C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. 2

VOL. LXII 1991 FASC. 2

ON A COMPACTIFICATION OF THE HOMEOMORPHISM GROUP OF THE PSEUDO-ARC

BY

KAZUHIRO K A W A M U R A (TSUKUBA)

1. Introduction. A continuum means a compact connected metric space. For a continuum X, H(X) denotes the space of all homeomorphisms of X with the compact-open topology. It is well known that H(X) is a com- pletely metrizable, separable topological group. J. Kennedy [8] considered a compactification of H(X) and studied its properties when X has various types of homogeneity. In this paper we are concerned with the compact- ification G P of the homeomorphism group of the pseudo-arc P , which is obtained by the method of Kennedy. We prove that G P is homeomorphic to the Hilbert cube. This is an easy consequence of a combination of the re- sults of [2], Corollary 2, and [9], Theorem 1, but here we give a direct proof.

The author wishes to thank the referee for pointing out the above refer- ence [2]. We also prove that the remainder of H(P ) in G P contains many Hilbert cubes. It is known that H(P ) contains no nondegenerate continua ([10]).

Notation and basic definitions 1.1. Let X be a continuum. Let f : X → X be a map. The graph of f = {(x, f (x)) | x ∈ X} ⊂ X × X is denoted by gr f .

A map f : X → X is called a near-homeomorphism if, for each ε > 0, there exists a homeomorphism h : X → X such that d(f, h) = sup{d(f (x), h(x)) | x ∈ X} < ε.

The hyperspace C(X) is the space of all nonempty subcontinua of X with the Hausdorff metric d H . The ε-neighbourhood of K ∈ C(X) is denoted by N ε (K). The map ϕ : H(X) → C(X × X) defined by ϕ(f ) = gr f is an imbedding ([8], p. 43).

A compactification G X of H(X) is defined by cl _C(X×X) im ϕ.

The space C π (X × X) is defined by C π (X × X) = {K ∈ C(X × X) | π 1 (K) = π 2 (K) = X}, where π i is the projection onto the ith factor (i = 1, 2).

A surjective map f : X → Y induces f ^∗ : C π (X × X) → C π (Y × Y )

defined by f ^∗ (K) = (f × f )(K).

A continuum is called arc-like if it is represented as the limit of an inverse sequence of arcs. A continuum X is called hereditarily indecomposable if, for each pair A, B of subcontinua of X such that A ∩ B 6= ∅, either A ⊂ B or A ⊃ B holds.

A hereditarily indecomposable arc-like continuum is topologically unique and is called the pseudo-arc (denoted by P ). It is known that P is the unique homogeneous arc-like continuum ([1]).

In what follows, the Hilbert cube is denoted by Q.

The following theorem is fundamental in this paper.

Theorem 1.2 ([13]). G P = C π (P × P ).

2. G P is homeomorphic to Q. First we prove the following result.

Theorem 2.1. Let X be an arc-like continuum. Then C π (X × X) is homeomorphic to Q.

P r o o f. Let X = lim

← (I n , p n,n+1 ), where each I n is an arc and each p n,n+1 : I n+1 → I _n is surjective. Let p n : X n → I _n be the projection onto the nth factor.

S t e p 1. First we prove that C π (X × X) = lim

← (C π (I n × I n ), p ^∗ _n,n+1 ).

Notice that p ^∗ _n,n+1 ◦ p ^∗ _n+1 = p ^∗ _n . So the limit of p ^∗ _n ’s, lim

← p ^∗ _n : C π (X × X) → lim ← (C π (I n × I _n ), p ^∗ _n,n+1 ), is defined.

By [6], Proposition 1.2, p ^∗ _n,n+1 : C π (I n+1 × I _n+1 ) → C π (I n × I _n ) and p ^∗ _n : C π (X × X) → C π (I n × I _n ) are surjective for each n. Using this fact, we can see that lim

← p ^∗ _n is a homeomorphism.

S t e p 2. Next we show that if I is an arc, then C π (I ×I) is homeomorphic to Q. It is clear that C π (I × I) has the following property:

(1) If K and L are subcontinua of I × I such that K ⊂ L and K ∈ C π (I × I), then L ∈ C π (I × I).

