3
Hilbert Manifolds and Hilbert Cube Manifolds
This chapter is a continuation of the previous chapter. Consequently,
• letΓ be an infinite set (or an infinite discrete space) withcardΓ =τ.
We now concentrate our attention on Hilbert manifolds (i.e.,ℓ2(Γ)-manifolds) and Hilbert cube manifolds (i.e.,Q-manifolds). We will prove the Toru´nczyk characterizations of these manifolds. By the characterization of Hilbert space, it is shown that every Fr´echet space with weightτ is homeomorphic toℓ2(Γ).
In particular, it follows thatRN≈ℓ2. To prove the characterization ofℓ2(Γ )-manifolds, we require the Toru´nczyk Factor Theorem 2.2.13. That is restated that, for each completely metrizable ANR X with w(X) 6 τ, the product X×ℓ2(Γ) is anℓ2(Γ)-manifold. For the characterization ofQ-manifolds, we need to prove itsQ-version, the so-called Edwards Factor Theorem 3.7.1, that is, for each locally compact ANRX, the productX×Qis aQ-manifold.
146 3 Hilbert Manifolds and Hilbert Cube Manifolds For eachf ∈C(X, Y) andU ∈cov(Y), we define
U(f) ={
g∈C(X, Y)g isU-close tof} .
Observe that, if V ∈ cov(Y) is a star-refinement of U, then V(g) ⊂ U(f) for each g ∈ V(f). Then, when Y is paracompact, C(X, Y) has a topology such that {U(f) | U ∈cov(Y)} is a neighborhood basis of f. Such a topol-ogy is called the limitation topology. By Emb(X, Y), we denote the sub-space of C(X, Y) consisting of all closed embeddings of X into Y. Then, f ∈cl Emb(X, Y) means that f can be approximated by closed embeddings.
Thus, we can define thatX isC-universal if Emb(Y, X) is dense in C(Y, X) for everyY ∈ C. — For the limitation topology, refer to Sect. 2.9 in [GAGT].
For each metrizable space Y, let MetrB(Y) be the set of all admissible bounded metric forY. WhenY is completely metrizable, let Metrc(Y) be the set of all admissible bounded complete metric for Y. For each f ∈C(X, Y) andd∈MetrB(Y), let
Ud(f) ={
g∈C(X, Y)supx∈Xd(f(x), g(x))<1} .
[2.9.1] Proposition 3.1.1 For each metrizable spaceY,{Uα(f)|d∈MetrB(Y)} is a neighborhood basis off ∈C(X, Y)in the spaceC(X, Y)with the limitation topology. When Y is completely metrizable, {Uα(f) | d ∈ Metrc(Y)} is a neighborhood basis off ∈C(X, Y). ⊓⊔
In the caseY = (Y, d) is a metric space, for eachf ∈C(X, Y) and α∈ C(Y,(0,∞)), let
Nα(f) ={
g∈C(X, Y)∀x∈X, d(f(x), g(x))< αf(x)} .
[2.9.3] Proposition 3.1.2 When Y = (Y, d) is a metric space, {Nα(f) | α ∈ C(Y,(0,∞))} is a neighborhood basis of f ∈ C(X, Y) in the space C(X, Y) with the limitation topology. ⊓⊔
As can be seen from the following proposition, the limitation topology is favorable.
[2.9.9] Proposition 3.1.3 — Some Properties of the Limitation Topology (1) For each paracompact spaceY, the evaluation map
ev :X×C(X, Y)∋(x, f)7→f(x)∈Y is continuous with respect to the limitation topology.
(2) WhenY andZ are paracompact, the composition
C(X, Y)×C(Y, Z)∋(f, g)7→gf∈C(X, Z) is continuous with respect to the limitation topology.
3.1 Universality andU-Maps 147 (3) For every paracompact spaceX, the inverse operation
Homeo(X)∋h7→h−1∈Homeo(X)
is continuous with respect to the limitation topology. Combining this with (2)above, the groupHomeo(X)with the limitation topology is a topological group.
Hereafter, we assume that the space C(X, Y) and its subspace have the limitation topology.
[2.9.4]
Theorem 3.1.4 For a completely metrizable space Y, the space C(X, Y) is a Baire space, that is, the intersection of countably many dense open sets (or denseGδ-sets) inC(X, Y)is also dense.
Sketch of Proof.LetGn,n∈N, be open dense sets in C(X, Y). For eachf∈ C(X, Y) and a complete bounded metricd∈Metr(Y), we can inductively choosegn∈C(X, Y) anddn∈Metr(Y),n∈N, so that
gn∈U2dn−1(gn−1)∩Gn, Udn(gn)⊂Gn, and dn>dn−1,
whereg0 =f andd0 =d. Then, it is easy to see that (gn)n∈N is Cauchy with respect to the sup-metric induced byd0. By the completeness, (gn)n∈N
is uniformly convergent tog∈C(X, Y). It is easy to calculated0(f, g)<1 anddn(gn, g)<1, which means g∈Ud(f) andg∈Udn(gn)⊂Gn. Hence, g∈Ud(f)∩∩
n∈NGn.
