136 2 Fundamental Results on Infinite-Dimensional Manifolds (d) Ais contained in some collared set in M.
Proof. The implications (c)⇒ (d)⇒(a) are easy. We shall show the impli-cations (a)⇒(b)⇒(c).
(a)⇒ (b): Eacha∈Ahas an open neighborhood U in M with an open embedding φ : U → Q. Take another open neighborhood V of a in M so that clV is compact and clV ⊂ U. By Theorem 2.8.7(ii), φ(A∩U) is a Z-set in φ(U), hence φ(A∩clV) is also aZ-set inφ(U). Since φ(A∩clV) is compact, it is a Z-set in Q by Theorem 2.8.7(iii). It follows from Theorem 2.10.6 thatφ(A∩clV) isQ-deficient inQ. By Corollary 2.3.10(ii),φ(A∩V) = φ(A∩clV)∩φ(V) isQ-deficient inφ(V). Hence,A∩V isQ-deficient inV. This means that A is locally Q-deficient in M. It follows from Proposition 2.3.14 thatAisQ-deficient inM.
(b)⇒(c): Let φ:M →M×Qbe a homeomorphism such that φ(x) = (x,0) for eachx∈A (Corollary 2.3.10(iv)). On the other hand, Q×I≈Q is homogeneous (Theorem 2.1.1), hence we have a homeomorphismψ:Q→ Q×Isuch thatψ(0) = (0,0). The following is the desired homeomorphism:
h= (φ−1×idI)(idM ×ψ)φ:M →M ×I.
Using Theorem 2.10.9, the Strong Universality Theorem 2.3.16 and the Closed Embedding Approximation Theorem 2.3.17 can be rewritten as follows:
Theorem 2.10.10 (Strong Universality) Let M be a Q-manifold and A ∈ M0 with a closed set B ⊂ A and a map f : A → M such that f|B is a Z-embedding. Then, for each U ∈ cov(M), there exists a Z-embedding h:A→M such that h|B =f|B andhisU-homotopic tof. ⊓⊔
Theorem 2.10.11 (Z-Embedding Approximation) LetM be a Q-mani-fold and A be a locally compact separable metrizable space with a closed set B ⊂ A and a proper map f : A → M such that f|B is a Z-embedding.
Then, for eachU ∈cov(M), there exists aZ-embeddingh:A→M such that h|B=f|B andh≃pUf (i.e.,his properlyU-homotopic to f). ⊓⊔
2.11 Complementary Results forQ-Manifolds 137 Theorem 2.11.1 (Collaring) Let M andN be Q-manifolds such that N is closed inM. Then,N is a Z-submanifold ofM if and only ifN is collared inM.
Proof. The “if” part is trivial. We show the “only if” part. Since a locally collared set is collared (Theorem 2.5.7), it suffices to show that N is locally collared in M. Eachx∈N has an open neighborhoodU in M with an open embedding φ: U → Q. Let W = pr−11 (1) ⊂Q, where pr1 : Q→ [−1,1] is the projection onto the first factor. We can easily find an open neighborhood V of x in N and an embedding ψ : clV → W such that clV is compact, clV ⊂ U, and ψ(V) is open in W. By Theorem 2.8.7(ii), clV is aZ-set in U, hence φ(clV) is also a Z-set in φ(U). Since φ(clV) is compact, it is a Z-set in Qby Theorem 2.8.7(iii). On the other hand, since W is a Z-set in Q, ψ(clV) is also a Z-set in Q. Applying Corollary 2.10.7, we can obtain a homeomorphismh:Q→Q such thathφ|clV =ψ. Then,hφ(U) is an open set inQandψ(V) =hφ(V) is open inhφ(U)∩W. Hence,hφ(V) is collared in hφ(U) (cf. (G-1) in the proof of Theorem 2.5.7), which implies that V is collared inU. Thus,N is locally collared inM. ⊓⊔
The Open Embedding Theorem 2.5.10 does not hold for Q-manifolds, because any compactQ-manifold cannot be embedded intoQas an open set unless it is homeomorphic toQ. However, the following can be obtained as a corollary of the Collaring Theorem 2.11.1 above:
Theorem 2.11.2 (Open Embedding) For every connectedQ-manifoldM, M ×[0,1)can be embedded in Q as an open set.
