3. Lemma. Let U ⊂ E be open and let A ⊂ B be subsets of U satisfying the lens condition

d(x, A) ≤ cd(x, ∂U)

for some c ≥ 1 and for all x ∈ B. Then d(A, ∂U) ≤ 2cd(B, ∂U).

Proof. We may assume that A ≠ ∅ and U ≠ E. Write r = d(A, ∂U) and let x ∈ B. If d(x, A) ≤ r/2, then d(x, ∂U) ≥ r/2 ≥ r/2c. If d(x, A) ≥ r/2, then d(x, ∂U) ≥ d(x, A)/c ≥ r/2c. Hence d(x, ∂U) ≥ r/2c.

3.4. Properties HT and HT^{∗}. Let G ⊂ E be a domain and let p ∈ N, c ≥ 1, c′ ≥ 1. We recall from [Al, 4.3] that G is said to have property HT (p, c′, c) if for each (reduced) singular p-cycle z in G with d(|z|) ≤ c′d(|z|, ∂G) there is g ∈ S_{p+1}(G) such that

(3.5)  ∂g = z, d(|g|) ≤ cd(|z|), d(x, |z|) ≤ cd(x, ∂G)

for all x ∈ |g|. This was proved in [Al, 4.5] to be quantitatively equivalent to another condition HT′(p, c′, c). For our present needs, it is convenient to introduce a third condition, still quantitatively equivalent to HT (p, c′, c). We say that G has property HT^{∗}(p, c′, c) if for each p-cycle z in G with d(|z|) ≤ c′d(|z|, ∂G) there is g ∈ S_{p+1}(G) such that

(3.6)  ∂g = z, d(|g|) ≤ cd(|z|), d(|z|, ∂G) ≤ cd(|g|, ∂G).

3.7. Lemma. For a domain G ⊂ E, property HT(p, c′, c) implies HT^{∗}(p, c′, 2c), and HT(p, c′, c) implies HT(p, c′, c′c^{2}).

Proof. The first part of the lemma follows from 3.3. Suppose that G has property HT^{∗}(p, c′, c). Let z be a p-cycle with d(|z|) ≤ c′d(|z|, ∂G). Choose g satisfying (3.6), and let x ∈ |g|. Then

d(x, |z|) ≤ d(|g|) ≤ cd(|z|) ≤ c′cd(|z|, ∂G) ≤ c′c^{2}d(|g|, ∂G) ≤ c′c^{2}d(x, ∂G).

Hence (3.5) is true with c replaced by c′c^{2} .

3.8. Lemma. Suppose that p ≥ 1 is an integer, that c, c_{1} ≥ 1 and that G is a completely hlog (p − 1, c)-uniform domain in E. Write c′ = c′(c, p) = 8(2c)^{2p+1}(1 + 2(4c)^{p}). Suppose also that G has property HT^{∗}(p, c′, c_{1}) and that z is a p-cycle in G such that d(|z|, ∂G) ≥ r > 0 and d(|z|) > c′r. Then there is a chain g ∈ S_{p+1}(G) such that writing z_{1} = ∂g + z, c_{0} = 4(4c)^{p}(2c)^{p+1} = c_{0}(c, p) and c_{2} = c_{0} + 3c_{0}c_{1} = c_{2}(c, c_{1}, p) we have

(1) |z_{1}| ⊂ \overline{B}(|z|, c_{0}r),

(2) d(|z_{1}|, ∂G) ≥ 2r,

(3) |g| ⊂ \overline{B}(|z|, c_{2}r),

(4) d(|g|, ∂G) ≥ r/c_{2} .

Proof. Let z = ∑_{σ<z} n_{σ}σ be the normal representation. Let Sdz be the barycentric subdivision of z; see [Ro, p. 114]. Then |Sdz| ⊂ |z| and z − Sdz bounds in |z|. Hence it suffices to prove the lemma with z replaced by Sdz. Iterating this argument we see that we may assume that d(|σ|) ≤ r′ = 4(2c)^{p+1}r for all σ < z.

Following the method described in 2.5 we construct a prism φ = ∑_{σ<z} n_{σ}φ_{σ} of z. Starting with 0-chains, assume that v is a vertex of P Δ^{p}. If v ∈ Δ^{p}, we set of course φ_{σ}v = σv. Suppose that v = e′_{j} = (e_{j}, 1). Since G is (0, c)-uniform, it is 4c-plump by [Vä_{1} , 2.15]. Since r′ < c′r < d(|z|) < d(G), we can choose the point φ_{σ}v ∈ G such that

|φ_{σ}v − σe_{j}| ≤ r′, d(φ_{σ}v, ∂G) ≥ r′/4c.

Moreover, the points φ_{σ}e′_{j} are chosen in a compatible way; see 2.5.

Proceeding inductively, assume that 0 ≤ q ≤ p − 1 and that the homomorphisms φ_{σ}: C_{q}(P Δ^{p}) → S_{q}(G) have been defined in such a way that the following three conditions are satisfied for every q -simplex s ∈ P Δ^{p} and for every σ < z:

(i) |φ_{σ}s| ⊂ \overline{B}(|σ|, (4c)^{q}r′),

(ii) d(|φ_{σ}s|, ∂G) ≥ r/(2c)^{q},
(iii) d(|φ_{σ}s|, ∂G) ≥ r′/2(2c)^{q+1} whenever s ⊂ Δ^{p} × {1}.

The conditions are easily verified for q = 0.

Let σ < z. To define φ_{σ}: C_{q+1}(P Δ^{p}) → S_{q+1}(G) assume that s is a (q + 1)- simplex of P Δ^{p}. If s ⊂ Δ^{p}, we set of course φ_{σ}s = σε_{s}; see 2.5. The conditions (i), (ii), (iii) are easily verified for such s. Assume that s is not in Δ^{p}. Now φ_{σ}∂s is defined, and it is a q -cycle in G. If φ_{σ}∂s = 0, we set φ_{σ}s = 0. If φ_{σ}∂s ≠ 0, we apply the hlog (q, c)-uniformity of G and Lemma 3.3 to find a chain φ_{σ}s ∈ S_{q}(G) such that

∂φ_{σ}s = φ_{σ}∂s, d(|φ_{σ}s|) ≤ cd(|φ_{σ}∂s|), d(|φ_{σ}∂s|, ∂G) ≤ 2cd(|φ_{σ}s|, ∂G).

Moreover, the chains φ_{σ}s are chosen in a compatible way; see 2.5.

We show that the conditions (i)–(iii) hold for each (q + 1)-simplex s. We may assume that φ∂s ≠ 0. By the inductive hypothesis we have |φ_{σ}∂s| ⊂ \overline{B}(|σ|, (4c)^{q}r′), which implies that |φ_{σ}s| ⊂ \overline{B}(|σ|, r′′) where

r′′ = (4c)^{q}r′ + cd(|φ_{σ}∂s|).

Since d(|σ|) ≤ r′, we have d(|φ_{σ}∂s|) ≤ r′ + 2(4c)^{q}r′, and hence r′′ ≤ (4c)^{q+1}r′, which proves (i).

By the inductive hypothesis we have d(|φ_{σ}∂s|, ∂G) ≥ r/(2c)^{q}, which implies that

d(|φ_{σ}s|, ∂G) ≥ r/(2c)^{q+1},

and hence (ii) is true. Finally, if s ⊂ Δ^{p} × {1}, we apply (iii) of the inductive hypothesis and obtain d(|φ_{σ}∂s|, ∂G) ≥ r′/2(2c)^{q+1} , which yields

d(|φ_{σ}s|, ∂G) ≥ r′/2(2c)^{q+2},

and hence (iii) is true for s.

As the last step of the induction we obtain chain maps φ_{σ}: C_{p}(P Δ^{p}) → S_{p}(G), σ < z. We shall continue by one step to φ_{σ}: C_{p+1}(P Δ^{p}) → S_{p+1}(G), but instead of uniformity, we shall apply the property HT^{∗}(p, c′, c).