Using (1), we can see that C π (I × I) is an AR in the same way as in [7], Theorem 4.4 (see also Remark, p. 29 of [7]). Using the method of [5], Lemma 4.4, we have

(2) for each ε > 0, there exists a map g : C π (I × I) → C π (I × I) such that d(g, id) < ε and im g is a Z-set in C π (I × I).

Hence C π (I ×I) has the disjoint n-cell property for each n, so by Toru´ nczyk’s characterization theorem [14], C π (I × I) is homeomorphic to Q.

S t e p 3. p ^∗ _n,n+1 : C π (I n+1 × I _n+1 ) → C π (I n × I _n ) is a cell-like map. To show this, first we prove that

(3) p ^∗ _n,n+1 is a monotone map.

Take K ∈ C π (I n × I _n ) and let Λ K = p ^∗−1 _n,n+1 (K). For each A, B ∈ Λ K , A ∩ B 6= ∅, because π 1 (A) = π 1 (B) = π 2 (A) = π 2 (B) = I n+1 . So there exist order arcs α A , β B from A to A ∪ B and from B to A ∪ B respectively.

It is easy to see that α A ∪ α _B ⊂ Λ _K . Hence Λ K is an arcwise connected continuum.

Consider the hyperspace C(Λ K ) (note that C(Λ K ) ⊂ C(C(I n × I _n ))).

Since Λ K is a continuum, C(Λ K ) has the trivial shape ([11], p. 180). Let σ : C(C(I n ×I _n )) → C(I n ×I _n ) be the union function defined by σ(A) = S A for each A ∈ C(C(I n × I _n )).

Take any A ∈ C(Λ K ). Then p n,n+1 (A) = K for each A ∈ A, and hence p ^∗ _n,n+1 (σ(A)) = p n,n+1 (S A) = K. This means σ(C(Λ K )) ⊂ Λ K , and it is easy to see that σ({A}) = A for each A ∈ Λ K . Hence σ(C(Λ K )) = Λ K

and σ|C(Λ K ) is a retraction onto Λ K . The trivial shape is preserved under any retraction, so Λ K has the trivial shape. (See [12], Lemma 2.1, for that argument.)

R e m a r k. In fact, Λ K is locally connected, and so C(Λ K ) and Λ K are AR’s.

By Steps 2 and 3, each p ^∗ _n,n+1 is a near-homeomorphism (see [4], pp. 105–106). Hence by [3] and Step 1, C π (X × X) is homeomorphic to Q.

Combining Theorem 1.2 and Theorem 2.1, we have Corollary 2.2. G P is homeomorphic to Q.

3. The remainder of G P

Definition 3.1. Let X be a continuum. A continuous map µ : C(X) → [0, 1] is called a Whitney map if it satisfies the following conditions:

1) µ(X) = 1 and µ({x}) = 0 for each x ∈ X.

2) If A, B ∈ C(X) satisfy A B, then µ(A) < µ(B).

Definition 3.2. Let X be a hereditarily indecomposable continuum, and fix a Whitney map µ : C(X) → [0, 1].

1) Let p be a point of X. The order arc α p : [0, 1] → C(X) is defined by α p (0) = {p} and µ(α p (t)) = t for each 0 ≤ t ≤ 1. By the hereditary indecomposability of X, α p is uniquely determined ([7], (8.4), or [11], (1.61)).

2) Let α : X × [0, 1] → C(X) be the map defined by α(p, t) = α p (t) for (p, t) ∈ X × [0, 1]. Then α is continuous ([11], (1.63), pp. 113–114).

Lemma 3.3. Let ϕ : H(P ) → C(P × P ) be the map defined in 1.1. Then

im ϕ = {K ∈ C π (P ×P ) | f or each p ∈ P, #(P ×p∩K) = #(p×P ∩K) = 1},

where #A denotes the cardinality of a set A.