Proof. LetαM =M∪{∞}be the one-point compactification ofM. SinceαM is compact metrizable, we have a Z-embedding h: αM → Q. Then, h(M) is aZ-submanifold ofQ\ {h(∞)}, hence it is collared inQ\ {h(∞)} by the Collaring Theorem 2.11.1. Thus, we have an open embeddingg:M×[0,1)→ Q. ⊓⊔
In the above proof, we may assume h(∞) =0 by the homogeneity ofQ (Theorem 2.1.1). Then,h(M) has a collark:h(M)×[0,1)→Q\ {0}.
M ≈ M ×I ⊃ M ×[0,1) h×id≈ //h(M)×[0,1) k //Q\ {0}
For each Z-set A in M, h(A) is a Z-set in Q\ {0}. On the other hand, by Theorem 2.10.9, we have a homeomorphism f : M → M ×I such that f(x) = (x,0) for eachx∈A. Then,U =f−1(M×[0,1)) is a homotopy dense open set in M with A ⊂ U and k(h×id)f|U : U → Q\ {0} is an open embedding. Thus, we have the following:
Corollary 2.11.3 Let M be a connected Q-manifold and A a Z-set in M. Then, there is an open dense set U in M with an open embedding g : U → Q\ {0} such that A⊂U andg(A)is aZ-set inQ\ {0}. ⊓⊔
138 2 Fundamental Results on Infinite-Dimensional Manifolds
Modifying the proof of the Unknotting Theorem 2.5.11, we can prove the followingZ-Set Unknotting Theorem forQ-manifolds:
Theorem 2.11.4 (Z-Set Unknotting) Let M be a Q-manifold and h : A→Ba homeomorphism betweenZ-sets inM. Ifhis properly homotopic to idinM, thenhcan be extended to a homeomorphismh˜:M →M. Moreover, if f : A×I→ M is a proper homotopy with f0 = id andf1 = h andU is an open cover off(A×I) inM, then ˜hcan be chosen so as to be ambiently invertiblyU∗-isotopic toid, whereU∗={st(f({x} ×I),U)|x∈A}.
Proof. It suffices to consider the case whereM is connected. TakeV ∈cov(M) such that st3V ≺ U,V is locally finite, and clV is compact for eachV ∈ V.
Observe thatV[clV] is finite for eachV ∈ V. Then, it follows that stiV,i= 1,2,3, are also locally finite and the closures of these members are compact.
First, we shall construct an ambient invertible stV-isotopyψ:M×I→M such that A∩ψ1(B) =∅. Since A∪B is a Z-set inM, it is Q-deficient in M. By Corollary 2.3.10(iv), we have a homeomorphismk:M×Q→M such that k(x,0) =x for each x∈A∪B and k is V-close to prM. We define an ambient invertible isotopyζ:Q×I→Qas follows:
ζt(x) =
{((1−t/2)x(1) +t/2, x(2), x(3), . . .) ifx(1)>0, ((1 +t/2)x(1) +t/2, x(2), x(3), . . .) ifx(1)60.
Then,ζ0= id andζ1(0) = (1/2,0,0, . . .). The desired isotopyψ:M×I→M can be defined by ψt = k(id×ζt)k−1. In the same way as in the proof of Theorem 2.5.11, we have a proper homotopyf′ :A×I→M from id toψ1h (cf. Proposition 2.1.18). Then, by Theorem 2.10.11, we have aZ-embedding f′′:A×I→M such thatf′′|A× {0,1}=f′|A× {0,1} andf′′ isV-close to f′.
On the other hand, we have aZ-embeddingj :A→Q\ {0} by Corollary 2.11.3. Then,j(A)×Iis aZ-set in (Q\{0})×[−1,2] and (Q\{0})×[−1,2]≈ Q\ {0} by the Stability Theorem 2.3.13. Using Corollary 2.11.3, we have an open neighborhoodU of f′′(A×I) inM with an open embeddingg′ :U → (Q\ {0})×[−1,2] such that g′f′′(A×I) is a Z-set in (Q\ {0})×[−1,2].