Let s be a (p + 1)-simplex of P Δ^{p} and let σ < z. We again have the p-cycle φ_{σ}∂s in G. Assuming that φ_{σ}∂s ≠ 0 we infer from (i) and (ii) that

\frac{d(|φ_{σ}∂s|)}{d(|φ_{σ}∂s|, ∂G)} ≤ \frac{(1 + 2(4c)^{p})r′}{r/(2c)^{p}} = c′/2 < c′.

Hence property HT^{∗}(p, c′, c_{1}) gives a (p + 1)-chain φ_{σ}s such that ∂φ_{σ}s = φ_{σ}∂s and such that

(3.9)  \begin{aligned}d(|φ_{σ}s|) & ≤ c_{1}d(|φ_{σ}∂s|) ≤ c_{1}(1 + 2(4c)^{p})r′, \\ d(|φ_{σ}s|, ∂G) & ≥ d(|φ_{σ}∂s|, ∂G)/c_{1} ≥ r/c_{1}(2c)^{p}. \end{aligned}

These chains are also chosen in a compatible way. The construction of the prism φ = ∑_{σ<z} n_{σ}φ_{σ} has now been completed.

Setting z_{1} = φα_{1}Δ^{p} and g = g(φ) we have ∂g = z_{1}−z by 2.10. We show that the conditions (1)–(4) of the lemma are true. Since z_{1} = φs with s = Δ^{p} × {1},

(1) follows from (i). The condition (2) follows from (iii), since r′/2(2c)^{p+1} = 2r. To prove (3) let x ∈ |g|. Then x ∈ |φ_{σ}t| for some (p + 1)-simplex t ∈ P Δ^{p} and some σ < z. Now (i) implies that

|φ_{σ}∂t| ⊂ \overline{B}(|σ|, (4c)^{p}r′) = \overline{B}(|σ|, c_{0}r).

Since d(|σ|) ≤ r′ < c_{0}r, this proves d(|φ_{σ}∂t)| ≤ 3c_{0}r. By (3.9) we obtain d(|φ_{σ}t|) ≤ 3c_{0}c_{1}r. It follows that x ∈ \overline{B}(|z|, c_{2}r), and (3) is proved.

Finally, (4) follows from (3.9), since d(|g|, ∂G) ≥ r/c_{1}(2c)^{p} > r/c_{2} .

We next extend [Al, 4.13] to uniform domains of all orders.

3.10. Theorem. Suppose that p ≥ 1 and that G ⊂ E is a completely hlog (p−1, c)-uniform domain with property HT^{∗}(p, c′, c_{1}), where c′ = c′(c, p) is given in 3.8. Then G is hlog (p, c_{2})-uniform with c_{2} = c_{2}(c, c_{1}, p).

The theorem is valid also in the case p = 0; the (−1, c)-uniformity is interpreted as c-plumpness.

Proof. The case p = 0 was given in [Vä_{1} , 2.15]. Assume that p ≥ 1, and let z_{0} be a p-cycle in G. We look for a chain g ∈ S_{p+1}(G) with ∂g = z_{0} and satisfying the uniformity conditions. Write

d_{0} = d(|z_{0}|), r_{0} = d(|z_{0}|, ∂G).

If d_{0} ≤ c′r_{0} , the desired g is given by HT^{∗}(p, c′, c) and Lemma 3.7. Suppose that d_{0} > c′r_{0} . By successive applications of 3.8 we get sequences

z_{0}, …, z_{k} ∈ Z_{p}(G), g_{1}, …, g_{k} ∈ S_{p+1}(G)

such that ∂g_{i} = z_{i} − z_{i−1} and such that writing d_{i} = d(|z_{i}|), r_{i} = d(|z_{i}|, ∂G) we have for all 1 ≤ i ≤ k

(1) |z_{i}| ⊂ \overline{B}(|z_{i−1}|, c_{0}r_{i−1}),

(2) r_{i} ≥ 2r_{i−1} ,
(3) |g_{i}| ⊂ \overline{B}(|z_{i−1}|, c_{2}r_{i−1}),
(4) d(|g_{i}|, ∂G) ≥ r_{i−1}/c_{2} ,

where the constants c_{0} = c_{0}(c, p) and c_{2} = c_{2}(c, c_{1}, p) are as in 3.8. We proceed with the construction as long as we get an index k such that d_{k} ≤ c′r_{k}. We first show that such an index exists. Assume that d_{i} > c′r_{i} for 0 ≤ i ≤ k − 1. We show that

(3.11)  d_{i} ≤ d_{0} + 4c_{0}r_{i−1} ≤ 2d_{0}

for all i = 1, …, k.

By (1) we obtain d_{i} ≤ d_{i−1} + 2c_{0}r_{i−1} . By iteration and (2) this gives

d_{i} ≤ d_{0} + 2c_{0}(r_{i−1} + ⋯ + r_{0}) ≤ d_{0} + 2c_{0}r_{i−1}(1 + 2^{−1} + ⋯ + 2^{1−i}) ≤ d_{0} + 4c_{0}r_{i−1},

which is the first inequality of (3.11). The second inequality follows by induction, since assuming d_{i−1} ≤ 2d_{0} and observing that c′ ≥ 8c_{0} we obtain

4c_{0}r_{i−1} < 4c_{0}d_{i−1}/c′ ≤ d_{i−1}/2 ≤ d_{0}.
If d_{k} < c′r_{k}, then (3.11) and (2) give
8c_{0} ≤ c′ < d_{k}/r_{k} ≤ d_{0}/r_{k} + 4c_{0}r_{k−1}/r_{k} ≤ 2^{−k}d_{0}/r_{0} + 2c_{0}.

Since this gives a contradiction for large k, we can choose the smallest k ≥ 1 for which d_{k} ≤ c′r_{k}. Then (3.11) holds for 1 ≤ i ≤ k.

By property HT^{∗}(p, c′, c_{1}) there is g_{k+1} ∈ S_{p+1}(G) such that

(3.12)  ∂g_{k+1} = −z_{k}, d(|g_{k+1}|) ≤ c_{1}d_{k}, d(|g_{k+1}|, ∂G) ≥ r_{k}/c_{1}.

Setting

g = − ∑_{i=1}^{k+1} g_{i}

we have ∂g = z_{0} . We show that g satisfies the uniformity conditions in G with some constant c_{3} = c_{3}(c, c_{1}, p).

To prove the turning condition let x ∈ |g|. Then x ∈ |g_{i}| for some 1 ≤ i ≤ k + 1. If i ≤ k, then (1), (2) and (3) yield

(3.13)  d(x, |z_{0}|) ≤ c_{2}r_{i−1} + c_{0}(r_{0} + ⋯ + r_{i−2}) ≤ (c_{2} + c_{0})r_{i−1}.

Since r_{i−1} < d_{i−1}/c′ < d_{i−1} ≤ 2d_{0} by (3.11) and since c_{2} ≥ c_{0} , we get d(x, |z_{0}|) ≤ 4c_{2}d_{0} . If i = k + 1, then (3.12) and (3.11) give

d(x, |z_{k}|) ≤ d(|g_{k+1}|) ≤ c_{1}d_{k} ≤ 2c_{1}d_{0}.

Since |z_{k}| ⊂ |g_{k}| ⊂ \overline{B}(|z_{0}|, 4c_{2}d_{0}) by the first case, we have d(x, |z_{0}|) ≤ (2c_{1} + 4c_{2})d_{0} in both cases. This gives the turning condition d(|g|) ≤ (1 + 4c_{1} + 8c_{2})d_{0} .

To prove the lens condition let again x ∈ |g_{i}| for some 1 ≤ i ≤ k + 1. If i ≤ k, then (4) implies d(x, ∂G) ≥ r_{i−1}/c_{2} . By (3.13) we obtain

(3.14)  d(x, |z_{0}|) ≤ c_{2}(c_{2} + c_{0})d(x, ∂G).

Assume that i = k + 1. Now (3.12) implies that

(3.15)  d(x, ∂G) ≥ d(|g_{k+1}|, ∂G) ≥ r_{k}/c_{1}.

