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Topology Proceedings

ON THE ASPHERICITY OF ONE-POINT UNIONS OF CONES

KATSUYA EDA AND KAZUHIRO KAWAMURA

Abstract. We prove that the one-point union of two copies of the cone over the Hawaiian earring is asperical.

1. Introduction and definitions

The one-point union Cℍ ∨ Cℍ of two copies of the cone over the Hawaiian earring ℍ is not simply connected [9]. This is a wellknown example of a non-contractible one-point union of two contractible spaces [15, p.59]. The non-triviality of its fundamental group follows from the presentation of the group given by H.B. Griffiths in [10], a flaw in which was remedied in [13]. Another proof was suggested by R.H. Fox in his review of [9] and is proved in detail in [3, Theorem 2].

On the other hand, the Hawaiian earring and, more generally, every planar or one-dimensional space is aspherical in the sense that all homotopy groups of dimension at least 2 is trivial [17],[2] and [1]. In [6], the authors constructed a 2-dimensional, simply-connected, cell-like Peano continuum SC(𝕊^{1}) such that the second homotopy group π_{2}(SC(𝕊^{1})) is non-trivial. In [7] the authors provided a construction which, on one hand, produces a space homotopy equivalent to SC(𝕊^{1}) and, on the other hand, produces a space homotopy equivalent to Cℍ ∨ Cℍ. This leads to a question whether the space Cℍ ∨ Cℍ is aspherical. The present paper answers this question in the affirmative.

2000 Mathematics Subject Classification. Primary 55Q20, 55Q70; Secondary 57Q99, 54F50, 54F15.

Key words and phrases. Hawaiian earring, one-point union of cones, asphericity.

The first author was supported in part by the Japanese- Slovenian research grant BI–JP/03–04/2 and JSPS Grants-in-Aid for Scientific research (C) No.20540097.

The second author was supported in part by JSPS Grants-in-Aid for scientific research (C) No.18540066.

For a Hausdorff space X, CX denotes the cone over X:

CX = X × [0, 1]/X × {1},

with the quotient topology. The peak point of CX is the point represented by X × {1} and is denoted by p. The space X is identified with the subspace X × {0}. Let X_{0} and X_{1} be two Hausdorff spaces with two points o_{0} ∈ X_{0} and o_{1} ∈ X_{1} being specified. For i = 0, 1, the peak point of CX_{i} is denoted by p_{i}. The one-point union CX_{0} ∨ CX_{1} is the space obtained from the topological sum CX_{0} ⊕ CX_{1} with the points o_{0} and o_{1} being identified with a point o.

Theorem 1.1. Let X_{0} and X_{1} be one-dimensional compact metric spaces. Then, π_{n}(CX_{0} ∨ CX_{1}) is trivial for each n ≥ 2.

Consequently we have an answer to the question above.

Corollary 1.2. Let ℍ_{0} and ℍ_{1} be copies of the Hawaiian earring ℍ. Then, π_{n}(Cℍ_{0} ∨ Cℍ_{1}) is trivial for each n ≥ 2.

Since the cone construction makes a space contractible, it does not seem that ”the coning” adds any complexity to one point unions.

Question 1.3. Let X_{0} and X_{1} be path-connected (Hausdorff) spaces such that the n-th homotopy group π_{n}(X_{0} ∨ X_{1}) is trivial. Then, is the group π_{n}(CX_{0} ∨ CX_{1}) also trivial?

At the time of this writing, we can answer the above question only for n = 2.

Theorem 1.4. Let X_{0} and X_{1} be path-connected Hausdorff spaces such that the second homotopy group π_{2}(X_{0} ∨ X_{1}) is trivial. Then, the group π_{2}(CX_{0} ∨ CX_{1}) is also trivial.

All spaces are assumed to be Hausdorff and all maps are assumed to be continuous unless otherwise stated. The word “components” means “path-connected components”. The reader is referred to [15] for undefined notions.

2. Proofs of Theorems 1.1 and 1.4

Let K be a polyhedron with a triangulation T . By abuse of notation, the subcomplex of T that defines a subpolyhedron L of K is denoted by the same symbol L. For an n-dimensional PL submanifold Q of 𝕊^{n} (with the standard triangulation), the manifold boundary of Q coincides with the topological boundary of Q in 𝕊^{n} and is denoted by ∂Q. Also IntQ = Q \ ∂Q.

