Generalizing n-shape to locally compact spaces, the author [1] introduced the proper n-shape, which is defined by using embed- dings of spaces into locally compact AR-spaces

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VOL. 77 1998 NO. 2

HOMEOMORPHIC NEIGHBORHOODS IN µⁿ⁺¹-MANIFOLDS

BY

Y ¯UJI A K A I K E (IBARAKI)

1. Introduction. The notion of n-shape for compact spaces was introduced by Chigogidze [5]. Generalizing n-shape to locally compact spaces, the author [1] introduced the proper n-shape, which is defined by using embeddings of spaces into locally compact AR-spaces. In the case of dim ≤ n + 1, the proper n-shape of locally compact spaces can also be defined by using their embeddings into locally compact (n + 1)-dimensional LCⁿ∩ Cⁿ-spaces (cf. [2]). In this paper, we prove a µⁿ⁺¹-manifold version of the result of [11], that is, if X and Y are Z-sets in µⁿ⁺¹-manifolds M and N respectively, and n-Shp(X) = n-Shp(Y ), then X and Y have arbitrarily small homeomorphic µⁿ⁺¹-manifold closed neighborhoods. As a corollary, if X is a connected Z-set in a µⁿ⁺¹-manifold and X ∈ SU Vⁿ, then there exists a tree T such that X has arbitrarily small closed neighborhoods homeomorphic (≈) to the

∆n+1-product T ∆n+1µⁿ⁺¹ of T and µⁿ⁺¹. Here, property SU Vⁿ is a non- compact variant of property U Vⁿ, and the ∆n+1-product is defined in [10];

it plays the role of the Cartesian product in the category of µⁿ⁺¹-manifolds.

For a locally finite polyhedron P , P ∆n+1µⁿ⁺¹ is the µⁿ⁺¹-manifold having the same proper n-homotopy type of P .

2. Preliminaries. In this paper, spaces are separable metrizable and maps are continuous. The (n+1)-dimensional universal Menger compactum is denoted by µⁿ⁺¹ and a manifold modeled on µⁿ⁺¹ is called a µⁿ⁺¹- manifold . We define µⁿ⁺¹_∞ = µⁿ⁺¹\ {∗}, where ∗ ∈ µⁿ⁺¹. Recall that two proper maps f, g : X → Y are properly n-homotopic (written f 'ⁿ_p g) if, for any proper map α : Z → X from a space Z with dim Z ≤ n into X, the com- positions f α and gα are properly homotopic in the usual sense (f α 'pgα).

A µⁿ⁺¹-manifold M lying in a µⁿ⁺¹-manifold N is said to be n-clean in N (cf. [8]) if M is closed in N and there exists a closed µⁿ⁺¹-manifold δ(M ) in M such that

(i) (N \ M ) ∪ δ(M ) is a µⁿ⁺¹-manifold;

1991 Mathematics Subject Classification: 54C56, 55P55.

Key words and phrases: µⁿ⁺¹-manifold, proper n-shape, n-clean, ∆n+1-product.

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(ii) δ(M ) is a Z-set in both M and (N \ M ) ∪ δ(M ); and (iii) M \ δ(M ) is open in N .

Remark 2.1.Let P be a PL-manifold and L a submanifold in P such that Bd L = Bd(P \L). By [7, Theorem 1.6], there exists a proper U Vⁿ-surjection f : N → P from a µⁿ⁺¹-manifold N satisfying the following conditions:

(a) f⁻¹(K) is a µⁿ⁺¹-manifold for any closed subpolyhedron K of P ; and (b) f⁻¹(Z) is a Z-set in f⁻¹(K) for any Z-set Z in a closed subpolyhedron K of P .

Then it is easy to see that M = f⁻¹(L) is an n-clean submanifold of N with δ(M ) = f⁻¹(BdL).

Lemma 2.2. Let Y be closed in a locally compact Cⁿ∩ LCⁿ-space N.

Assume that r : V0→ Y is a proper retraction of a closed neighborhood V₀ of Y in N. Then for each closed neighborhood V of Y in N there exists a closed neighborhood V⁰ of Y in N such that V⁰ ⊂ V ∩ V0 and idV⁰ 'ⁿ_p r|V⁰

in V.