Choose a point y ∈ |z_{k}|. Then (1) and (2) yield

d(y, |z_{0}|) ≤ c_{0}(r_{0} + ⋯ + r_{k−1}) ≤ c_{0}r_{k}.

By (3.12) and (3.15) this gives

d(x, |z_{0}|) ≤ |x − y| + d(y, z_{0}) ≤ d(|g_{k+1}|) + c_{0}r_{k}
≤ c_{1}d_{k} + c_{0}r_{k} ≤ (c_{1}c′ + c_{0})r_{k}.
≤ c_{1}(c_{1}c′ + c_{0})d(x, ∂G).

This and (3.14) prove the lens condition.

3.16. Uniformity and compactness. With the aid of Theorem 3.10, we shall generalize the results of [Al, §6] on hlog (1)-uniform domains to hlog (p)-uniform domains with arbitrary p. For p = 0 these results were known earlier.

We recall that an open set U ⊂ E is c-plump if for each x ∈ U and 0 < r < d(U) there is z ∈ \overline{B}(x, r) with B(z, r/c) ⊂ U. If, in addition, int(E \ U) = ∅, the closed set E \ U is said to be c-porous.

We recall some notation and terminology of [Al, §6]. Let K_{∞}^{n} be the family of all compact sets A with ∞ ∈ A ⊂ Ṙ^{n} = R^{n} ∪ {∞}. Each closed subset A of R^{n} will be identified with A ∪ {∞} and considered as an element of K_{∞}^{n}. With the Hausdorff metric induced by the spherical metric, K_{∞}^{n} is a compact metric space. For 0 ≤ p ≤ n−2, let HU(p, c) denote the family of all A ∈ K_{∞}^{n} such that R^{n} \A is a completely hlog (p, c)-uniform domain, and let HU(−1, c) be the family of all sets A ∈ K_{∞}^{n} such that R^{n} \ A is c-plump. We set HU(p) = ∪{HU(p, c) : c ≥ 1}.

We fix a unit vector e_{1} ∈ R^{n}. A family H ⊂ K_{∞}^{n} is stable if H is invariant under similarities and if H^{2} = {A ∈ H : {0, e_{1}} ⊂ ∂A} is compact in K_{∞}^{n}. To each H ⊂ K_{∞}^{n} we associate the subfamily

σ(H) = ∪{L ⊂ H : L stable}.

A filtration of σ(H) is a function c ↦ M_{c}, defined for c ∈ [1, ∞), such that (1) c < d implies M_{c} ⊂ M_{d},

(2) each M_{c} is contained in a stable subfamily of H,

(3) each stable subfamily of H is contained in some M_{c}.

For −1 ≤ p ≤ n − 2 let D_{p} be the family of all A ∈ K_{∞}^{n} such that H_{k}(R^{n} \ A) = 0 for k ≤ p. By Alexander duality, a set A ∈ K_{∞}^{n} is in D_{p} if and only if H^{q}(A) = 0 for n − p − 1 ≤ q ≤ n, where H^{q}(A) denotes the q^{th} reduced Čech cohomology group of A. Observe that D_{−1} = K_{∞}^{n} \ {Ṙ^{n}}.

3.17. Theorem. For every −1 ≤ p ≤ n − 2 we have σ(D_{p}) = HU(p), and the function c ↦ HU(p, c) is a filtration of σ(D_{p}).

Proof. The case p = −1 is [Vä_{1} , 3.4]. Let p ≥ 0. The inclusion HU(p) ⊂ σ(D_{p}) was proved in [Al, 6.6]. In fact, the proof shows that each HU(p, c) is contained in a stable subfamily of D_{p}. Conversely, assume that M ⊂ D_{p} is a stable family. Using inductively [Al, 6.4], 3.7 and 3.10 we see that M is contained in some HU(p, c).

3.18. Corollary. A domain G ⊂ R^{n} is not completely hlog (p)-uniform if and only if there is a sequence (α_{j}) of similarities of R^{n} such that

(1) {0, e_{1}} ⊂ α_{j}∂G,

(2) Ṙ^{n} \ α_{j}G → A ∈ K_{∞}^{n}, and either 0 ∈ intA or H_{k}(R^{n} \ A) ≠ 0 for some 0 ≤ k ≤ p.

3.19. NUD sets. We recall that a closed set A ⊂ R^{n} is hlog (p, c)-NUD (null-set for uniform domains) if intA = ∅ and if R^{n} \ A is a hlog (p, c)-uniform domain. For 0 ≤ p ≤ n − 2 let HN(p, c) denote the family of all completely hlog (p, c)-NUD sets in R^{n}, and let HN(−1, c) be the family of all c-porous sets. We set HN(p) = ∪{HN(p, c) : c ≥ 1}. Let L_{p} be the family of all closed sets in R^{n} with topological dimension at most p.

3.20. Theorem. Let p ≥ −1 and q ≥ 0 be integers with p+q = n−2. Then σ(L_{q}) = HN(p), and c ↦ HN(p, c) is a filtration of σ(L_{q}).

Proof. The case p = −1, q = n − 1, is [Vä_{2} , 5.4]. Let p ≥ 0. In [Al, 6.10] it was proved that the closure of HN(p, c) is a stable subfamily of L_{q}. Conversely, let M ⊂ L_{q} be stable. By [HW, VIII.4F, p. 137], we have L_{q} ⊂ D_{p}. Hence M ⊂ HU(p, c) for some c by 3.17, which implies that M ⊂ HN(p, c).

3.21. Theorem. Suppose that A ∈ HU(p, c) and that f: A → Ṙ^{n} is a θ-quasimöbius map with ∞ ∈ fA. Then fA ∈ HU(p, c_{1}) with c_{1} = c_{1}(c, θ).

Proof. See [Al, 6.11]. The condition ∞ ∈ fA is unnecessary if we allow the possibility ∞ ∈ G for uniform domains; see [Vä_{3} , 5.4].

3.22. Theorem. Let E be a normed space of dimension at least n with n ≥ 3. Let γ ⊂ E be an arc of c-bounded turning. Then γ is completely hlog (n − 3, c_{1})-NUD with c_{1} = c_{1}(c, n).

Proof. This follows inductively from 3.7, 3.10, and [Al, 5.4].

3.23. Remark. Theorem 3.22 is valid in the case dim E = ∞. However, one can show that in this case, every compact set in E is htop (p, c_{0})-NUD for every p ≥ 0 with a universal constant c_{0} . This will be proved in Part III.

4. Relations between homotopical and homological uniformity

4.1. Summary of Section 4. Roughly speaking, homotopical properties imply the corresponding homological properties. For example, if a space X is pconnected, it is also p-acyclic, that is, H_{k}(X) = 0 for 0 ≤ k ≤ p. The converse is not true, but it is true for 1-connected spaces. This follows directly from the Hurewicz theorem [Sp, 7.5.5].

In this section we consider quantitative versions of these results. We show in 4.2 that a completely htop (p, c)-uniform domain is completely hlog (p, c_{1})- uniform with c_{1} = c_{1}(c, p). The case p = 0 was given in [Al, p. 7] and the case p = 1 in [Al, 1.7], both with c_{1} = 2c + 1. The converse is true for p = 0 [Al, p. 7] but not for p ≥ 1, since for n ≥ 4, the complement of a BT arc in R^{n} is completely hlog (n − 3)-uniform by 3.22, but it need not be simply connected and hence not htop (1)-uniform; see [Al, pp. 32–33].

We conjecture that the converse of 4.2 is true for htop (1, c)-uniform domains. In 4.5 we prove this conjecture in R^{n}, but the proof makes use of compact families of sets, and hence the bound for the htop uniformity depends also on n.

4.2. Theorem. If a domain G ⊂ E is completely htop (p, c)-uniform, it is completely hlog (p, c_{1})-uniform with c_{1} = c_{1}(c, p).