The following result seems to be well-known and a proof is provided for completeness. Let n be an integer such that n ≥ 2. Note that, for n = 2, we make no assumption on the space X other than its path-connectivity.

Lemma 2.1. Let X be a path-connected space such that π_{i}(X, o) = 0 for each i = 2, ⋯, n − 1. Let P be a compact n-dimensional PL submanifold of 𝕊^{n} and let f : P → X be a map such that

(1) for each map g : 𝕊^{1} → ∂P, the composition f ∘ g : 𝕊^{1} → X

is null homotopic. Then the map f admits an extension to a map f̄: 𝕊^{n} → X.

Proof. Let P_{0}, ⋯, P_{k} be the components of P and let {C_{ij}|j = 0, ⋯ l_{i}} be the components of ∂P_{i}. We take a sufficiently fine triangulation T of 𝕊^{n} such that

(2) each P_{i} is a subpolyhedron with respect to T , and

(3) no 1-simplex of T connects distinct components C_{ij} and C_{i′j′}.

We define an extension f̄ of f by an induction on the skeleton T^{(m)}. At the outset, we fix a maximal tree T_{ij} ⊆ C_{ij} ⊆ (∂P_{i})^{(1)} and a vertex v_{ij} ∈ T_{ij} for each C_{ij}. Also we choose and fix a path p_{ij} from f(v_{ij}) to o. For a 1-simplex with vertices u and v, (u, v) denotes the 1-simplex endowed with the orientation from u toward v.

Define f̄(v) = f(v) for each vertex v ∈ P and f(v) = o for v ∉ P. For a 1-simplex σ ∉ P with vertices v_{0} and v_{1}, we define f̄ on σ as follows:

(1.1) if σ ∩ P = ∅, then let f̄ on σ be the constant map c_{o} to the

point o, and (1.2) if v_{0} ∈ C_{ij} and v_{1} ∉ P, take the unique path q_{v_{0}} in T_{ij}

from v_{0} to v_{ij} and let f̄|(v_{0}, v_{1}) be a map defined by the concatenation (f ∘q_{v_{0}})∗p_{ij} of the paths f ∘q_{v_{0}} (from f(v_{0}) to f(v_{ij})) and p_{ij} (from f(v_{ij}) to o). Notice that f̄(v_{0}) = f(v_{0})

and f̄(v_{1}) = o.

Next we take a 2-simplex σ with vertices v_{0}, v_{1} and v_{2}. If σ∩P = ∅, then let f̄|σ be the constant map c_{o}. Assume that σ intersects with P.

(2.1) If v_{0}, v_{1} ∉ P and v_{2} ∈ C_{ij}, then the restriction f̄|∂σ = f|(v_{0}, v_{1}, v_{2}) is null homotopic, because it is represented by the concatenation c_{o} ∗ (f ∘ q_{v_{2}} ∗ p_{ij})^{−1} ∗ (f ∘ q_{v_{2}} ∗ p_{ij}). Thus f̄|∂σ admits an extension on σ.

(2.2) If v_{0} ∉ P and v_{1}, v_{2} ∈ C_{ij}, then let g : ∂σ → C_{ij} ⊂ P_{i} ⊂ P be a map defined by the loop q_{v_{1}}^{−1} ∗ (v_{1}, v_{2}) ∗ q_{v_{2}} at v_{ij}. Then f|∂σ is a map defined by the path p_{ij}^{−1} ∗ (f ∘ q_{v_{1}})^{−1} ∗ f|(v_{1}, v_{2}) ∗ (f ∘ q_{v_{2}}) ∗ p_{ij} which is freely homotopic to the map f ∘ (q_{v_{1}}^{−1} ∗ (v_{1}, v_{2}) ∗ q_{v_{2}}) ≃ f ∘ g ≃ 0 by the hypothesis (1). Hence f|∂σ is null homotopic and it extends to a map on σ.

The above completes an extension procedure of f to the 2-skeleton T^{(2)} and thus completes the proof for n = 2. For n > 2, we can make use of the triviality of π_{i}(X, o) to continue the extension process and, at the n-th step, obtain the desired extension f̄ on 𝕊^{n}. □

The proof of Theorem 1.1 relies on the following lemma. The idea of using the monotone-light factorization theorem is due to Curtis and Fort, Jr [2] and was applied in [4]. A local dendrite (resp. a dendrite) is a one-dimensional locally connected compact connected metric space containing at most finitely many (resp. no) simple closed curves. A map h : S → T between compact metric spaces is said to be monotone (light resp.) if every point inverse of h is connected (zero-dimensional resp.).