Using Corollary 2.10.8, we can obtain a homeomorphism g′′: (Q\ {0})×[−1,2]→(Q\ {0})×[−1,2]
such thatg′′g′f′′(x, t) = (j(x), t) for each (x, t)∈A×I. Then, note that {j(x)} ×I=g′′g′f′′({x} ×I)⊂g′′g′(st(f′({x} ×I),V)).
LetW0 be an open neighborhood ofg′′g′f′′(A×I) in (Q\ {0})×[−1,2]
such that clW0⊂g′′g′(U). For eachx∈A, define W(x) =W0∩g′′g′(st(f′({x} ×I),V))
2.11 Complementary Results forQ-Manifolds 139 and letW={W(x)|x∈A}. In the same way as the proof of Theorem 2.5.11, we can construct an ambient invertibleW-isotopy
φ′: (Q\ {0})×[−1,2]×I→(Q\ {0})×[−1,2]
such that φ′0 = id, φ′({x} ×I×I)⊂ W(x), and φ′1(j(x),0) = (j(x),1) for every x∈A, where the last condition means thatφ′1g′′g′|A=g′′g′ψ1h. Since
φ′t|(Q\ {0})×[−1,2]\W0= id,
we can define an ambient invertible (g′′g′)−1(W)-isotopy φ:M×I→M by φt|U = (g′′g′)−1φ′tg′′g′ and φt|M\U = id.
Then,φ0= id andφ1|A= (g′′g′)−1φ′1g′′g′|A=ψ1h. Hence,hcan be extended to a homeomorphism ˜h=ψ1−1φ1:M →M. In the same way as for Theorem 2.5.11, we can define an ambient invertibleU∗-isotopyηfrom id to ˜h=ψ1−1φ1.
⊓
⊔
Remark 2.12 In the above, the properness of the homotopy is necessary.
In fact, A = R× {0,1} ×Q is a Z-set in the Q-manifold M = R×I×Q.
We define a homeomorphismh:A→Abyh(t, i, x) = ((−1)it, i, x) for each (t, i, x)∈A=R× {0,1} ×Q. Then,h≃id inM =R×I×Q, buthcannot be extended to a homeomorphism of M onto itself. Indeed, assume that h is extended to a homeomorphism ˜h : M → M. Let U+ = (0,∞)×I×Q, U− = (−∞,0)×I×Q and L = {0} ×I×Q. Choose β > 0 so that β ̸∈
prR˜h(L). Then, {β} ×I×Q⊂˜h(U+)∪˜h(U−), (β,0,0) =h(β,0,0)∈˜h(U+) and (β,1,0) = h(−β,1,0)∈h(U˜ −), which contradicts the connectedness of {β} ×I×Q.
Suppose that f : M → N is a proper U-homotopy equivalence between Q-manifolds, where U ∈ cov(N). As observed in Sect. 2.6, Remark 2.7, we can apply the same construction as in the proof of Theorem 2.6.2 to Q-manifolds, where Theorem 2.4.9, the Strong Universality Theorem 2.3.15, the Collaring Theorem 2.5.9, and the Unknotting Theorem 2.5.11 are replaced by Theorem 2.10.9, the Closed Embedding Approximation 2.3.17, the Collaring Theorem 2.11.1, theZ-set-Unknotting Theorem 2.11.4, respectively. Thus, we can obtain the homeomorphism
h=ψ−1φ:M ×R+ →N×R+
such that prNh is st(st2U,V)-close to fprM, where V is an open star-refinement ofU. Thus, we have the following:
Proposition 2.11.5 Let M and N be Q-manifolds and U be an open cover of N. For each proper U-homotopy equivalence f : M → N, there exists a homeomorphism h:M ×R+ →N ×R+ such that prNh isst5U-homotopic tofprM andh(M × {0})⊂N× {0}. ⊓⊔
140 2 Fundamental Results on Infinite-Dimensional Manifolds
Recall that a spaceX isR+-stable ifX×R+≈X. For everyQ-manifold M,M ×R+is anR+-stableQ-manifold because
M×R+×R+≈M×I×R+≈M ×R+, whereR+≈[0,1) and [0,1)2≈[0,1)×Iby Lemma 2.1.2.
Lemma 2.11.6 For each R+-stable Q-manifold M, there is a homeomor-phism h:M×R+→M that is homotopic to the projection.