Proof. The easy case p = 0 was given in [Al, p. 7] with c_{1} = 2c + 1. Proceeding inductively, we see that it suffices to prove that if G is completely htop (p, c)-uniform, it is hlog (p, c_{1})-uniform with c_{1} = c_{1}(c, p). By 3.10, it suffices to show that G has property HT^{∗}(p, c′, c_{1}) where c′ = c′(c, p) is given in 3.8.

Let z ∈ Z_{p}(G) be such that d ≤ c′r where d = d(|z|) and r = d(|z|, ∂G). We must show that z = ∂g for some g such that

(4.3)  d(|g|) ≤ c_{1}d,  d(|g|, ∂G) ≥ r/c_{1}.

Let z = ∑_{σ<z} n_{σ}σ be the normal representation. Fix a point a ∈ |z|. We use the method described in 2.5 and 2.6 to construct maps h_{σ}: Δ^{p} × I → G such that φ = ∑_{σ<z} n_{σ}\overline{h}_{σ} is a simple prism of z. For 0 ≤ q ≤ p + 1, let K_{q} be the subcomplex of P Δ^{p} consisting of the simplexes s ∈ P Δ^{p} such that either s ⊂ Δ^{p} × {0, 1} or dim s ≤ q. For σ < z and y ∈ |K_{0}|, define h_{σ}(y) by h_{σ}(x, 0) = σ(x) and h(x, 1) = a. The compatibility condition of 2.6 is clearly satisfied for the 0-simplexes in K_{0} . Proceeding inductively, assume that for some 0 ≤ q ≤ p, the maps h_{σ} have been defined in |K_{q}| in such a way that setting A_{q} = ∪{h_{σ}|K_{q}| : σ < z} we have

(4.4)  d(A_{q}) ≤ (2c + 1)^{q}d,  d(A_{q}, ∂G) ≥ r/(2c)^{q}.

Since A_{0} = |z|, (4.4) holds for q = 0. We extend these maps to |K_{q+1}| as follows: Let s ∈ K_{q+1} \ K_{q} and let σ < z. Then the map w_{s}^{σ} = h_{σ}|∂s is defined. By (4.4) we have d(|w_{s}^{σ}|) ≤ (2c + 1)^{q}d and d(|w_{s}^{σ}|, ∂G) ≥ r/(2c)^{q}. Since G is htop (q, c)-uniform, we can extend w_{s}^{σ} to a map u_{s}^{σ}: s → G such that

d(|u_{s}^{σ}|) ≤ cd(|w_{s}^{σ}|) ≤ cd(A_{q}),
d(|u_{s}^{σ}|, ∂G) ≥ r/(2c)^{q+1},

where the second inequality follows from 3.3. These maps are chosen for all s ∈ K_{q+1} \ K_{q} and for all σ < z in a compatible way, that is, u_{s}^{σ} = u_{t}^{τ} whenever w_{s}^{σ} = w_{t}^{τ}. It follows that

d(A_{q+1}) ≤ d(A_{q}) + 2cd(A_{q}) ≤ (2c + 1)^{q+1}d,

d(A_{q+1}, ∂G) ≥ r/(2c)^{q+1},

and hence (4.4) holds for q ↦ q + 1.

After the step q = p we have the maps h_{σ}: Δ^{p} × I → X, which give a simple prism φ = ∑_{σ<z} n_{σ}\overline{h}_{σ} of z. By 2.10 we have ∂g(φ) = z_{1} −z where z_{1} is the cycle φα_{1}Δ^{p}. Thus |z_{1}| ⊂ {a}, and hence z_{1} = ∂g_{1} for a chain g_{1} with |g_{1}| ⊂ {a}. Setting g = g_{1} − g(φ) we have ∂g = z and |g| ⊂ A_{p+1} . From (4.4) it follows that g satisfies (4.3) with c_{1} = (2c + 1)^{p+1} .

4.5. Theorem. If a domain G ⊂ R^{n} is completely hlog (p, c)-uniform and htop (1, c)-uniform, then G is completely htop (p, c_{1})-uniform with c_{1} = c_{1}(c, n).

Proof. We use the notation of [Al, §6], partially recalled in 3.16. The complement of G belongs to the family H = HU(p, c) ∩ U(1, c). By 3.17 and by [Al, 6.8], there are stable families L ⊂ D_{p} and L′ ⊂ C_{1} such that H is contained in L ∩ L′, which clearly is a stable subfamily of D_{p} ∩ C_{1} . Since D_{p} ∩ C_{1} = C_{p} by the Hurewicz theorem, the theorem follows from [Al, 6.8].

5. Turning conditions

5.1. Summary of Section 5. A metric space X is of c-bounded turning or briefly c-BT if each pair a, b of points in X can be joined by an arc γ with d(γ) ≤ c|a − b|. More generally, let p ≥ 0 be an integer. The space X is htop (p, c)-BT if each map f: S^{p} → X extends to a map g: \overline{B}^{p+1} → X such that d(|g|) ≤ cd(|f|). Thus c-BT is equivalent to htop (0, c)-BT.

In this section we give some improvements on the results of [Al] concerning htop (p, c)-BT, and also study the corresponding homological property.

5.2. Basic concepts. We say that a metric space X is hlog (p, c)-BT if each pcycle z in X bounds a chain g with d(|g|) ≤ cd(|z|). This implies that H_{p}(X) = 0. Furthermore, let a ∈ X and r > 0. It is easy to see that the homomorphism H_{p}(B(a, r)) → H_{p}(B(a, c_{1}r)) is zero for c_{1} = 2c + 1. Conversely, if this is true for all a and r, then X is hlog (p, c)-BT for any c > 2c_{1} . Consequently, the property hlog (p, c)-BT is quantitatively equivalent to the property hlog outer (p, c)-joinable [Vä_{3} ] together with the condition H_{p}(X) = 0.

In some cases these quantitative properties are unnecessarily strong, and they can be replaced by the corresponding topological conditions. Let G be an open set in E, and let p ≥ 0 be an integer. We say that G is p-LC rel E if each neighborhood U of each point a ∈ ∂G contains a neighborhood V such that every map f: S^{p} → V ∩ G is null-homotopic in U ∩ G. If this holds for all 0 ≤ p ≤ m, G is said to be LC^{m} rel E.

Replacing homotopy by homology, we obtain the concepts p-lc rel E and lc^{m} rel E. For example, G is p-lc rel E if for each a ∈ ∂G and for each neighborhood U of a there is a neighborhood V of a such that H_{p}(V ∩ G) → H_{p}(U ∩ G) is zero.

These and related concepts have been extensively studied in the topological literature since the thirties. See, for example, [EW] and [Wi].

5.3. Remarks. 1. The quantitative properties imply the topological properties in the obvious way. For example, if G ⊂ E is completely hlog (p, c)-BT, G is lc^{p} rel E.

2. If X is hlog (0, c)-BT, X is c-BT. Conversely, a c-BT space is hlog (0, 2c + 1)-BT. The proofs are elementary.

3. If X is completely htop (p, c)-BT, then X is completely hlog (p, c_{1})-BT with c_{1} = c_{1}(c, p). This can be proved by using similar techniques as in 4.2.

4. Let F ⊂ R^{n} be closed with intF = ∅, and let 0 ≤ p ≤ n − 2 and c ≥ 1. Set G = R^{n} \ F. Then the following conditions are equivalent:

(1) dim F ≤ n − p − 2,

(2) G is completely hlog (p, \sqrt{2} )-BT,

(3) G is completely hlog (p)-BT,

(4) G is lc^{p} rel R^{n}. The implications (2) ⇒ (3) ⇒ (4) are trivial, and (4) ⇒ (1) ⇒ (2) follows from [Ku, Th. 2′, p. 9]. Indeed, assume that (1) is true and that z is a k -cycle in G with k ≤ p. By Jung’s theorem [Fe, 2.10.41], |z| is contained in an open ball of radius d(|z|)/\sqrt{2}. Since k < n − dimF − 1, it follows from [Ku, Th. 2′, p. 9] that z = ∂g for some chain g in B \ F. Since d(|g|) < d(|z|)\sqrt{2}, (2) is true.