Lemma 2.2. Let f : N → X be a map of a compact polyhedron N to a compact metric space X such that dim X ≤ 1. Then there exist a compact metric space G, maps m : N → G and l : G → X such that

(1) f = l ∘ m,

(2) the map m is monotone and the map l is light, and

(3) the space G has finitely many components, each of which is a local dendrite or a singleton.

Proof. Applying the monotone-light factorization [16, p.184] to the map f, we find a monotone map m : N → G and a light map l : G → X satisfying the conditions (1) and (2). We show that the space G satisfies the condition (3). Since l is a light map, by [8, Theorem 3.3.10] and the hypothesis, we see dim G ≤ dim X + 0 = 1. By the monotonicity of m, every component of N is of the form m^{−1}(S), where S is a component of G. The space N has finitely many components and so does G. Enumerate the components of G as {G_{j}} and let N_{j} = m^{−1}(G_{j}). Each N_{j} is a component of N and the restriction m|N_{j} : N_{j} → G_{j} is monotone. By the Hahn-Mazurkiewicz Theorem, G_{j} - as a continuous image of a locally connected compact connected metric space N_{j} - is locally connected. Furthermore, by the monotonicity of m|N_{j}, the induced homomorphisms (m|N_{j})^{∗} : Ȟ^{1}(G_{j}; ℤ) → Ȟ^{1}(N_{j}; ℤ) is a monomorphism [12]. Hence Ȟ^{1}(G_{j}; ℤ) is finitely generated. By [11, section 52], every one-dimensional locally connected compact connected metric space with finitely generated first Čech cohomology is a local dendrite. Hence we obtain the desired conclusion (3). □

Proof of Theorem 1.1. Fix an integer n ≥ 2 and take a map f : 𝕊^{n} → CX_{0} ∨ CX_{1}. Notice that the set f^{−1}(CX_{0} ∨ CX_{1} \ {o}) consists of at most countably many connected components, each of which is open in 𝕊^{n}. Among these components, at most finitely many of them meet f^{−1}({p_{0}, p_{1}}) and neither of them intersects both of f^{−1}(p_{0}) and f^{−1}(p_{1}).

We construct a map g : 𝕊^{n} → CX_{0} ∨ CX_{1} such that

(1) g is homotopic to f, and

(2) g(𝕊^{n}) ⊂ CX_{0} ∨ CX_{1} \ {p_{0}, p_{1}}. Let us assume, for a moment, that we have the above map g. Since X_{0}∨X_{1} is a strong deformation retract of CX_{0}∨CX_{1}\{p_{0}, p_{1}}, g is homotopic to a map from 𝕊^{n} to X_{0} ∨ X_{1}. Since dim(X_{0} ∨ X_{1}) = 1, π_{n}(X_{0} ∨ X_{1}) is trivial by [2] and [1] and hence g is null homotopic. Consequently, we conclude that f is null homotopic, as desired.

The map g and the homotopy between f and g are defined on each component of f^{−1}(CX_{0} ∨ CX_{1} \ {o}). If a component O does not meet f^{−1}({p_{0}, p_{1}}), then g|O = f|O and the homotopy H_{O} : O × [0, 1] → CX_{0} ∨ CX_{1} is given by H(x, t) = f(x) for each point (x, t) ∈ O × [0, 1].

Next take a component O of f^{−1}(CX_{0}∨CX_{1}\{o}) such that O∩ f^{−1}({p_{0}, p_{1}}) ≠ ∅. Without loss of generality, we may assume that O ∩f^{−1}(p_{0}) ≠ ∅ = O ∩f^{−1}(p_{1}). Take a compact PL submanifold N of 𝕊^{n} such that 𝕊^{n} \O ⊂ IntN and N ∩(f^{−1}({p_{0}})∩O) = ∅. Define a map f_{O} : 𝕊^{n} → CX_{0} by f_{O}(x) = f(x) for x ∈ O and f_{O}(x) = o otherwise. Let r : CX_{0} ∨CX_{1} \{p_{0}, p_{1}} → X_{0} ∨X_{1} be the standard retraction which is a homotopy equivalence.

