88 2 Fundamental Results on Infinite-Dimensional Manifolds
For eachx∈C, h′(x,0) =xbecausef(x) =f′(x)∈X × {0}. For each x∈ X\C, letf(x) = (x′, t). Then,t >0. Sincef′h′(x,0) = (x′, h(t,0)) = (x′, t,0) andf′(x,0) = (f(x),0) = (x′, t,0), we have h′(x,0) = (x,0).
Applying Lemma 2.3.2 (A = B, A0 = C, A1 = X), we have a homeo-morphism g: (X×E)C →X such thatg(x,0) =xfor each x∈B\C and g|C = id. Then, gh′ : X ×E → X is a homeomorphism. For each x ∈ C, gh′(x,0) = g(x) = x. For each x ∈ B \C, gh′(x,0) = g(x,0) = x. Thus, gh′(x,0) =xfor eachx∈A∪B. ⊓⊔
2.5 The Collaring and Unknotting Theorems 89
b
0 a
β
∪ α U {x} ×[a, b]
A
U N
X R
Fig. 2.9. The mapsαandβ
Then, γ′ is l.s.c. Indeed, if γ′(x) > t > s then {x} ×[α(x)−t, β(x) +t] is contained in someU ∈ U, soxhas a neighborhoodV inX such that
V ×[α(x)−t, β(x) +t]⊂U.
Sinceαandβare continuous,xhas a neighborhoodW inXsuch thatW ⊂V and|α(x)−α(y)|,|β(x)−β(y)|< t−sfory∈W. Hence,
{y} ×[α(y)−s, β(y) +s]⊂U for every y∈W, that is,γ′(y)> sfory∈W.
Note thatU0 is paracompact because X is hereditarily paracompact by Theorem 1.3.9. Applying Theorem 1.3.10, we have a map γ0 : U0 → (0,1) such that 0 < γ0(x) < γ′(x) for each x∈ U0. Since X is perfectly normal, X\U0 =k0−1(0) for some mapk0 :X →I. We can define a map γ:X →I byγ(X\U0) ={0} andγ(x) =k0(x)γ0(x) for eachx∈U0. It is easy to see that γsatisfies conditions (3), (4), and (5). ⊓⊔
Remark 2.5 In Lemma 2.5.1 above, if X is not assumed to be perfectly normal, we define a map γ using k instead of k0. This does not satisfy (4), but satisfies the following weaker condition:
(4)’ γ(x)>0 ifα(x)< β(x) (equivalentlyα(x)<0 orβ(x)>0).
Whena=b= 0 in Lemma 2.5.1, we have the following:
Lemma 2.5.2 Let X be a perfectly normal paracompact space.15 For each open collection U inX×I, there exists a map γ:X →Isuch that γ(x)>0 if and only if (x,0)∈∪U, and
{{x} ×[0, γ(x)]x∈X, γ(x)>0}
≺ U.
15If U is an open cover (orU is an open neighborhood) ofX× {0}inX×I, the perfect normality ofX need not to be assumed.
90 2 Fundamental Results on Infinite-Dimensional Manifolds
In particular, for an open setU inX×I, there is a map γ:X →Isuch that γ(x)>0 if and only if(x,0)∈U, and
{(x, t)∈X×Iγ(x)>0, t6γ(x)}
⊂U.
In the above, if U is an open cover (or U is an open neighborhood) of X× {0}inX×I, without assuming the perfect normality ofX, we can prove the same result, that is,
Lemma 2.5.3 LetX be a paracompact space. For each open coverU ofX× {0} inX×I, there exists a mapγ:X→Isuch that γ(x)>0 if and only if
{{x} ×[0, γ(x)]x∈X, γ(x)>0}
≺ U.
In particular, for an open neighborhoodU ofX× {0}inX×I, there is a map γ:X →Isuch thatγ(x)>0 if and only if(x,0)∈U, and
{(x, t)∈X×Iγ(x)>0, t6γ(x)}
⊂U.
Lemma 2.5.4 LetX be a hereditarliy paracompact space, that is, every open set in X is paracompact (cf. Theorem 1.3.9). If A is a collared set in X and U is an open set in X with A∩U ̸=∅, thenA∩U is collared in U.
Proof. Letk:A×[0,1)→X be a collar of Ain X. Applying Lemma 2.5.3, we have a mapγ:A∩U →(0,1) such that
{(x, t)∈(A∩U)×[0,1)t < γ(x)}
⊂k−1(U).
We defineh: (A∩U)×[0,1)→U byh(x, t) =k(x, γ(x)t). Then,his a collar ofA∩U inU. ⊓⊔
For a collared closed setAin a space X, a collark:A×[0,1)→X may have a defect such that k(A×[0, t]) need not be closed in X for 0< t < 1.