5.4. Lemma. Suppose that U is open in E and LC^{p} rel E. Suppose also that X is a compact polyhedron with dim X ≤ p + 1. Then for each map f: X → \overline{U} and for each ε > 0 there is a map g: X → U with ∥g − f∥ < ε.

Proof. Using the compactness of fX and the LC^{p} -condition it is easy to show that for every s > 0 there is t > 0 such that if y ∈ fX and 0 ≤ q ≤ p, then each map φ: S^{q} → U ∩ B(y, t) is null-homotopic in U ∩ B(y, s). The map g is found by choosing a sufficiently fine triangulation of X, starting with the vertices and proceeding by induction to skeletons of higher dimension.

5.5. Plumpness and porosity conditions. The definition of a c-plump open set U ⊂ E was recalled in 3.16. This property could also be called (0, c)-plumpness, since it is a special case of (p, c)-plumpness defined in [Al, 3.1]. In this paper we shall call this property htop (p, c)-plumpness, since a hlog version will be given in §6. We recall the definition. Let p ≥ 0 be an integer and let c ≥ 1. An open set U ⊂ E is htop (p, c)-plump if for each map f: Δ^{p} → \overline{U} and for each 0 < r < d(U) there is a map g: Δ^{p} → U such that ∥g − f∥ ≤ r and d(|g|, ∂U) ≥ r/c.

It is essential that we consider maps f: Δ^{p} → \overline{U} and not only maps into U. If the condition above is valid for every map f: Δ^{p} → U, we say that U is htop inner (p, c)-plump. This is an absolute concept, which makes sense in any metric space.

A closed set F ⊂ E is htop (p, c)-porous or htop inner (p, c)-porous if intF = ∅ and if U = R^{n} \ F satisfies the appropriate plumpness condition.

Trivially, htop (p, c)-plumpness and (p, c)-porosity imply the corresponding inner condition. The converse is true for p = 0 with an arbitrary small change of c, but not for p ≥ 1. For example, let F be a line segment in R^{3} . Then F is htop (1)-porous and htop inner (2)-porous but not htop (2)-porous.

We next give a result in the converse direction:

5.6. Theorem. Suppose that U ⊂ E is open, LC^{p−1} rel E and htop inner (p, c)-plump. Then U is htop (p, c_{1})-plump for every c_{1} > c.

Proof. Let f: Δ^{p} → \overline{U} be continuous and let 0 < r < d(U). Given c_{1} > c, we set ε = (1 − c/c_{1})r. By 5.4 there is a map f′: Δ^{p} → U such that ∥f′ − f∥ ≤ ε. Since U is htop inner (p, c)-plump, there is a map g: Δ^{p} → U such that ∥g−f′∥ ≤ r − ε and d(|g|, ∂U) ≥ (r − ε)/c = r/c_{1} . Since ∥g − f∥ ≤ r, the theorem follows.

We apply Theorem 5.6 to improve 4.8 and 3.10 of [Al]:

5.7. Theorem. Let G ⊂ E be a domain and let 1 ≤ p < dim E. Then the following conditions are quantitatively equivalent:

(1) G is htop (p, c)-plump and completely htop (p − 1, c)-BT.

(2) G is htop inner (p, c)-plump and completely htop (p − 1, c)-BT.

(3) G is completely htop (p − 1, c)-uniform.

Proof. The implication (1) ⇒ (2) is trivial, and (1) ⇔ (3) was proved in [Al, 4.8]. Since htop (p − 1, c)-BT implies (p − 1)-LC rel E, (2) implies (1) by 5.6.

5.8. Theorem. Let D ⊂ E be a domain and let F be closed in E with intF = ∅. Suppose that 0 ≤ p < dim E, that D is htop (p, c)-BT and that E \ F is LC^{p} rel E. Then G = D \ F is htop (p, c_{1})-BT for every c_{1} > c.

Proof. Let ε > 0. We show that G is htop (p, c + ε)-BT. Let f: S^{p} → G be continuous and nonconstant. Define f′: S^{p}(1/2) → D by f′(x) = f(2x). Since D is htop (p, c)-BT, there is an extension g′: \overline{B}^{p+1} (1/2) → D of f′ such that d(|g′|) ≤ cd(|f′|) = cd(|f|). Write 4δ = εd(|f|) ∧ d(|g′|, ∂D) ∧ d(|f|, F). By 5.4, there is a map g_{1}: \overline{B}^{p+1} (1/2) → E \ F such that ∥g_{1} − g′∥ ≤ δ. We extend g_{1} to a map g: \overline{B}^{p+1} → E by setting g((1 − t)e/2 + te) = (1 − t)g_{1}(e/2) + tf(e) for e ∈ S^{p} and 0 ≤ t ≤ 1. By the choice of δ we have |g| ⊂ D ∩ (E \ F) = G. Moreover, g | S^{p} = f, and

d(|g|) ≤ d(|g_{1}|) + 2δ ≤ d(|g′|) + 4δ ≤ cd(|f|) + 4δ ≤ (c + ε)d(|f|).

Thus g is the required extension of f.

5.9. The homological case. We shall give a homological version of 5.8 in 5.11. The proof is based on Lemma 5.10 below, which can be regarded as a homological version of Lemma 5.4. A homological version of 5.7 will be given in Section 6, where we discuss homological plumpness.

5.10. Lemma. Suppose that U is open in E and lc^{p−1} rel E. Let γ ∈ S_{p}(\overline{U}) and let ε > 0. Then there are g ∈ S_{p+1}(E) and γ_{1} ∈ S_{p}(U) such that

(1) |g| ⊂ \overline{B}(|γ|, ε),

(2) |γ − γ_{1} − ∂g| ⊂ \overline{B}(|∂γ|, ε).

If γ is a cycle, (2) means that ∂g = γ − γ_{1} , and hence γ_{1} is also a cycle.

Proof. Let γ = ∑_{σ<γ} n_{σ}σ be the normal representation. Since |γ| is compact, it follows from the lc^{p−1} -condition that for each r > 0 there is 0 < t(r) < r such that the map H_{k}(B(a, t(r))∩U) → H_{k}(B(a, r)∩U) is zero whenever a ∈ |γ| and 0 ≤ k ≤ p − 1. Set r_{p+1} = ε and define inductively the numbers r_{p}, …, r_{0} by r_{q} = t(r_{q+1})/2.

We first assume that d(|σ|) ≤ r_{0} for all σ < γ. We shall construct a prism φ = ∑_{σ<γ} n_{σ}φ_{σ} of γ by the method explained in 2.5. Thus we start with the 0-chains of P Δ^{p} and set φ_{σ}v = σv for the vertices v = e_{j}, 0 ≤ j ≤ p. For v = e′_{j} = (e_{j}, 1) we let φ_{σ}v be a point in U with |φ_{σ}v − σe_{j}| ≤ r_{0} . These points are chosen in a compatible way; see 2.5. Proceeding inductively, assume that 0 ≤ q ≤ p and that the chain maps φ_{σ}: C_{k}(P Δ^{p}) → S_{k}(E) have been defined for all 0 ≤ k ≤ q in such a way that for each k -simplex s of P Δ^{p} and for all σ < γ we have

(i) |φ_{σ}s| ⊂ \overline{B}(σpr_{1}s, r_{k}),

(ii) |φ_{σ}s| ⊂ U whenever s ⊂ Δ^{p} × {1}.

Let s be a (q + 1)-simplex of P Δ^{p} and let σ < γ. If s ⊂ Δ^{p}, we follow the procedure of 2.5 and set φ_{σ}s = σε_{s}, where ε_{s}: Δ^{q+1} → Δ^{p} is the order-preserving isometry onto s. Assume that s ⊄ Δ^{p}. Now φ_{σ}∂s is defined, and it is a cycle in E. By (i) we have |φ_{σ}∂s| ⊂ \overline{B}(σpr_{1}s, r_{q}). We consider two cases.