Let CP(X, Y) be the subspace of C(X, Y) consisting of all proper maps.
Note that when X and Y are metrizable, a mapf :X →Y is proper if and only if it is perfect (Proposition 1.3.1).3The following is the direct consequence of Proposition 2.1.17:
[2.9.7]
Proposition 3.1.5 IfY is locally compact and paracompact, then CP(X, Y) is clopen in the spaceC(X, Y), whereX is also locally compact if CP(X, Y)̸=
∅. ⊓⊔
Combining this with Theorem 3.1.4, we can obtain the following:
[2.9.8]
Theorem 3.1.6 For every locally compact metrizable spaces X and Y, the spaceCP(X, Y)is a Baire space. ⊓⊔
GivenU ∈cov(X), a mapf :X →Y is called aU-mapiff−1(V)≺ Ufor someV ∈cov(Y). By CU(X, Y), we denote the subspace of C(X, Y) consisting of allU-maps. — ForU-maps, refer to Sect. 5.8 of [GAGT].
[5.8.6]
Lemma 3.1.7 Let Y be paracompact and U ∈ cov(X). Then, CU(X, Y) is open in the spaceC(X, Y).
3 WhenY is locally compact, this equivalence also holds [GAGT, Proposition 2.1.5].
148 3 Hilbert Manifolds and Hilbert Cube Manifolds
Sketch of Proof.Letf∈CU(X, Y). Then,f−1(V)≺ Ufor someV ∈cov(Y).
For a star-refinementW ∈cov(Y) ofV,W(f)⊂CU(X, Y).
[5.8.7] Lemma 3.1.8∩ In caseX = (X, d)is a complete metric space,Emb(X, Y) =
n∈NCUn(X, Y), where Un ∈cov(X) withmeshUn<2−n. Thus, whenX is a completely metrizable space, Emb(X, Y)is aGδ-set in the space C(X, Y).
Sketch of Proof.Eachf∈∩
n∈NCUn(X, Y) is injective. Forxn∈X,n∈N, if (f(xn))n∈N is convergent, then (xn)n∈Nis Cauchy, so convergent. Hence, f∈Emb(X, Y).
By combining of the above two lemma with Theorem 3.1.4, the following can be obtained:
[5.8.8] Proposition 3.1.9 Let X and Y be completely metrizable spaces. Suppose that, for each U ∈cov(X), CU(X, Y) is dense in the space C(X, Y). Then, Emb(X, Y)is also dense inC(X, Y). ⊓⊔
For an open coverU ∈cov(X) consisting of open sets with compact closure, we have CU(X, Y)⊂CP(X, Y). Thus, the following locally compact version holds:
[5.8.9] Proposition 3.1.10 Let X and Y be locally compact metrizable. Suppose that, for each open cover U of X consisting of open sets with the compact closure, CU(X, Y) is dense in the spaceCP(X, Y). Then, Emb(X, Y) is also dense in CP(X, Y). ⊓⊔
Proposition 3.1.11 Let X andY be paracompact spaces, f : X →Y be a map, A⊂X be a closed set and U ∈cov(X). If f|A is a(U|A)-map, thenA has a neighborhood N inX such thatf|N is a(U|N)-map.
Proof. Choose a locally finite open cover V of Y so that (f|A)−1(Vcl)≺ U.
LetV0 be an open star-refinement ofV. By the continuity off and the local finiteness of (f|A)−1(Vcl), we can obtainW ∈cov(X) such that
f(W)≺ V0 and {
st(f−1(clV)∩A,W)V ∈ V}
≺ U.
Then, N = st(A,W) is a neighborhood of A in X. We shall show that (f|N)−1(V0)≺ U, which means thatf|N is a (U|N)-map. For eachV0∈ V0, choose V ∈ V so that st(V0,V0)⊂V. Then, it suffices to see
(f|N)−1(V0)⊂st(f−1(V)∩A,W).
For each x∈(f|N)−1(V0) =f−1(V0)∩N, we havea∈Asuch thatx, a∈W for some W ∈ W, hence f(x), f(a)∈ V1 for some V1 ∈ V0. Then, it follows that f(a)∈ st(V0,V0)⊂ V, so a∈f−1(V)∩A. Consequently, we havex∈ st(f−1(V)∩A,W). ⊓⊔
3.1 Universality andU-Maps 149 For a completely metrizable ANRX, the M1(τ)-universality implies the stronger property as below:
Proposition 3.1.12 Let X ∈M1(τ) be an ANR. If X isM1(τ)-universal, then each mapf :Y →X of everyY ∈M1(τ)can be approximated by strong Z-embeddings.