Applying Lemma 2.2 to the composition r∘f_{O}|N : N → X_{0}∨X_{1}, we obtain a compact metric space G, a monotone map m : N → G and a light map l : G → X_{0} ∨ X_{1} such that r ∘ f_{O}|N = l ∘ m and

(3) the space G has finitely many components G_{j}, each of which is a local dendrite or a singleton.

Let C_{j} = l^{−1}({o})∩G_{j} and note l^{−1}(o) = ⋃_{j} C_{j}. Since dim C_{j} = 0, the above condition (3) implies that there exists a closed neighborhood D_{j} of C_{j} such that D_{j} is the disjoint union of finitely many dendrites, each of which intersects with C_{j}. In particular, D_{j} contains no simple closed curve and hence

(4) the inclusion i_{j} : D_{j} → G_{j} is null homotopic.

Observe that 𝕊^{n} \ O ⊆ (f_{O}|N)^{−1}({o}) ⊆ (r ∘ f_{O}|N)^{−1}({o}) = (l∘m)^{−1}({o}). There exists a compact PL submanifold P of N with the components P_{0}, ⋯, P_{k} such that \overline{𝕊^{n} \ P} is a PL submanifold and also

(5) 𝕊^{n} \ O ⊆ (l ∘ m)^{−1}({o}) ⊆ IntP, (l ∘ m)^{−1}({o}) ∩ P_{i} ≠ ∅ for each i = 0, ⋯, k, and each m(P_{i}) is a subset of some D_{j}.

Then, for each h : 𝕊^{1} → P_{i}, the composition r∘(f_{O}|P)∘h = l∘m∘h is null homotopic by (4). Since r is a homotopy equivalence, the map (f_{O}|P) ∘ h is null homotopic as well. By Lemma 2.1, f_{O}|P extends to a map g_{0} : 𝕊^{n} → CX_{0} \ {p_{0}}. Define g_{1} : 𝕊^{n} → CX_{0} ∨ CX_{1} by: g_{1}(x) = g_{0}(x) for x ∈ O and g_{1}(x) = f(x) otherwise. Then g_{1} | P = f | P. Since P and \overline{𝕊^{n} \ P} are compact PL-submanifolds and CX_{0} is contractible, we see that f and g_{1} are homotopic relative to P and g_{1}(Ō) ∩ {p_{0}, p_{1}} = ∅.

We iterate this procedure at most finitely many times to obtain the desired map g : 𝕊^{n} → CX_{0} ∨ CX_{1} \ {p_{0}, p_{1}} satisfying the conditions (1) and (2). □

The next lemma is for the proof of Theorem 1.4.

Lemma 2.3. Let K_{0}, K_{1} be disjoint closed subsets of 𝕊^{2} and X be a path-connected space with a point o ∈ X specified. There exists a compact surface P ⊂ 𝕊^{2} with boundary such that

(1) K_{0} ⊂ IntP and K_{1} ∩ P = ∅,

(2) each component of the boundary ∂P is a polygonal simple closed curve, and

(3) for each map f : P → X with f(K_{0}) = {o}, the restriction f|∂P : ∂P → X is null homotopic.

Proof. Take a compact surface P satisfying (1),(2) above and let P_{0}, ⋯, P_{k} be the components of P. We may assume that

(4) each component of 𝕊^{2} \ (K_{0} ∩ P_{i}) contains at most one com-

ponent of 𝕊^{2} \ P_{i} for every i. Indeed, if a component of 𝕊^{2} \ (K_{0} ∩ P_{i}) contains two components of 𝕊^{2} \ P_{i}, then by cutting open P_{i} along an arc connecting these components, we have a smaller neighborhood P′_{i} ⊂ P_{i} of P_{i} ∩ K_{0} so that these components are contained in a single component of 𝕊^{n} \ P′_{i}. Iterating this procedure, we can make P satisfy the condition (4).

For a map f : P → X satisfying the hypothesis of (3), we show that f|∂P_{i} is null homotopic for each component P_{i}, which follows from

f|C is null homotopic for each component C of ∂P_{i}.

Let O be a component of 𝕊^{2}\K_{0} which intersects with the component of 𝕊^{2}\P_{i} whose boundary is equal to C. The curve C divides 𝕊^{2} into two components. Let U be the component of 𝕊^{2} \ Ccontaining Int(P_{i}).