Case 1. s ⊂ Δ^{p} × {1}. Now |φ_{σ}∂s| ⊂ U by (ii). Choose a point a ∈ σpr_{1}s. Then (i) implies that |φ_{σ}∂s| ⊂ \overline{B}(a, 2r_{q}), since r_{q} + d(|σ|) ≤ r_{q} + r_{0} ≤ 2r_{q}. Since 2r_{q} = t(r_{q+1}), there is a chain φ_{σ}s ∈ S_{q+1}(U) such that ∂φ_{σ}s = φ_{σ}∂s and |φ_{σ}s| ⊂ \overline{B}(a, r_{q+1}). Clearly φ_{σ}s satisfies (i) and (ii) for k = q + 1.

Case 2. s ⊄ Δ^{p} × {1}. Now we use the cone construction to define the chain φ_{σ}s ∈ S_{q+1}(E) such that ∂φ_{σ}s = φ_{σ}∂s and such that |φ_{σ}s| is contained in the convex hull of |φ_{σ}∂s|. Since |φ_{σ}∂s| ⊂ \overline{B}(σpr_{1}s, r_{q}), the condition (i) holds for k = q + 1.

The choices of the chains φ_{σ}s for σ < γ are made in a compatible way; see 2.5. The construction gives a prism φ = ∑_{σ<γ} n_{σ}φ_{σ} of γ. With the notation of 2.7 and 2.10 we then have ∂g(φ) + g(ψ) = γ_{1} − γ by 2.10. Moreover, (i) and 2.14 imply that |g(φ)| ⊂ \overline{B}(|γ|, ε) and |g(ψ)| ⊂ \overline{B}(|∂γ|, ε). Hence g = −g(φ) and γ_{1} are the desired chains, and we have proved the lemma in the case where d(|σ|) ≤ r_{0} for σ < γ.

The general case reduces to the case above by barycentric subdivision. Let γ ∈ S_{p}(\overline{U}). It suffices to show that if the lemma holds with γ replaced by Sdγ, it holds also for γ. Thus we may assume that there are g′ ∈ S_{p+1}(E) and γ_{1} ∈ S_{p}(U) such that

|g′| ⊂ \overline{B}(|Sdγ|, ε), |Sdγ − γ_{1} − ∂g′| ⊂ \overline{B}(|∂Sdγ|, ε).

Let T: S_{∗}(E) → S_{∗}(E) be the chain homotopy given in [Ro, 6.13] satisfying ∂T + T ∂ = id − Sd and |T β| ⊂ |β| for every singular chain β. Setting g = g′ + T γ we show that g and γ_{1} satisfy (1) and (2). First,

|g| ⊂ |g′| ∪ |T γ| ⊂ \overline{B}(|Sdγ|, ε) ∪ |γ| ⊂ \overline{B}(|γ|, ε),

which gives (1). Next, we have T ∂γ = γ − ∂T γ − Sdγ, and hence

|γ − γ_{1} − ∂g| = |γ − γ_{1} − ∂g′ − ∂T γ − Sdγ + Sdγ|
⊂ |γ − ∂T γ − Sdγ| ∪ |Sd γ − γ_{1} − ∂g′|
⊂ |T ∂γ| ∪ \overline{B}(|∂Sdγ|, ε).

Since |T ∂γ| ⊂ |∂γ| and |∂Sdγ| = |Sd∂γ| ⊂ |∂γ|, this gives (2).

5.11. Theorem. Let D ⊂ E be a domain and let F be closed in E with intF = ∅. Suppose that 0 ≤ p < dim E, that D is hlog (p, c)-BT and that E \ F is lc^{p} rel E. Then G = D \ F is hlog (p, c_{1})-BT for every c_{1} > c.

Proof. Let ε > 0. We show that G is hlog (p, c_{1})-BT for c_{1} = c + ε. Let z ∈ Z_{p}(G), and assume that d(|z|) > 0. Since D is hlog (p, c)-BT, there is g′ ∈ S_{p+1}(D) such that ∂g′ = z and d(|g′|) ≤ cd(|z|). Choose δ with 0 < 2δ < εd(|z|) ∧ d(|z|, F) ∧ d(|g′|, ∂D). By 5.10 there are h ∈ S_{p+2}(E) and g_{1} ∈ S_{p+1}(E \ F) such that |h| ⊂ \overline{B}(|g′|, δ) and |g′ − g_{1} − ∂h| ⊂ \overline{B}(|z|, δ). Write β = g′ − g_{1} − ∂h and g = g_{1} + β = g′ − ∂h. Then ∂g = ∂g′ = z. Moreover, |g| ⊂ |g′| ∪ |∂h| ⊂ \overline{B}(|g′|, δ) ⊂ D and |g| ⊂ |g_{1}| ∪ |β| ⊂ |g_{1}| ∪ \overline{B}(|z|, δ) ⊂ E \ F. Since d(|g|) ≤ d(|g′|) + 2δ ≤ (c + ε)d(|z|), the theorem follows.

6. Homological plumpness

6.1. Summary of Section 6. We shall define a homological version of (p, c)- plumpness. It turns out that most results of [Al] and §5 on htop (p, c)-plumpness have homological versions.

6.2. Definitions. Let 0 ≤ p < dim E be an integer and let c ≥ 1. An open set U ⊂ E is hlog (p, c)-plump if for each chain γ ∈ S_{p}(\overline{U}) and for each 0 < r < d(U) there is γ_{1} ∈ S_{p}(U) such that

(1) |γ_{1}| ⊂ \overline{B}(|γ|, r),

(2) d(|γ_{1}|, ∂U) ≥ r/c,
(3) ∂γ_{1} ∼ ∂γ in \overline{B}(|∂γ|, r).

If the condition holds for all γ ∈ S_{p}(U), U is hlog inner (p, c)-plump. If F ⊂ E is closed with intF = ∅ and if E \ F is hlog (p, c)-plump, F is hlog (p, c)-porous. Hlog inner (p, c)-porosity is defined in the obvious way.

A slightly stronger condition will be considered in 6.8.

6.3. Remarks. 1. The condition in 6.2 is relevant only if ∂γ ≠ 0. Indeed, if γ is a cycle, the condition holds with c = 1 and γ_{1} = 0.

2. We remind the reader that we are using the reduced theory. For p = 0, the condition (3) of 6.2 should be understood as ∂γ_{1} = ∂γ ∈ Z. It is easy to see that hlog (0, c)-plumpness is equivalent to htop (0, c)-plumpness, that is, ordinary c-plumpness.

3. In [Al, 3.3.3] it was proved by an easy argument that htop (p, c)-plumpness implies htop (q, c)-plumpness for all 0 ≤ q ≤ p. We do not know whether the corresponding hlog result is true.

We first give homological versions of [Al, 3.4, 3.8, 3.9].

6.4. Lemma. Let U ⊂ E be open and hlog (p, c)-plump, let 0 < r < d(U), and let γ ∈ S_{p}(\overline{U}) be a chain such that d(|∂γ|, ∂U) ≥ (c + 1)r/c. Then there is γ_{1} ∈ S_{p}(U) satisfying ∂γ_{1} = ∂γ and the conditions (1) and (2) of 6.2.

Proof. Let γ′_{1} be the chain given by 6.2. By 6.2(3), there is g ∈ S_{p}(E) such that ∂g = ∂γ − ∂γ′_{1} and |g| ⊂ \overline{B}(|∂γ|, r). We show that γ_{1} = γ′_{1} + g is the desired chain. Clearly |γ_{1}| ⊂ \overline{B}(|γ|, r) and ∂γ_{1} = ∂γ. Moreover, for every x ∈ |g| we have

d(x, ∂U) ≥ d(|∂γ|, ∂U) − d(x, |∂γ|) ≥ (c + 1)r/c − r = r/c.

6.5. Theorem. Let U ⊂ E be hlog (p, c_{1})-plump and let F ⊂ E be hlog (p, c_{2})-porous. Then V = U \ F is hlog (p, 3c_{1}c_{2})-plump.