Proof. In the following, we replaceR+with [0,1). Letφ:M →M×[0,1) be a homeomorphism. SinceM×[0,1) is aQ-manifold andQ×I≈Q, we apply the Stability Theorem 2.3.13 to obtain a homeomorphism
f :M×[0,1)×I→M×[0,1)
that is homotopic to the projection. On the other hand, as is observed in the above, we have a homeomorphism ψ : [0,1)2 → [0,1)×I by Lemma 2.1.2.
Then, we can define a homeomorphismh:M×[0,1)→M so as to make the following diagram commutative:
M×[0,1)
h
φ×id[0,1)
//M ×[0,1)×[0,1)
idM×ψ
M×[0,1)×I
f
M φ //M×[0,1)
Since [0,1) is contractible, it follows that pr[0,1)ψ≃pr[0,1), hence the following diagram is commutative up to homotopy:
M×[0,1)×[0,1)
idM×ψ
prM×[0,1)
**T
TT TT TT TT TT TT TT T
M×[0,1)×I pr
M×[0,1) //M×[0,1) Note that the bottom prM×[0,1)is homotopic to f. Thus, we have
h=φ−1◦f◦idM×ψ◦φ×id[0,1)
≃φ−1◦prM×[0,1)◦idM ×ψ◦φ×id[0,1)
≃φ−1◦prM×[0,1)◦φ×id[0,1)=φ−1◦φ◦prM = prM.
By Proposition 2.11.5 and Lemma 2.11.6, we can prove the following Clas-sification Theorem forR+-stableQ-manifolds:
2.11 Complementary Results forQ-Manifolds 141 Theorem 2.11.7 (Classification) Let M and N be R+-stable Q-mani-folds. Then, every proper homotopy equivalencef :M →N is homotopic to a homeomorphism. Thus, twoR+-stableQ-manifolds are homeomorphic if they have the same proper homotopy type. ⊓⊔
By the Open Embedding Theorem 2.11.2, we have the following:
Theorem 2.11.8 (Open Embedding) Every connectedR+-stable Q-man-ifold can be embedded in Qas an open set. ⊓⊔
Due to the Classification Theorem 2.11.7 and the Open Embedding The-orem 2.11.8 above,R+-stableQ-manifolds are tractable amongQ-manifolds.
In order to characterize suchQ-manifolds, we introduce the notion of proper contractibility to infinity. A spaceX is said to beproperly contractible to infinity provided, for each compact setC in X, there is a proper homotopy h:X×I→X withh0= idX andh1(X)⊂X\C.
Theorem 2.11.9 A connectedQ-manifoldM isR+-stable if and only ifM is properly contractible to infinity.
Proof. The “only if” part follows from the fact that M ×R+ is properly contractible to infinity. This fact can be shown as follows: For each compact set C ⊂M ×R+, choosea >0 so that C ⊂M ×[0, a). The desired proper homotopyh:M×R+×I→M×R+ can be defined byh(x, s, t) = (x, s+at) for every (x, s, t)∈M×R+×I.
To see the “if” part, it suffices to prove thatM×I≈M ×[0,1), because M ≈M×Iby the Stability Theorem 2.3.13. SinceM is locally compact and σ-compact, there are compact setsXn,n∈N, inM such thatXn ⊂intXn+1
and M =∪
n∈NXn. Take 0 < t1 < t2 <· · · <1 so that supn∈Ntn = 1. We will inductively construct homeomorphisms hn :M ×I→M ×I, n∈N, so as to satisfy the following:
(1) hnhn−1· · ·h1(M× {1})⊂(M \Xn)× {1}, (2) hnhn−1· · ·h1|M ×[0, tn] =hn−1· · ·h1|M ×[0, tn], (3) hn|Xn−1×I= id.
In the following, keep in mind that every proper map ofM into itself can be approximated by closed embeddings (the Z-Embedding Approximation Theorem 2.10.11). Since M is properly contractible to infinity, we have a closed embeddingf1:M× {1} →(M\X1)× {1}that is properly homotopic to id. By Theorem 2.10.9, M × {1} is a Z-set in M ×[t1,1]. Applying the Z-Set Unknotting Theorem 2.11.4, we can extend f1 to a homeomorphism h1 : M ×I→ M ×I such that h1|M ×[0, t1] = id. Then, h1(M × {1}) ⊂ (M\X1)× {1}.