P r o o f. Let W be an open cover of V ∩ V0 such that if one of any two W-close maps from an arbitrary locally compact space is proper, then the other is also proper. Since int(V ∩ V0) is LCⁿ, there exists an open cover U of int(V ∩ V0) such that any two U -close maps from a space with dim ≤ n to int(V ∩ V0) are W-homotopic. By the continuity of r, for any U ∈ U ∩ Y = {U ∈ U | U ∩ Y 6= ∅} and x ∈ U ∩ Y there exists a closed neighborhood Vx of x in N such that Vx ⊂ U and r(V_x) ⊂ U ∩ Y . Since Y is locally compact, {Vx | x ∈ Y } has a locally finite refinement V⁰. Then V⁰=S V⁰ is the desired neighborhood.

Let X and Y be closed sets in locally compact Cⁿ∩ LCⁿ-spaces M and N respectively. Recall that a proper n-fundamental net f = {fλ | λ ∈ Λ} : X → Y in (M, N ) is generated by a proper map f : X → Y (or f generates f ) provided f = fλ|_X for all λ ∈ Λ. The proper n-homotopy class [f ] of f is generated by f if f generates some f⁰ ∈ [f ].

Proposition 2.3. If Y is a locally compact LCⁿ-space, then the proper n-homotopy class of f : X → Y in (M, N ) is generated by a proper map f : X → Y .

P r o o f. Since Y is LCⁿ, there exist a closed neighborhood V0of Y in N and a proper retraction r : V0 → Y by [3, Lemma 3.2]. By Lemma 2.2, for each closed neighborhood V of Y in N there exists a closed neighborhood V⁰ of Y in N such that r|V⁰ 'ⁿ_p idV⁰ in V . Then there exist a closed neighborhood U⁰ of X in M and λ0 ∈ Λ such that f_λ|_U0 'ⁿ_p fλ0|_U0 in V⁰ for all λ ≥ λ0. Let r⁰ : N → N be an extension of r and f_λ⁰ = r⁰fλ0.

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Note that f⁰ = {f_λ⁰} is generated by f = rf_λ₀|_X, i.e., f_λ⁰|_X = f . Since f_λ⁰|_U⁰ = r⁰fλ0|_U⁰ 'ⁿ_p fλ0|_U⁰ 'ⁿ_p fλ|_U⁰ in V for all λ ≥ λ0, we have f⁰'ⁿ_p f .

3. Homeomorphic neighborhoods in µⁿ⁺¹-manifolds

Lemma 3.1. Let X be a Z-set in a µⁿ⁺¹-manifold M. Then there exists a closed embedding F : M → µⁿ⁺¹_∞ such that F (M ) is a neighborhood of F (X) in µⁿ⁺¹_∞ and F (M ) is n-clean in µⁿ⁺¹_∞ with δ(F (M )) ≈ M .

P r o o f. By [6, Theorem 9], there exists a proper (n + 1)-invertible U Vⁿ- surjection f : M → P from M to a locally finite (n + 1)-dimensional polyhedron P . We can assume that P is a closed subpolyhedron in (I^2(n+1)+1× {0}) \ {∗}, where ∗ = (0, . . . , 0) ∈ I^2(n+1)+2. Then there exists a proper (n + 1)-invertible U Vⁿ-surjection f : N → I^2(n+1)+2\ {∗} from a µⁿ⁺¹- manifold N as in Remark 2.1. Since I^2(n+1)+2 \ {∗} 'ⁿ_p µⁿ⁺¹_∞ and f is proper U Vⁿ, we have N ≈ µⁿ⁺¹_∞ (cf. [7, Theorem 1.3]).

Let N (P ) be a regular neighborhood of P . By Remark 2.1, M⁰ = f⁻¹(N (P )) and M⁰⁰ = f⁻¹(Bd N (P )) are µⁿ⁺¹-manifolds in N and M⁰ is n-clean with δ(M⁰) = M⁰⁰. Since P is a Z-set in N (P ), it follows that N (P ) 'ⁿ_p Bd N (P ). By [7, Theorem 1.4], there exist homeomorphisms g : M → M⁰ and g⁰ : M → M⁰⁰. By the Z-set unknotting theorem [4], there exists a homeomorphism h : M⁰→ M⁰ such that h(g(X)) ∩ M⁰⁰ = ∅.