For example, (0,∞)× {0} is a closed set in [0,∞)2\ {(0,0)} that has the natural collar with the image (0,∞)×[0,1). On this account, we introduce a collar without such a defect. Aclosed collarofAinX is a closed embedding k:A×I→X such thatk|A×[0,1) is a collar ofA inX.
Proposition 2.5.5 Every collared closed setAin a paracompact spaceX has a closed collar inX. Moreover, for each open coverU of X, there is a closed collar k:A×I→X such that {k({x} ×I)|x∈A} ≺ U.
Proof. Leth:A×[0,1)→X be a collar ofAinX. Choose an open setU in X so that
A⊂U ⊂clU ⊂h(A×[0,1)).
By Lemma 2.5.3, we have a mapγ:A→(0,1) such that
2.5 The Collaring and Unknotting Theorems 91 Aγ ={
(x, t)∈A×[0,1)t6γ(x)}
⊂h−1(U).
Since Aγ is closed inA×[0,1) and h(Aγ) ⊂clU ⊂h(A×[0,1)), it follows that h(Aγ) is closed in clU, hence it is closed in X. Then, a closed collar k:A×I→X ofAcan be defined byk(x, t) =h(x, γ(x)t).
When an open open coverU ofX is given in the above,γ can be taken by Lemma 2.5.3 so as to satisfy
{{x} ×[0, γ(x)]x∈A}
≺h−1(U).
Then, we have {k({x} ×I)|x∈A} ≺ U. Thus, the additional statement is also true. ⊓⊔
WhenX is metrizable, we can show the following:
Lemma 2.5.6 Let A be a collared set in a metrizable space X and U be an open set inX such thatA⊂U. Then, there is a closed maph: clA×I→X such that h|A×Iis an embedding into U,h|A×[0,1) is a collar ofA inX, andh({x} ×I) ={x}for each x∈clA\A.
Proof. Since A is locally closed in X, we can assume that A =U ∩clA by replacing U with a small open set. We have a collar k : A×[0,1) → U by Lemma 2.5.4. Then, note that
k(A×[0,1))∩(clA\A) =∅.
Taked∈Metr(X) and let V ={
k(x, t)(x, t)∈A×[0,1),
d(k(x, t), x)<2−1d(x, X\k(A×[0,1)))} .
Then,A⊂V ⊂U andV is open inX becausek(A×[0,1)) is open inX. By Lemma 2.5.2, there is a mapγ :A→(0,1) (as illustrated in the left-side of Fig. 2.10) such that
{(x, t)∈A×[0,1)t6γ(x)}
⊂k−1(V).
For each (x, t)∈A×Iandy∈X\k(A×[0,1)), sincek(x, tγ(x))∈V, it follows that d(k(x, tγ(x)), x)<2−1d(x, y), hence
d(x, y)6d(k(x, tγ(x)), x) +d(k(x, tγ(x)), y)
<2−1d(x, y) +d(k(x, tγ(x)), y) and
d(k(x, tγ(x)), y)6d(k(x, tγ(x)), x) +d(x, y)<2−1d(x, y) +d(x, y).
Thus, we have the following:
2−1d(x, y)< d(k(x, tγ(x)), y)<2d(x, y).
92 2 Fundamental Results on Infinite-Dimensional Manifolds
A
k(A×[0,1))
A k 0
1
γ
V
X k−1(V)
A×[0,1)
h(A×I)
Fig. 2.10.Refining a collark:A×I→X
By this inequality, the desired maph: clA×I→X can be defined as follows (see Fig. 2.10):
h(x, t) =
{k(x, tγ(x)) ifx∈A, x ifx∈clA\A.
Indeed, the continuity ofhat (x, t)∈(clA\A)×Ican be seen as follows:
Take (xn, tn) ∈ A×I, n ∈ N, so that (xn, tn) → (x, t) as n → ∞. Since x∈clA\A, it follows that
d(h(xn, tn), h(x, t)) =d(k(xn, tnγ(xn)), x)<2d(xn, x)→0.
To show that h is closed, let (xn, tn) ∈ clA ×I, n ∈ N, and assume h(xn, tn) →y for some y ∈ X. Then, it suffices to show that (xn, tn) has a convergent subsequence. Since Iis compact, we may assume thattn →tfor some t∈I.
When y ∈ k(A×[0,1)), since k is an open embedding, it follows that (xn, tn) is convergent inA×[0,1).
Wheny ̸∈ k(A×[0,1)), if (xn, tn) ̸∈ A×[0,1) for infinitely many n ∈ N, then (xn)n∈N has a subsequence that converges to y. Otherwise, we may assume that (xn, tn)∈A×[0,1) for everyn∈N. Then, it follows that
2−1d(xn, y)< d(k(xn, tnγ(xn)), y) =d(h(xn, tn), y)→0, hencexn→y.