Suppose that homeomorphisms h1, . . . , hn−1 have been defined so as to satisfy (1), (2), and (3). As is easily observed,
142 2 Fundamental Results on Infinite-Dimensional Manifolds
Xn−1
hn−1· · ·h1(M× {1}) Xn
1
0 M
(hn−1· · ·h1)−1(Xn−1×I) M (hn−1· · ·h1)−1(Xn−1×I) gn(M× {1})
tn
s 1
0 t1
hn−1· · ·h1
t1
Fig. 2.26.homeomorphismshn−1· · ·h1 andgn
(hn−1· · ·h1)−1(Xn−1×I)∩(M × {1}) =∅.
Chooses∈(tn,1) so that
(hn−1· · ·h1)−1(Xn−1×I)⊂M×[0, s].
SinceM is properly contractible to infinity, we have a closed embedding fn:M× {1} →(M× {1})\(hn−1· · ·h1)−1(Xn×I)
that is properly homotopic to id. Applying the Z-Set Unknotting Theorem 2.11.4, we can extendfnto a homeomorphismgn:M×I→M×Isuch that gn|M×[0, s] = id. Then,
gn|(hn−1· · ·h1)−1(Xn−1×I) = id. (∗) Sincegn is an extension offn, we also have
(hn−1· · ·h1)gn(M× {1})⊂(M\Xn)× {1}.
The following homeomorphism satisfies (1), (2), and (3):
hn= (hn−1· · ·h1)gn(hn−1· · ·h1)−1:M ×I→M ×I Indeed, this can be verified as follows:
hnhn−1· · ·h1(M× {1}) = (hn−1· · ·h1)gn(M× {1})⊂(M \Xn)× {1}.
Sincegn|M×[0, s] = id andtn< s, we have
hnhn−1· · ·h1|M×[0, tn] = (hn−1· · ·h1)gn|M×[0, tn]
=hn−1· · ·h1|M×[0, tn].
Moreover, it follows from (∗) that
Notes for Chapter 2 143 hn|Xn−1×I= (hn−1· · ·h1)gn(hn−1· · ·h1)−1|Xn−1×I= id.
Now, by induction, we have homeomorphisms hn, n ∈N, satisfying (1), (2), and (3). Then, by virtue of (2), we can defineh:M×[0,1)→M×Iby
h(x) = lim
n∈Nhnhn−1· · ·h1(x) for each (x, t)∈M×[0,1).
SinceM ×[0,1) =∪
n∈NM×[0, tn) and
h|M×[0, tn) =hn−1· · ·h1|M×[0, tn)
is an open embedding into M ×I for eachn ∈N, it is easy to see thathis continuous, open, and injective. For eachy ∈M×I=∪
n∈NXn×I, choose n∈Nso that y∈Xn×I. Then, we havex= (hn· · ·h1)−1(y)∈M×[0,1).
Indeed, hn· · ·h1(x) = y ̸∈ (M \Xn)× {1}, which impliesx ̸∈M × {1} by (1). Sincehn· · ·h1(x)∈Xn×Iandhi|Xn×I= id for everyi > n, it follows that h(x) = hn· · ·h1(x) = y. Hence, h is surjective. Consequently, h is a homeomorphism. ⊓⊔
Notes for Chapter 2
For related topics, refer to the book of Bessaga and Pe lczy´nski:
• C. Bessaga and A. Pe lczy´nski, Selected Topics in Infinite-Dimensional Topology, MM58, Polish Sci. Publ., Warsaw, 1975.
ConcerningQ-manifolds, the follwing literatures are excellent:
• J. van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library43, Elsevier Sci. Publ. B.V., Amsterdam, 1989.
• T.A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser.
in Math.28, Amer. Math. Soc., Providence, 1975.
In this chapter, we have treated almost all contents in the book of van Mill except for the characterization of compactQ-manifolds. Without compactness, the charac-terization ofQ-manifolds will be treated in the next chapter togather with the one of ℓ2(Γ)-manifols. As a well-known application to Shape Theory, Chapman estab-lished in [16] the Complement Theorem, that is, twoZ-setsX andY in the Hilbert cubeQhave the same shape type if and only if their complementsQ\X andQ\Y are homeomorphic. As another remarkable application to Simple Homotopy Theory, Chapman proved in [18] the topological invariance of Whitehead torsion. For this subject, refer to the following lecture notes due to Cohen:
• M.M. Cohen, A Course in Simple-Homotopy Theory, GTM10, Springer-Verlag, New York, 1973.