Then F = hg is the desired closed embedding.

Theorem 3.2. Let X and Y be Z-sets in µⁿ⁺¹-manifolds M and N respectively, such that n-Shp(X) ≤ n-Shp(Y ). Then, for each neighborhood U of X in M and each neighborhood V of Y in N , there exists an open neighborhood V⁰ of Y such that for every µⁿ⁺¹-manifold closed neighborhood S of Y in V⁰∩ V , there exists a µⁿ⁺¹-manifold closed neighborhood R of X in U which is homeomorphic to S.

P r o o f. By Lemma 3.1, we can assume that M = N = µⁿ⁺¹_∞ and U is n-clean. Let f : X → Y in (µⁿ⁺¹_∞ , µⁿ⁺¹_∞ ) and g = {gδ | δ ∈ ∆} : Y → X in (µⁿ⁺¹_∞ , µⁿ⁺¹_∞ ) be proper n-fundamental nets such that gf 'ⁿ_p iX. Let U⁰ be a closed neighborhood of X such that U⁰ ⊂ int U . Then there exist δ0 ∈ ∆, λ₀ ∈ Λ and a closed neighborhood W of Y with W ⊂ V such that gδfλ|_X 'ⁿ_p idX in U⁰, gδ|_W 'ⁿ_p gδ0|_W in U⁰ for each δ ≥ δ0, λ ≥ λ0. By the Z-set approximation theorem [4], there exists a Z-embedding g⁰_δ₀ : W → int U approximating gδ0. Note that g_δ⁰₀|_Y is properly n-homotopic to the inclusion in µⁿ⁺¹_∞ . By the Z-set unknotting theorem [4], there exists a homeomorphism h : µⁿ⁺¹_∞ → µⁿ⁺¹_∞ such that hg_δ⁰₀|_Y = idY.

Let V⁰= h(int U ) and S ⊂ V ∩ V⁰ be a closed µⁿ⁺¹-manifold neighborhood of Y . Then S⁰ = h⁻¹(S) is a µⁿ⁺¹-manifold closed neighborhood of

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g⁰_δ₀(Y ) lying in int U . Let W⁰be a closed neighborhood of Y lying in int S so that g_δ⁰₀(W⁰) ⊂ int S⁰. Then there exists λ ≥ λ0such that fλ(X) ⊂ W⁰. By the Z-set approximation theorem, we can assume that fλis a Z-embedding.

Note that g⁰_δ₀fλ(X) ⊂ g⁰_δ₀(W⁰) ⊂ int S⁰ and g⁰_δ₀fλ|_X 'ⁿ_p gδ0fλ|_X 'ⁿ_p idX in U⁰ ⊂ int U . By the Z-set unknotting theorem, there exists a homeomorphism h⁰ : µⁿ⁺¹_∞ → µⁿ⁺¹_∞ such that h⁰g_δ⁰₀fλ|_X = idX and h⁰|_µn+1

∞ \int U = id_µⁿ⁺¹

∞ \int U. Then R = h⁰(S⁰) is the desired neighborhood.

For the ∆n+1-product, refer to [10].

Lemma 3.3. Let P be a locally finite polyhedron embedded in µⁿ⁺¹∞ as a closed set and U a neighborhood of P in µⁿ⁺¹_∞ . Then there exists a µⁿ⁺¹- manifold closed neighborhood V of P such that V ⊂ U , V is n-clean in µⁿ⁺¹_∞ and V ≈ δ(V ) ≈ P ∆n+1µⁿ⁺¹.

P r o o f. We can assume that P ⊂ (I^2(n+1)+1× {0}) \ {∗} ⊂ I^2(n+1)+2\ {∗} = M as a closed subpolyhedron and µⁿ⁺¹_∞ is obtained from M by the Lefschetz construction [9, 2.1, II]. Let L be a combinatorial triangulation of M and eU be a neighborhood of P in M such that U = µⁿ⁺¹_∞ ∩ eU . By Whitehead’s theorem [12], there exists a subdivision L⁰ of L such that L⁰ refines {M \ P } ∪ { eU }.