P r o o f. It is clear that for each f ∈ H(P ) and for each p ∈ P , #(P × p∩

gr f ) = #(p × P ∩ gr f ) = 1. Conversely, take any K ∈ C π (P × P ) such that for each p ∈ P , #(P × p ∩ K) = #(p × P ∩ K) = 1. By Theorem 1.2, C π (P × P ) = G P , hence there exists a sequence (f n ) ⊂ H(P ) such that gr f n → K (convergence in the Hausdorff metric). We claim that

(1) (f n ) is equicontinuous.

Suppose not. Then there exists an ε 0 > 0 such that for each n ≥ 1, there exist x n , y n ∈ P and a subsequence (f _k

) such that d(x n , y n ) < 1/n and d(f k

(x n ), f k

(y n )) ≥ ε 0 . We may assume that lim x n = lim y n = p and lim f k

(x n ) = x, lim f k

(y n ) = y. Then (p, x) = lim(x n , f k

(x n )) ∈ K and similarly (p, y) ∈ K. But x 6= y, which contradicts the hypothesis.

By (1) and the Ascoli–Arzel` a theorem, the sequence (f n ) converges uni- formly to a continuous map f . So K = gr f . Since #(P × p ∩ K) = 1, we have f ∈ H(P ). This completes the proof.

Theorem 3.4. For each ε > 0, there exists a homotopy H : G P ×[0, 1] → G P which satisfies the following conditions.

(1) H is an ε-homotopy and H 0 = id.

(2) H(G P × (0, 1]) ⊂ G _P − H(P ).

P r o o f. Fix a Whitney map µ : C(P ) → [0, 1]. Take a small t 0 > 0 such that

(3) 0 < diam A < ε for each A ∈ µ ⁻¹ (t 0 ).

Then H : G P × [0, 1] → G _P is defined by H(K, t) = [

{x × α _y (t · t 0 ) | (x, y) ∈ K}.

We prove that H(K, t) ∈ G P for each (K, t). Take (x n , z n ) ∈ H(K, t) and assume that (x n , z n ) → (x, z). There exist (x n , y n ) ∈ K such that (x n , z n ) ∈ x n × α _y

(t · t 0 ). We may assume that y n → y. Then (x _n , y n ) → (x, y) and by the continuity of α, (x, z) ∈ x × α y (t · t 0 ) ⊂ H(K, t). Hence H(K, t) is compact. It is clear that H(K, t) is connected and contains K.

So H(K, t) ∈ C π (P × P ) = G P . Using the continuity of α again, we see that H is continuous. By (3), H is an ε-homotopy, and by Lemma 3.3, condition (2) is satisfied.

Theorem 3.5. For each open subset U of G P − H(P ), there exists an imbedding i : Q → U of Q into U .

P r o o f. Let V be any open subset of G P − H(P ). There exists an open

subset V of G P such that V ∩ (G P − H(P )) = U . Since H(P ) is dense

in G P , we can find f ∈ H(P ) ∩ V . Take ε > 0 sufficiently small so that

N ε (gr f ) ⊂ V .

Let (p n ) be a sequence in P such that p n → p ∈ P . Take a sequence (K n ) of subcontinua of P such that

(1) f (p n ) ∈ K n and diam K n → 0 as n → ∞.

For each n ≥ 0, let α n : [0, 1] → C(K n ) be the order arc such that (2) α n (0) = {f (p n )} and α n (1) = K n .

Let Q ⁰ = I ^∞ . We define a map i : Q ⁰ → V by i((t n )) = gr f ∪ [

n≥0

{p _n } × α _n (t n ) for (t n ) n≥0 ∈ Q ⁰ .

Then in the same way as in [7], Theorem 5.1, i is an imbedding. But im i ∩ H(P ) = {gr f } by Lemma 3.3, and we can take a Hilbert cube Q ⊂ Q ⁰ such that i(Q) ⊂ V ∩ (G P − H(P )) = U . This completes the proof.

R e m a r k 3.6. H(P ) has no interior points in G P by Theorem 3.4.

Therefore G P − H(P ) is not completely metrizable, and hence is not a Q- manifold.

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INSTITUTE OF MATHEMATICS UNIVERSITY OF TSUKUBA TSUKUBA-CITY, IBARAKI 305 JAPAN

Re¸ cu par la R´ edaction le 24.10.1989 ;

en version modifi´ ee le 30.8.1990