The closure \overline{U} = U ∪ C is the closed disk such that \overline{U} ⊃ P_{i} ∩ O. Define g : \overline{U} → X by

g(u) = \{ \begin{matrix}f(u) & for u ∈ P_{i} ∩ O, \\ o & for u ∈ U \ O. \end{matrix}

By (4) g is actually defined on \overline{U} and is a continuous extension of f|C and hence f|C is null homotopic. □

Proof of Theorem 1.4. Let f : 𝕊^{2} → CX_{0} ∨CX_{1} be a map. As in the proof of Theorem 1.1, we construct a map g : 𝕊^{2} → CX_{0} ∨ CX_{1} such that (i) g(𝕊^{2}) ⊂ CX_{0} ∨ CX_{1} \ {p_{0}, p_{1}}, and (ii) g ≃ f : 𝕊^{2} → CX_{0} ∨ CX_{1}. Having constructed such map g, the proof is completed as follows:

Let r : CX_{0} ∨CX_{1} \{p_{0}, p_{1}} → X_{0} ∨X_{1} be the standard retraction. Then the hypothesis π_{2}(X_{0} ∨ X_{1}) = 0 together with (ii) imply f ≃ g ≃ r ∘ g ≃ 0.

Choose a component O of f^{−1}(CX_{0} ∨ CX_{1} \ {o}) such that O ∩ f^{−1}({p_{0}, p_{1}}) ≠ ∅ and assume, without loss of generality, O ∩ f^{−1}({p_{0}}) ≠ ∅. We construct a map g_{1} : 𝕊^{2} → CX_{0} ∨ CX_{1} such that

(1) g_{1} is homotopic to f,

(2) g_{1}|𝕊^{2} \ O = f|𝕊^{2} \ O and g_{1}(\overline{O}) ∩ \{p_{0}, p_{1}\} = ∅.

First we apply Lemma 2.3 to K_{0} = 𝕊^{2} \ O, K_{1} = f^{−1}({p_{0}}) ∩ O and obtain a compact surface P with polygonal boundary such that 𝕊^{2} \ O ⊂ IntP and

(3) for each map φ : P → CX_{0} \ {p_{0}} with φ(𝕊^{2} \ O) = {o} the restriction φ|∂P : ∂P → X is null homotopic.

Define f_{O} : P → CX_{0} by:

f_{O}(x) = \{ \begin{matrix}f(x) & for x ∈ P ∩ O, \\ o & for x ∈ 𝕊^{2} \ O. \end{matrix}

The condition (3) above guarantees that f_{O} : P → CX_{0} \ {p_{0}} satisfies the hypotheses (1) and (2) of Lemma 2.1, and hence admits an extension g_{0} : 𝕊^{2} → CX_{0} \ {p_{0}}. Define g_{1} : 𝕊^{2} → CX_{0} ∨ CX_{1} by: g_{1}(x) = g_{0}(x) for x ∈ O and g_{1}(x) = f(x) otherwise. Then g_{1} | P = f | P. Since P and \overline{𝕊^{2} \ P} are compact surfaces and CX_{0} is contractible, we see that f and g_{1} are homotopic relative to P. By the definition, we also have g_{1}(Ō) ∩ {p_{0}, p_{1}} = ∅.

As in the proof of Theorem 1.1, we obtain the desired map g : 𝕊^{2} → CX_{0}∨CX_{1}\{p_{0}, p_{1}} by iterating the above procedure at most finitely many times on each component of f^{−1}(CX_{0} ∨ CX_{1} \ {o}) such that O ∩ f^{−1}({p_{0}, p_{1}}) ≠ ∅. □

For more information on homology and homotopy groups on onepoint unions of cones, see [3] and [5].

3. Remark on Lemma 2.3

The proof of Lemma 2.3 shows the following: for each compact subset K_{0} of ℝ^{2} and for for each neighborhood U of K_{0}, there exists a compact surface P such that

(1) K_{0} ⊂ IntP ⊆ U,

(2) for each map f : P → X with f(K_{0}) = {o}, the restriction f|∂P : ∂P → X is null homotopic.

The following result illustrates that the 3-dimensional analogue of the above result does not hold.