Proof. Let 0 < r < d(V) and let γ ∈ S_{p}(\overline{V}) = S_{p}(\overline{U}). Then there is γ_{2} ∈ S_{p}(U) such that |γ_{2}| ⊂ \overline{B}(|γ|, 2r/3), ∂γ_{2} ∼ ∂γ in \overline{B}(|∂γ|, 2r/3), and d(|γ_{2}|, ∂U) ≥ 2r/3c_{1} . By the hlog (p, c_{2})-porosity of F, there is γ_{1} ∈ S_{p}(E \ F) such that |γ_{1}| ⊂ \overline{B}(|γ_{2}|, r/3c_{1}), ∂γ_{1} ∼ ∂γ_{2} in \overline{B}(|∂γ_{2}|, r/3c_{1}), and d(|γ_{1}|, F) ≥ r/3c_{1}c_{2} .

We show that γ_{1} is the desired chain. First, we have |γ_{1}| ⊂ \overline{B}(|γ|, 2r/3 + r/3c_{1}) ⊂ \overline{B}(|γ|, r), which proves (1) of 6.2. Next,

d(|γ_{1}|, ∂U) ≥ d(|γ_{2}|, ∂U) − r/3c_{1} ≥ 2r/3c_{1} − r/3c_{1} = r/3c_{1}.

Since d(|γ_{1}|, F) ≥ r/3c_{1}c_{2} , we obtain d(|γ_{1}|, ∂V) ≥ r/3c_{1}c_{2} . Finally, ∂γ ∼ ∂γ_{2} ∼ ∂γ_{1} in \overline{B}(|∂γ|, 2r/3 + r/3c_{1}) ⊂ \overline{B}(|∂γ|, r).

6.6. Corollary. The union of a hlog (p, c_{1})-porous set and a hlog (p, c_{2})- porous set is hlog (p, 3c_{1}c_{2})-porous.

6.7. Plumpness and uniformity. We recall that htop (p, c)-uniformity and htop (p + 1, c)-plumpness are closely connected; see 5.7. In particular, a closed set F ⊂ E is completely htop (p, c)-NUD if and only if, quantitatively, it is htop (p + 1, c)-porous [Al, 4.9]. We shall give homological versions of these and some related results. More precisely, the homological versions of [Al, 4.2], 5.6, 5.7, [Al, 4.9], [Al, 4.10] and [Al, 4.11] will be given as 6.9, 6.12, 6.13, 6.15, 6.16 and 6.17, respectively.

6.8. Strong hlog plumpness. We shall also consider the following variation of hlog plumpness: An open set U ⊂ E is said to be strongly hlog (p, c)-plump if for each γ ∈ S_{p}(\overline{U}) and for each 0 < r < d(U) there are γ_{1} ∈ S_{p}(U) and g ∈ S_{p+1}(E) such that

(1) |g| ⊂ \overline{B}(|γ|, r),

(2) d(|γ_{1}|, ∂U) ≥ r/c,

(3) |∂g − γ + γ_{1}| ⊂ \overline{B}(|∂γ|, r).

It is easy to show that this property implies hlog (p, c)-plumpness, defined in 6.2. Indeed, (2) above is the same as 6.2(2). Setting h = ∂g − γ + γ_{1} we have ∂h = ∂γ_{1} − ∂γ, and hence ∂γ_{1} ∼ ∂γ in \overline{B}(|∂γ|, r). Finally, |γ_{1}| ⊂ |h| ∪ |γ| ∪ |g| ⊂ \overline{B}(|γ|, r).

We do not know whether the converse is true. In 6.15 we show that it is true in the special case int(E \ U) = ∅.

6.9. Theorem. If G ⊂ E is a completely hlog (p, c)-uniform domain, then G is strongly hlog (p + 1, c_{1})-plump and hence hlog (p + 1, c_{1})-plump with c_{1} = c_{1}(c, p).

Proof. The proof is a variation of the proof of 5.10. Let γ ∈ S_{p}(\overline{U}) with normal representation γ = ∑_{σ<γ} n_{σ}σ. Let 0 < r < d(G) and set r′ = r/2(4c)^{p+1} . As in 5.10, we may assume that d(|σ|) ≤ r′ for all σ < γ, using barycentric subdivision. We apply the method of 2.5 to construct a prism φ = ∑_{σ<γ} n_{σ}φ_{σ} of γ. Thus we set φ_{σ}v = σv if v = e_{j} is a vertex of Δ^{p}. Since G is hlog (0, c)- uniform, it is 4c-plump; cf. [Vä_{1} , 2.15]. If v = e′_{j} = (e_{j}, 1), we can therefore choose the point φ_{σ}v ∈ G such that |φ_{σ}v − σe_{j}| ≤ 2r′ and d(φ_{σ}v, ∂G) ≥ r′/2c. These points are chosen in a compatible way; see 2.5.

Proceeding inductively, assume that 0 ≤ q ≤ p and that the chain maps φ_{σ}: C_{k}(P Δ^{p}) → S_{k}(E) have been defined for 0 ≤ k ≤ q in such a way that for each k -simplex s ∈ P Δ^{p} and for each σ < γ we have

(i) |φ_{σ}s| ⊂ \overline{B}(σpr_{1}s, 2(4c)^{k}r′),
(ii) d(|φ_{σ}s|, ∂G) ≥ r′/(2c)^{k+1} whenever s ⊂ Δ^{p} × {1}.

These conditions are clearly true for k = 0. Let s ∈ P Δ^{p} be a (q+1)-simplex and let σ < γ. If s ⊂ Δ^{p}, we set φ_{σ}s = σε_{s} according to 2.5. Assume that s ⊄ Δ^{p}. Now φ_{σ}∂s is defined, and it is a q -cycle in E. By (i) we have

(6.10)  |φ_{σ}∂s| ⊂ \overline{B}(σpr_{1}s, 2(4c)^{q}r′).

We consider two cases.

Case 1. s ⊂ Δ^{p} × {1}. Now |φ_{σ}∂s| ⊂ G. Since G is hlog (q, c)-uniform, φ_{σ}∂s = ∂φ_{σ}s for some chain φ_{σ}s ∈ S_{q+1}(G) satisfying the uniformity conditions

(6.11)  d(|φ_{σ}s|) ≤ cd(|φ_{σ}∂s|),  d(|φ_{σ}∂s|, ∂G) ≤ 2cd(|φ_{σ}s|, ∂G),

where the second inequality follows from 3.3. We show that φ_{σ}s satisfies (i) and (ii) with k = q + 1.

From (6.10) and (6.11) it follows that |φ_{σ}s| ⊂ \overline{B}(σpr_{1}s, r′′), where

r′′ = 2(4c)^{q}r′ + cd(|φ_{σ}∂s|) ≤ 2(4c)^{q}r′ + c(r′ + 4(4c)^{q}r′) < 2(4c)^{q+1}r′.

This gives (i) for k = q+1. By (ii) of the inductive hypothesis we have d(|φ_{σ}∂s|, ∂G) ≥ r′/(2c)^{q+1} , which implies (ii) for k = q + 1 by (6.11).

Case 2. s ⊄ Δ^{p} ×{1}. As in 5.10, we use the cone construction to get a chain φ_{σ}s ∈ S_{q+1}(E) such that ∂φ_{σ}s = φ_{σ}∂s and such that |φ_{σ}s| is contained in the convex hull of |φ_{σ}∂s|. Since d(|σ|) ≤ r′, (6.10) implies that

d(|φ_{σ}s|) ≤ d(|φ_{σ}∂s|) ≤ (1 + 4(4c)^{q})r′ < 2(4c)^{q+1}r′,

and hence (i) is true for k = q + 1.

The choices of the chains φ_{σ}s for σ < γ are made in a compatible way. The construction gives a prism φ = ∑_{σ<γ} n_{σ}φ_{σ} of γ. With the notation of 2.7 and 2.10 we have ∂g(φ)+g(ψ) = γ_{1}−γ by 2.10. Since r = 2(4c)^{p+1}r′, it follows from (i) and 2.14 that |g(φ)| ⊂ \overline{B}(|γ|, r) and |g(ψ)| ⊂ \overline{B}(|∂γ|, r). Furthermore, (ii) implies that d(|γ_{1}|, ∂G) ≥ r/c_{1} with c_{1} = 2(2c)^{p+1}(4c)^{p+1} . Since −∂g(φ)−γ+γ_{1} = g(ψ), the strong hlog (p + 1, c_{1})-plumpness condition holds with g = −g(φ).