It is easy to see thathsatisfies the other conditions. ⊓⊔
A subsetAin a spaceX is said to belocally collaredinX if eachx∈A has a neighborhood in A that is collared in X. Obviously, a collared set is locally collared, but the converse is also true, that is, the following theorem holds:
Theorem 2.5.7 (M. Brown) Every locally collared set A in a metrizable spaceX is collared in X.
2.5 The Collaring and Unknotting Theorems 93 Proof. First, observe thatAis locally closed inX, that is, closed in some open set in X. Since a collared set in an open set in X is collared in X, we may assume that Ais closed inX. To apply Michael’s Theorem 1.3.16, it suffices to show the following:
(G-1) If U is open in A and collared inX, then each open set V in U is also collared inX;
(G-2) IfU andV are open inAand collared inX, thenU∪V is collared inX;
(G-3) If {Uλ | λ∈Λ} is a discrete collection of open sets inA such that everyUλis collared in X, then∪
λ∈ΛUλis also collared in X.
Note that (G-1) easily follows from Lemma 2.5.4. For (G-3), we have a discrete collection {Wλ | λ∈ Λ} of open sets in X such that Uλ ⊂Wλ (cf.
the definition of collectionwise normality and its remark before Proposition 1.3.8(1)). Then, (G-3) can be obtained by Lemma 2.5.4.
To show (G-2), letU andV be open set inAthat are collared inX. Take an open set Ue in X so that U = A∩Ue and apply Lemma 2.5.6 to obtain a closed map h : clU ×I → X such that h|U ×Iis an embedding into Ue, h|U ×[0,1) is a collar of U in X, and h({x} ×I) = {x} for x ∈ clU \U. Observe
A∩h(clU×I) = (A∩h(U ×I))∪(A∩clU)
⊂(A∩Ue)∪clU = clU,
henceA∩h(clU×I) = clU. Then, we have the homeomorphism φ:X →X\h(U×[0,1/2))⊂X
defined byφ|X\h(U ×[0,1)) = id andφ(h(x, t)) =h(x, t/2 + 1/2) for each (x, t)∈clU×I, whereφ(x) =h(x,1/2) for eachx∈clU.
We have a collark:V ×[0,1)→X ofV in X, where it should be noted that φkφ−1|φ(V)×id[0,1) is a collar ofφ(V) in φ(X). Letα:U∪V →Ibe a map withα−1(0) =V \U andα−1(1) =U\V. Then, we can define a map k∗: (U∪V)×[0,1)→X as follows (see Fig. 2.11):
k∗(x, t) =
h
( x, t
2α(x) )
ifx∈U andt6α(x), φk(x, t−α(x)) ifx∈V andt>α(x).
Then, k∗ is injective andk∗(x,0) =xfor everyx∈U ∪V. To see that k∗ is a collar ofU∪V inX, it remains to verify thatk∗ is an open map.
Evidently,k∗is continuous at every point of the following open sets (U ∪V)×(0,1), U×[0,1), (V \clU)×[0,1).
We shall show the continuity of k∗ at (x,0), wherex ∈V ∩(clU \U). Let W be a neighborhood of x=k∗(x,0) inX. Since h({x} ×I) ={x}, we have
94 2 Fundamental Results on Infinite-Dimensional Manifolds
U
V [0,1)
U V
U h(U×I)
V
φk(V ×[0,1))
A α
0
1
2 0
1
φk k∗ h
1−α
1
Fig. 2.11.A collar ofU∪V
some neighborhood N of xin V such that h((N∩clU)×I)⊂W. Observe thatφk(x,0) =φ(x) =h(x,1/2) =x. Then, we have a neighborhoodN′ ofx in V with 0< δ <1 such thatφk(N′×[0, δ))⊂W. It follows that
k∗((N∩N′)×[0, δ))⊂h((N∩clU)×[0,1/2])∪φk(N′×[0, δ))⊂W.
Finally, we shall prove thatk∗is open. LetW be a neighborhood of (x, t)∈ (U∪V)×[0,1).
Whent < α(x), it follows that x∈U. Then,
W−={(x′, t′/2α(x′))|(x′, t′)∈W, t′< α(x′)}
is a neighborhood of (x, t/2α(x)) inU×[0,1/2) andh(W−)⊂k∗(W). Hence, k∗(W) is a neighborhood ofh(x, t/2α(x)) =k∗(x, t) in X.
Whent > α(x), we have x∈V. Then,
W+ ={(x′, t′−α(x′))|(x′, t′)∈W, t′> α(x′)}
is a neighborhood of (x, t−α(x)) inV×(0,1) andφk(W+)⊂k∗(W). Hence, k∗(W) is a neighborhood ofφk(x, t−α(x)) =k∗(x, t) inX.