Proof. It suffices to prove that for anyα∈C(X,(0,1)), there is a strong Z-embeddingh: X →X such thatd(h(x), x)< α(x) for everyx∈ X, where d∈Metr(X). For simplicity, we denote
C= C(X×(0,1], X×(0,1]),
whereC is the space with the limitation topology. For eachn∈N, let Gn ={
f ∈CprXf|X×[2−n,1]∈Emb(X×[2−n,1], X)} .
Then, Gn is a dense Gδ set in C. Indeed, since Emb(X ×[2−n,1], X) is a Gδ-set in C(X×[2−n,1], X) and the correspondencef 7→prXf|X×[2−n,1]
is continuous (3.1.3(2)), it follows that Gn is a Gδ-set in C. To see thatGn
is dense in C, let f ∈ C and U ∈ cov(X ×(0,1]). By the compactness of [2−n−1,1], we haveW ∈cov(X) such that
{W× {t}W ∈ W, t∈[2−n−1,1]}
≺ U.
SinceX is anM1(τ)-universal ANR, prXf|X×[2−n,1] isW-homotopic to a closed embeddingg:X×[2−n,1]→X. By the Homotopy Extension Theorem 1.8.10,g can be extended to a map ˜g:X×(0,1]→X such that
˜
g|X×(0,2−n−1] = prXf|X×(0,2−n−1] and
˜
g≃W prXf rel.X×(0,2−n−1].
Then,f isU-close to the mapf′∈Gn defined as follows:
f′(x, t) = (˜g(x, t),pr(0,1]f(x, t)) for each (x, t)∈X×(0,1], i.e., prXf′= ˜g and pr(0,1]f′= pr(0,1]f. Thus, Gn is dense in C.
On the other hand, id∈Chas the following neighborhood:
V ={
f ∈C∀(x, t)∈X×(0,1], d(prXf(x, t), x)< t} . Since C is a Baire space (Proposition 3.1.4), we have φ ∈ V ∩(∩
n∈NGn).
Then, prXφ is clearly injective. For anyα∈C(X,(0,1)), the desired strong Z-embeddinghα:X→X can be defined as follows:
hα(x) = prXφ(x, α(x)) for eachx∈X.
Indeed, d(hα(x), x) < α(x) for all x ∈ X and hα is an injection. To see that hα is closed, let xi ∈ X, i ∈ N, such that (hα(xi))i∈N converges to
150 3 Hilbert Manifolds and Hilbert Cube Manifolds
y∈X. If infi∈Nα(xi) = 0, then (xi)i∈N has a subsequence (xij)j∈Nsuch that limj→∞α(xij) = 0. In this case, limj→∞xij = y because φ ∈ V. Then, α(y) = limj→∞α(xij) = 0, which is a contradiction. Therefore, we have infn∈Nα(xn)>0. Choose n ∈N so that 2−n <infi∈Nα(xi). Sinceφ ∈Gn, prXφ|X×[2−n,1] is a closed embedding, hence (xn)n∈Nis convergent. Thus, hαis a closed embedding.
To see thath(X) is a strongZ-set inX, for eachβ ∈C(X,(0,1)), choose γ ∈C(X,(0,1)) so that γ < α, β. Replacingα with thisγ, we can obtain a closed embeddinghγ such thatd(hγ(x), x)< γ(x)< β(x) for allx∈X. Since prXφis injective, hα(X)∩clhγ(X) =hα(X)∩hγ(X) =∅. Therefore,hα(X) is a strongZ-set inX. ⊓⊔
By the same proof as the above, the following can be proved:
Proposition 3.1.13 LetX be a locally compact separable ANR. Suppose that Emb(Y, X)is dense inCP(Y, X)for every locally compact separable metrizable spaceY. Then, for suchY, each proper mapf :Y →X can be approximated by strongZ-embeddings.
For completely metrizable ANRs, we can give a characterization ofM1 (τ)-universality as follows:
Theorem 3.1.14 For a completely metrizable ANRX, in order to beM1 (τ)-universal (resp. M1(ℵ0)-universal or M0-universal), the following condition is necessary and sufficient:
• For any locally finite-dimensional simplicial complex K with cardK(0) 6 τ (resp. any countable locally finite simplicial complex K or any finite simplicial complex K), each map f : |K| → X can be approximated by OK-maps.4
Proof. (Necessity) When X is M1(ℵ0)-universal or M0-universal, let K be a countable locally finite simplicial complex or a finite simplicial complex, respectively. Then,|K| ∈M1(ℵ0) or |K| ∈M0 for anyK, respectively. Since every closed embedding from|K|toX is anOK-map, each mapf :|K| →X can be approximated byOK-maps.