Proposition 3.1. Let ST be a solid torus in ℝ^{3} which contains Antoine’s necklace K_{0} in its interior in the standard way ([14, p.71-72]). Let P be a compact 3-manifold-neighborhood of K_{0} in ST and Y_{0} be the quotient space P/K_{0} with the quotient map q : P → Y_{0}. Then, the restriction q|∂P : ∂P → Y_{0} is not null-homotopic.

To prove the above, it is convenient to make the following remark.

Lemma 3.2. Let X be a simply-connected PL manifold and Y be a connected PL submanifold of X. Then, for each component Z of X \ Y, the topological boundary of Z is path-connected.

Proof. It suffices to verify the conclusion when dim Y = dim X. Suppose that the topological boundary ∂Z of Z is not path-connected. Then we have two points p and q in ∂Z which are not joined by arcs in ∂Z. We have, on one hand, an arc A in Y connecting p and q and, on the other hand, an arc B in \overline{Z} connecting p and q such that A ∩ B = {p, q}. The union A ∪ B is a simple closed curve in X which is not null homotopic. This contradicts the assumption. □

For simplicity, ST \ K_{0} is regarded as a subspace of Y_{0} via the homeomorphism q|ST \K_{0}. Let y_{0} be the point with {y_{0}} = q(K_{0}).

Suppose that there exists a homotopy H : ∂P ×𝕀 → Y_{0} such that

H(x, 0) = x and H(x, 1) = y_{0} for x ∈ ∂P.

Let S be a component of ∂P. By making use of the homotopy H|S × 𝕀 between the inclusion S → Y_{0} and the constant map to y_{0}, we show that

(∗) there exists a homotopy \overline{H} : S × 𝕀 → P between the inclusion S → P to a constant map.

Take the component O of (H|S × 𝕀)^{−1}(Y_{0} \ {y_{0}}) which contains S × {0}. Define

H_{0}(x, t) = H(x, t) if (x, t) ∈ O, H_{0}(x, t) = y_{0} otherwise.

Then H_{0} is also a homotopy from the inclusion S → Y_{0} to the constant map. Hence we assume that O := (H|S × 𝕀)^{−1}(Y_{0} \ {y_{0}}) is connected and let C = S × 𝕀 \ O = (H|S × 𝕀)^{−1}({y_{0}}). Then, we have S × {0} ⊆ O and S × {1} ⊆ C. In the next lemma, H|S × 𝕀 is abbreviated to H.

Lemma 3.3. Let C_{0} be a component of C. Then there exists a unique u ∈ K_{0} such that H_{1} : O ∪C_{0} → P defined by: H_{1}|O = H|O and H_{1}(x, t) = u for (x, t) ∈ C_{0} is continuous.

Proof. We show that there exists a unique point u ∈ K_{0} such that, for each sequence {p_{n}} ⊂ O with lim_{n→∞} p_{n} ∈ ∂C_{0}, the sequence {H(p_{n})} accumulates to u. It is easily seen that u is the desired point.

To show this by contradiction, suppose that there exist two points a, b ∈ ∂C_{0} and sequences {a_{n}}, {b_{n}} ⊂ O such that lim_{n→∞} a_{n} = a, lim_{n→∞} b_{n} = b, and further lim_{n→∞} H(a_{n}) and lim_{n→∞} H(b_{n}) are distinct points of K_{0}. Since K_{0} is 0-dimensional, we have open sets U and V in P such that lim_{n→∞} H(a_{n}) ∈ U, lim_{n→∞} H(b_{n}) ∈ V, U ∩ V = ∅ and K_{0} ⊆ U ∪ V. There exists a PL-manifold-neighborhood P_{1} of C such that H(P_{1}) ⊆ {y_{0}}∪(U \K_{0})∪(V \K_{0}). Let P_{2} be the component of P_{1} containing C_{0}. Choose a_{n} and b_{n} so that a_{n}, b_{n} ∈ P_{2}. Notice that

(♯) there is no arc connecting a_{n} and b_{n} in P_{2} ∩ O,

because H(P_{2} ∩ O) ⊆ (U ∪ V) \ K_{0}. In other words, C separates the connected manifold P_{2}.

As S is a surface in ST, S × 𝕀 is naturally embedded in ℝ^{3} by ”thickening S.” Under this embedding the topological boundary of S × 𝕀 in ℝ^{3} is S × {0, 1}. We apply Lemma 3.2 to X = ℝ^{3} and Y = P_{2}. If P_{2} does not intersect with S × {1}, then the topological boundary of P_{2} in ℝ^{3} is contained in O. If P_{2} meets ∂P × {1}, then it contains S × {1} and the topological boundary of P_{2} in ℝ^{3} is contained in the disjoint union of S × {1} and O. Hence we have the following remark.