Let N (P, sd L⁰) be a regular neighborhood of P obtained from the barycentric subdivision sd L⁰ of L⁰ and let V be a µⁿ⁺¹-manifold obtained from N (P, sd L⁰) by the Lefschetz construction. Then V is n-clean in µⁿ⁺¹_∞ and such that δ(V ) = µⁿ⁺¹_∞ ∩ Bd N (P, sd L⁰) and V \ δ(V ) = µⁿ⁺¹_∞ ∩ int N (P, sd L⁰). Now there exists a proper deformation retraction r : N (P, sd L⁰) → P , and we have a proper U Vⁿ-retraction r|V : V → P . Since there exists a proper U Vⁿ-surjection P ∆n+1µⁿ⁺¹→ P (see [10]), and by [7, Theorem 1.4], V and P ∆n+1µⁿ⁺¹ are homeomorphic. Since P is a Z-set in N (P, sd L⁰), we have N (P, sd L⁰) 'ⁿ_p Bd N (P, sd L⁰), which implies δ(V ) ≈ V by [7, Theorem 1.4] again.

Theorem 3.4. Let X be a Z-set in a µⁿ⁺¹-manifold M and P an (n + 1)- dimensional locally finite polyhedron such that n-Shp(X) ≤ n-Shp(P ). Then X has arbitrarily small closed neighborhoods Uα, α ∈ A, such that

(1) each Uα is n-clean in M ; (2) Uα≈ δ(U_α) ≈ P ∆n+1µⁿ⁺¹; and

(3) for each α, β ∈ A there exists a homeomorphism h : Uα → U_β fixing X.

P r o o f. By Lemma 3.1, we can assume that X and P are closed sets in µⁿ⁺¹_∞ . Let f : X → P and g : P → X be proper n-fundamental nets in

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(µⁿ⁺¹_∞ , µⁿ⁺¹_∞ ) such that gf 'ⁿ_p iX. By Proposition 2.3, f is generated by a proper map f : X → P . Let A = {α | α is a closed neighborhood of X in µⁿ⁺¹_∞ }. For each α ∈ A, there exist δ_α ∈ ∆ and a closed neighborhood W of P which is homeomorphic to P ∆n+1µⁿ⁺¹, such that gδ|_W 'ⁿ_p gδα|_W and gδf 'ⁿ_p idX in α for all δ ≥ δα by Lemma 3.3. By the same argument as in Theorem 3.2, we may assume that gδα|_W is a Z-embedding of P ∆n+1µⁿ⁺¹ into α and X ⊂ gδα(W ). Then g_δ⁻¹_α|_X 'ⁿ_p g_δ⁻¹_αgδαf |X 'ⁿ_p f in P ∆n+1µⁿ⁺¹.

Let α, β ∈ A. Since g⁻¹_δ

α|X 'ⁿ_p f 'ⁿ_p g_δ⁻¹

β |X in P ∆n+1µⁿ⁺¹, by the Z-set unknotting theorem, there exists a homeomorphism G : P ∆n+1µⁿ⁺¹ → P ∆n+1µⁿ⁺¹ such that Gg⁻¹_δ

α |_X = g_δ⁻¹

β |_X. Then h = gδβGg_δ⁻¹

α is the desired homeomorphism.

In [1], it is proved that if X is connected, then X ∈ SU Vⁿ if and only if n-Shp(X) = n-Shp(T ) for some tree T . So we have the following:

Corollary 3.5. Let X be a connected Z-set in a µⁿ⁺¹-manifold and X ∈ SU Vⁿ. Then X has an arbitrarily small closed µⁿ⁺¹-manifold neighborhood V such that V ≈ T ∆n+1µⁿ⁺¹ for some tree T.

Acknowledgments. The author wishes to thank Professor K. Sakai for his helpful comments.

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Institute of Mathematics University of Tsukuba 305, Ibaraki, Japan

E-mail: akaike@math.tsukuba.ac.jp

Received 2 December 1996;

revised 15 December 1997