144 2 Fundamental Results on Infinite-Dimensional Manifolds
The proof of Lemma 2.1.9 is due to Cutler (cf. Lemma 1.1 in [39]). Theorem 2.1.10 is due to Henderson [39] (cf. [44] and [40]). In this section, we need not assume thatE is normable.
The non-invertible isotopy in the proof of Proposition 2.1.11 is due to Anderson and Bing [5]. The one of Proposition 2.1.12 is a modification of the one constructed by Crowell in [21]. Note that the Crowell’s example is not locally connected. Lemma 2.1.15 is due to Anderon, Henderson and West [7].
The results of§2.2 is due to Toru´nczyk [67] (the compact case was proved in [66]). In [68], the results have been generalized to metric linear spaces. The mapsf andgin§2.2 was first defined onX×lim
−→Rnand lim
−→Rnby D.W. Henderson in [41]
so as to prove thatX×lim
−→Rn≈lim
−→Rn.
The Stability Theorem 2.3.7 was first established forℓ2-manifolds by Anderson and Schori [8] and generalized as Theorem 2.3.7 by Schori [63] (cf. [44]). For the Stability Theorem, we can assert more (cf. [30] and [53]). The results can be extended to local trivial bundles withE-manifold fibers (cf. [54]). Reformulating Lemma 2.3.1 by Yamashita, the proofs of Lemmas 2.3.1 and 2.3.2 were made slim. The proof of the Strong Universality Theorem 2.3.15 is due to [53]. In this section, the assumption thatE≈EN or≈EfN is only used.
Proposition 2.4.1 is due to Anderson and Bing [5], which is generalized to an infinite-dimensional normed linear space E, that is, there exists an I-preserving homeomorphism ζ : E×I\ {(0,0)} → E×I. This can be proved by using non-complete norm. Refer to Bessaga-Pe lczy´nski’s book, Ch.III,§5 (cf. [12]). The Neg-ligibily Theorem was first established by Anderson, Henderson and West [7] and extended to Theorem 2.4.3 by Cutler [22]. In [1], Anderson proved thatσ-compact sets are negligible inℓ2 andRN, that is a key in his proof of ℓ2 ≈RN. The proof of Proposition 2.4.13 is due to [54].
Theorem 2.5.7 was established by Brown [13]. The Open Embedding Theorem 2.5.10 and the Unknotting Theorem 2.5.11 were respectively established by Hen-derson [38] (cf. [40]) and AnHen-derson and McCharen [6] forℓ2-manifolds. They were extended in [43] (cf. [40]) and [15], respectively. The proof is due to [53]. These results can be extended to local trivial bundles withE-manifold fibers (cf. [55]).
The Classification Theorem 2.6.1 was established by Henderson and Schori [43]
(cf. [44] and [40]). Theorem 2.6.2 is due to Ferry [29].
The Bing Shrinking Criterion arose in [11] and has been generalized by several peoples. Refer to [27]. Theorem 2.7.1 here is due to Toru´nczyk [73].
The condition (z) in Remark on Theorem 2.8.6 is called PropertyZ in [2]. In [6], a closed set with Property Z is called a Z-set. The notation of “closed over a set” was introduced by Bestvina and Mogilski [10], who adopt Lemma 2.8.9 as the definition. Propositions 2.8.11, 2.8.12 and 2.8.13 are proved in [10]. Yamashita simplified the proof of Proposition 2.8.11, which is presented here.
Proposition 2.9.1 is due to Toru´nczyk [70] (cf. [73]). The author refer the reader to [42]. In caseEis locally convex,Z-sets in anE-manifold is strongZ-sets. The non-locally convex case is not known. In [42], by using proper subdivisions of simplicial complexes, Henderson proved the implication (a) ⇒(b) in Theorem 2.9.3. In our proof, we avoid proper subdivisions.
R.Y.T. Wong characterized in [77] theR+-stability ofQ-manifolds as in Theorem 2.11.9.
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.