S × 𝕀 \ P_{2} consists of finitely many components and the topological boundary of each component in S×𝕀 is path-connected and is contained in O.

We have a polygonal arc A in O which connects a_{n} and b_{n}. There exist finitely many pairwise disjoint subarcs B_{1}, …, B_{r} of A such that the endpoints of each B_{j} belong to ∂P_{2}, each B_{j} is contained in the union of P_{2} and a unique component of the complement of P_{2}, and A \ ∪B_{j} ⊂ P_{2}. By the preceding remark, for each B_{j}, we have an arc on the boundary of P_{2} which connects the endpoints of B_{j}. Hence we obtain an arc in P_{2} ∩ O connecting a_{n} and b_{n}, which contradicts (♯).

□

Applying the above lemma to each component of C, we have a map \overline{H}_{S} : S × 𝕀 → P such that \overline{H}_{S}|O ∪ C_{0} is continuous for each component C_{0} of C. To see the continuity of \overline{H}_{S} on S ×𝕀, it suffices to show the following.

(∗∗) Let {p_{n}} be a sequence of C such that lim_{n→∞} p_{n} = p ∈ C and let C_{n} (resp. C_{0}) be the component of C containing p_{n} (resp. p). Take the unique point u_{n} for C_{n} and u for C_{0} as in the previous lemma. Then lim_{n→∞} u_{n} = u.

To show the above, we may assume that p_{n} ∈ ∂C_{n} and p ∈ ∂C. By the definition of \overline{H}_{S}, \overline{H}_{S}(p_{n}) = u_{n} and \overline{H}_{S}(p) = u. By the continuity of \overline{H}_{S}|O ∪ C_{n}, we may take a_{n} ∈ O so close to p_{n} that lim_{n→∞} a_{n} = p and lim_{n→∞} \overline{H}_{S}(a_{n}) = lim_{n→∞} \overline{H}_{S}(p_{n}). Then, by the uniqueness of u, we obtain lim_{n→∞} \overline{H}_{S}(a_{n}) = u. Thus lim_{n→∞} u_{n} = lim_{n→∞} \overline{H}_{S}(p_{n}) = lim_{n→∞} \overline{H}_{S}(a_{n}) = u. This proves (∗∗) and hence completes the proof of (∗).

Taking the union \overline{H} := ∪\overline{H}_{S} over all components S of ∂P, we have a homotopy \overline{H} : ∂P × 𝕀 → P such that \overline{H}(x, 0) = x, x ∈ ∂P, \overline{H}(∂P × \{1\}) ⊆ K_{0}. It is easy to see that, for each component P_{0} of P, we have \overline{H}(∂P_{0} × 𝕀) ⊆ P_{0}.

By the construction of Antoine’s necklace, there exists a component P_{0} of P such that the inclusion ∂P_{0} → P_{0} is not nullhomotopic. Then there exists a component S of ∂P_{0} such that the inclusion S → P_{0} is not null-homotopic. However, the restriction \overline{H}|S × 𝕀 provides a homotopy between the inclusion S → P_{0} and a constant map, because \overline{H}(S × \{1\}), as a connected set of the zero-dimensional K_{0}, is a singleton. This contradiction completes the proof. □

For the tame Cantor set K in ℝ^{3}, there exists an arbitrarily small neighborhood P of K which is the disjoint union of 3-balls. For the quotient map q : P → P/K, the restriction q|∂P : ∂P → P/K is null-homotopic, since the restriction q|∂P_{0} is easily seen to be null-homotopic for each component P_{0} of P. Acknowledgment. The authors are gratitude to G. Conner and D. Repovš for inspiring discussion. A stimulating conversation with Conner was one of the motivations of this research. Also the discussion of the first author with Conner and Repovš at the seminar in University of Ljubljana was very helpful in completing the proofs. The first author was supported by the Grant-in-Aid for Scientific research (C) of Japan No. 20540097.

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School of Science and Engineering, Waseda University, Tokyo 169-8555, JAPAN

   E-mail address: eda@logic.info.waseda.ac.jp

Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, JAPAN

   E-mail address: kawamura@math.tsukuba.ac.jp