6.12. Theorem. Suppose that U ⊂ E is open, lc^{p−1}rel E and hlog inner (p, c)-plump. Then U is hlog (p, c_{1})-plump for every c_{1} > c.

Proof. This is a homological version of 5.6. Assume that γ′ ∈ S_{p}(\overline{U}) and that 0 < r < d(U). Given c_{1} > c, we set ε = (1 − c/c_{1})r. By 5.10, there are g ∈ S_{p+1}(E) and γ′_{1} ∈ S_{p}(U) such that

|g| ⊂ \overline{B}(|γ|, ε), |γ − γ′_{1} − ∂g| ⊂ \overline{B}(|∂γ|, ε).

Hence ∂γ ∼ ∂γ′_{1} in \overline{B}(|∂γ|, ε) and |γ′_{1}| ⊂ \overline{B}(|γ|, ε). Since U is hlog inner (p, c)- plump, there is γ_{1} ∈ S_{p}(U) such that γ_{1} ⊂ \overline{B}(|γ′_{1}|, r−ε), d(|γ_{1}|, ∂U) ≥ (r−ε)/c = r/c_{1} , and ∂γ_{1} ∼ ∂γ′_{1} in \overline{B}(|∂γ′_{1}|, r − ε). It follows that |γ_{1}| ⊂ \overline{B}(|γ|, r) and ∂γ ∼ ∂γ_{1} in \overline{B}(|∂γ|, r).

6.13. Theorem. Let G ⊂ E be a domain and let 1 ≤ p < dim E. Then the following conditions are quantitatively equivalent:

(1) G is completely hlog (p, c)-plump and completely hlog (p − 1, c)-BT.

(2) G is completely hlog inner (p, c)-plump and completely hlog (p − 1, c)- BT.

(3) G is completely hlog (p − 1, c)-uniform.

Proof. The implication (1) ⇒ (2) is trivial. Since hlog (q, c)-BT implies q -lc rel E, (2) implies (1) by 6.12. The implication (3) ⇒ (1) follows from 6.9. It remains to show that (1) implies (3).

We use induction on p. Assume that m ≥ 1 and that the theorem holds for 1 ≤ p ≤ m − 1. Suppose that (1) is true for p = m. We show that G is hlog (m − 1, c_{1})-uniform with c_{1} = c_{1}(c, p). The argument will be valid also in the case m = 1 without any inductive hypothesis.

By 3.10 and the inductive hypothesis, it suffices to show that for every c′ ≥ 1, G has property HT^{∗}(m − 1, c′, c_{2}) of 3.4 with some c_{2} = c_{2}(c, c′). Let z be an (m − 1)-cycle in G with d ≤ c′r where d = d(|z|) and r = d(|z|, ∂G). Since G is hlog (m − 1, c)-BT, z = ∂g for some g ∈ S_{m}(G) with d(|g|) ≤ cd(|z|). Since (c + 1)/c ≤ 2, we can apply 6.4 with r ↦ r/2, p ↦ m, γ ↦ g to get a chain g_{1} ∈ S_{m}(G) such that

∂g_{1} = z, |g_{1}| ⊂ \overline{B}(|g|, r/2), d(|g_{1}|, ∂G) ≥ r/2c.

Then

\begin{aligned}d(|g_{1}|) ≤ d(|g|) + r & ≤ cd + r ≤ (cc′ + 1)r, \\ d(|z|, ∂G) & = r ≤ 2cd(|g_{1}|, ∂G). \end{aligned}

These inequalities give HT^{∗}(m − 1, c′, c_{2}) with c_{2} = 2cc′ + 1.

6.14. Lemma. Suppose that U ⊂ E is hlog (p − 1, c)-BT and that F ⊂ E is hlog (p)-porous. Then V = U \ F is hlog (p − 1, c_{1})-BT for all c_{1} ≥ c.

Proof. Let c_{1} > c and let z ∈ Z_{p−1}(V). Then there is g ∈ S_{p}(U) such that ∂g = z and d(|g|) ≤ cd(|z|). Applying 6.4 with the substitution U ↦ E \ F, γ ↦ g and with a sufficiently small r we find a p-chain g_{1} in V with ∂g_{1} = z and d(|g_{1}|) ≤ c_{1}d(|z|).

6.15. Theorem. Let F ⊂ E be closed and let 1 ≤ p < dim E. Then the following conditions are quantitatively equivalent:

(1) F is completely hlog (p − 1, c)-NUD.

(2) F is completely hlog (p, c)-porous.

(3) F is completely strongly hlog (p, c)-porous.

Proof. The implication (3) ⇒ (2) was explained in 6.8. The theorem follows from 6.9, 6.13 and 6.14.

Uniform domains of higher order II 437

6.16. Theorem. Suppose that D ⊂ E is a completely hlog (p, c)-uniform domain and that F ⊂ E is completely hlog (p, c)-NUD. Then D\F is a completely hlog (p, c_{1})-uniform domain with c_{1} = c_{1}(c, p).

Proof. By 6.13, the open sets D and E \ F are completely hlog (p + 1, c_{0})- plump and completely hlog (p, c_{0})-BT with c_{0} = c_{0}(c, p). Hence G = D \ F is completely hlog (p+1, 3c_{0}^{2})-plump by 6.5. Moreover, G is completely hlog (p, c_{0})- BT by 6.14. From 6.13 it follows that G is completely hlog (p, c_{1})-uniform with c_{1} = c_{1}(c, p).

6.17. Corollary. If F_{1} and F_{2} are completely hlog (p, c)-NUD, then F_{1} ∪F_{2} is completely hlog (p, c_{1})-NUD with c_{1} = c_{1}(c, p).

6.18. Relations between htop and hlog plumpness. In 4.2 we proved that complete htop (p, c)-uniformity quantitatively implies complete hlog (p, c)-uniformity. We do not know whether the corresponding result for plumpness is true. However, it follows from [Al, 4.9], 4.2 and 6.15 that a completely htop (p, c)-porous set F ⊂ E is completely hlog (p, c_{1})-porous with c_{1} = c_{1}(c, p).

References

[Al] Alestalo, P.: Uniform domains of higher order. - Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 94, 1994, 1–48.

[EW] Eilenberg, S., and R.L. Wilder: Uniform local connectedness and contractibility. - Amer. J. Math. 64, 1942, 613–622.

[Fe] Federer, H.: Geometric measure theory. - Springer-Verlag, 1969.

[HY] Heinonen, J., and S. Yang: Strongly uniform domains and periodic quasiconformal maps. - Ann. Acad. Sci. Fenn. Ser. A I Math. 20, 1995, 123–148.

[HW] Hurewicz, W., and H. Wallman: Dimension theory. - Princeton University Press, 1941.

[Ku] Kuzminov, V.I.: Homological dimension theory. - Russian Math. Surveys 23/5, 1968, 1–45.

[Ro] Rotman, J.J.: An introduction to algebraic topology. - Springer-Verlag, 1988.

[Sp] Spanier, E.H.: Algebraic topology. - McGraw-Hill, 1966. [Vä_{1} ] Väisälä, J.: Uniform domains. - Tôhoku Math. J. 40, 1988, 101–118. [Vä_{2} ] Väisälä, J.: Invariants for quasisymmetric, quasimöbius and bilipschitz maps. - J. Ana-

lyse Math. 50, 1988, 201–223.

[Vä_{3} ] Väisälä, J.: Metric duality in euclidean spaces. - Math. Scand. (to appear).

[Wi] Wilder, R.L.: Topology of manifolds. - Amer. Math. Soc. Colloquium Publ. 32, 1949.

Received 28 June 1995