WhenXisM1(τ)-universal, letKbe a locally finite-dimensional simplicial complex with cardK(0)6τand letf :|K| →X be a map. EachU ∈cov(X) has an open star-refinement V. We have a subdivision K′ ▹ K such that K′≺f−1(V) (Theorem 1.5.8). By Theorem 1.5.15, φ= id :|K| → |K′|m has a homotopy inverse ψ : |K′|m → |K| such that ψφ ≃K′ id and φψ ≃K′ id.
Then,f ψφisV-close tof and|K′|m∈M1(τ). By theM1(τ)-universality,f ψ isV-close to a closed embedding h:|K′|m→X. Then,hφisV-close tof ψφ, so U-close to f. Note thath(OK) = W|h(|K|) for some W ∈cov(X). Then, (hφ)−1(W) =OK, that is,hφ:|K| →X is anOK-map.
4 RecallOK is the open star cover ofKdefined in Sect. 1.5.
3.1 Universality andU-Maps 151 (Sufficiency) LetY ∈M1(τ) (resp.Y ∈M1(ℵ0) or Y ∈M0). By Propo-sition 3.1.9, it suffices to show that CV(Y, X) is dense in C(Y, X) for any V ∈cov(Y), that is, eachf ∈C(Y, X) can be approximated byV-maps. For each U ∈cov(X), take an open star-refinement U′ of U. Applying Theorem 1.8.16 with Remark 1.6, we can obtain aσ-discrete (resp. countable star-finite or finite) refinement W ≺ V (∈ cov(Y)) with the locally finite-dimensional nerve N(W) and a map ψ : |N(W)| → X such that ψφ ≃U′ f, where φ:Y → |N(W)|is a canonical map forW, that is,φ−1(ON(W)(W))⊂W for every W ∈ W. SinceW isσ-discrete (resp. countable star-finite or finite), it follows that cardN(W)(0)= cardW6ℵ0w(Y)6τ(resp.N(W) is countable locally finite or finite). Thus, we can apply the condition to obtain anON(W) -mapψ′ :|N(W)| →X that isU′-close toψ. Then,ψ′φis aW-map, so it is a V-map. Sinceψ′φisU′-close toψφ, it isU-close tof. ⊓⊔
The following is the locally compact version of Theorem 3.1.14 above:
Theorem 3.1.15 For a locally compact ANRX, in order that Emb(Y, X)is dense inCP(Y, X)for every locally compact separable metrizable spaceY, the following condition is necessary and sufficient:
• For any countable locally finite simplicial complex K, each proper map f :|K| →X can be approximated by OK-maps.
Proof. This can be proved by modifying the proof of Theorem 3.1.14. The necessity can be easily seen like theM1(ℵ0)-case.
To prove the sufficiency, we apply Proposition 3.1.10 instead of Proposition 3.1.9. LetY be a locally compact separable metrizable space,f :Y →X be a proper map, and U ∈ cov(X) such that clU is compact for everyU ∈ U.
Moreover, letU′ ∈cov(X) be an open star-refinement ofU. By Theorem 1.8.16 with Remark 1.6, every V ∈ cov(Y) has a countable star-finite refinement W ∈ cov(Y) with a map ψ : |N(W)| → X such that clW is compact for every W ∈ W and ψφ ≃U′ f, where φ : Y → |N(W)| is a canonical map for W. Then,φis proper because φ−1(ON(W)(W))⊂W for every W ∈ W.
The star-finiteness of W means the local compactness of|N(W)|, henceφis perfect (Proposition 1.3.1), so it is closed. SinceψφisU′-close to the proper map f, it is proper. Then, it follows thatψ|φ(Y) is proper. For eachU ∈ U, ψ−1(clU)∩φ(Y) is compact. It is easy to find an open set U∗ in |N(W)|
such thatU∗∩φ(Y) =ψ−1(U)∩φ(Y) and clU∗ is compact. Since φ(Y)⊂
∪
U∈UU∗, we have a subcomplex K of some small subdivision ofN(W) such that φ(Y)⊂ |K| ⊂ ∪
U∈UU∗. Then,ψ||K| is proper. Indeed, each compact setA⊂X is covered by finitely manyU1, . . . , Un∈ U. Since
ψ−1(A)∩ |K| ⊂
∪n i=1
ψ−1(Ui)∩ |K| ⊂
∪n i=1
clUi∗,
it follows thatψ−1(A)∩ |K|is compact. Thus, we can apply the condition to obtain anOK-mapψ′:|K| →Xthat isU′-close toψ. Then,ψ′φis aW-map, so aV-map. Since ψ′φisU′-close toψφ, it isU-close tof. ⊓⊔
152 3 Hilbert Manifolds and Hilbert Cube Manifolds