Ift=α(x)>0, thenx∈U∩V. Let
W−={(x′, t′/2α(x′))|(x′, t′)∈W, t′6α(x′)} and W+={(x′, t′−α(x′))|(x′, t′)∈W, t′ >α(x′)}.
Then, h(W−) is a neighborhood of k∗(x, t) in h(clU ×[0,1/2]) and φk(W+) is a neighborhood ofk∗(x, t) inφ(X) =X\h(U×[0,1/2)). Hence,k∗(W) = h(W−)∪φk(W+) is a neighborhood ofk∗(x, t) inX.
Whent =α(x) = 0 andx∈V \clU, we have (x,0)̸∈(φk)−1h(clU ×I) becauseh(clU×I)∩A= clU. Choose a neighborhoodN ofxinV \clU and 0< δ <1 so that
2.5 The Collaring and Unknotting Theorems 95 W0=N×[0, δ)⊂(
(V ×[0,1))\(φk)−1h(clU×I))
∩W.
Then,k(W0) =φk(W0) is a neighborhood ofk(x,0) =k∗(x, t) in the open set k((V \clU)×[0,1)) inX, hence it is a neighborhood ofk∗(x, t) inX. Since φk(W0)⊂k∗(W),k∗(W) is also a neighborhood ofk∗(x, t) in X.
Whent=α(x) = 0 and x∈V ∩bdU, choose an open neighborhood N of x in V and 0 < δ <1 so that N ×[0, δ)⊂ W and α(N)⊂ [0, δ). Then, h((N ∩clU)×[0,1/2]) is a neighborhood of xin h(clU×[0,1/2]). Observe that
M ={(x′, t′−α(x′))|x′ ∈N, α(x′)6t′< δ}
is a neighborhood of (x,0) inV ×[0,1), henceφk(M) is a neighborhood of x in φ(X) =X\h(U×[0,1/2)). Since
k∗(W)⊃h((N∩clU)×[0,1/2])∪φk(M), it follows that k∗(W) is a neighborhood ofx. ⊓⊔
Since E2 ≈ E, the following version of Klee’s Trick 1.8.4 can be easily obtained:
Theorem 2.5.8 (Klee’s Trick) Every homeomorphism h : A → B be-tween E-deficient closed sets in E can be extended to a homeomorphism
˜h:E→E. ⊓⊔
Using this form of Klee’s trick, we show the following:
Theorem 2.5.9 (Collaring) LetM andN beE-manifolds such thatN is E-deficient closed in M. Then, N is collared inM. Moreover, for each open coverU of M, there is a closed collark:N×I→M such that
{k({x} ×I)x∈N}
≺ U.
Proof. Since a locally collared set is collared (Theorem 2.5.7), it suffices to show thatN is locally collared inM. As is easily observed, eachx∈N has an open neighborhoodU inM such thatU∩N andUare homeomorphic to open sets inE. Letf :U∩N →E× {0}(⊂E×[0,1)) andg:U →E×[0,1) be open embeddings. Choose an open neighborhoodV ofxinU so that clV ⊂U andf(N∩clV),g(clV) are closed inE×{0}andE×[0,1), respectively. Since N∩clV isE-deficient inU by Corollary 2.3.4(ii),g(N∩clV) is anE-deficient closed set inE×[0,1) by Proposition 2.3.8. On the other hand,f(N∩clV) is also anE-deficient closed set inE×[0,1) by Theorem 2.4.9. Using Klee’s trick above,f(g|N∩clV)−1can be extended to a homeomorphismh:E×[0,1)→ E ×[0,1). Then, hg|V : V → E ×[0,1) is an open embedding such that hg(N ∩V) = f(N ∩V) is open in E× {0}. Since E × {0} is collared in E×[0,1),hg(N∩V) is also collared inhg(V) by Lemma 2.5.4. Hence,N∩V is collared inV. — Fig. 2.12.
Due to Proposition 2.5.5, the additional statement follows from the fact that N is a collared closed set inM. ⊓⊔
96 2 Fundamental Results on Infinite-Dimensional Manifolds 1
0
E f(N∩clV)
g(N∩clV) g(U)
f g−1 h
hg(U)
E×[0,1)
Fig. 2.12.A collar ofU∩V
Now, we have the following:
Theorem 2.5.10 (Open Embedding) Every connectedE-manifold M can be embedded inE as an open set. Moreover, ifAis anE-deficient closed set in M, then there is an open embeddingg:M →E such thatg(A)isE-deficient and closed inE.
Proof. Note thatM ∈ FE(Theorem 2.1.10). Then, by the Strong Universality Theorem 2.3.15, we have anE-deficient closed embeddingh:M →E, hence h(M) has a collar k : h(M)×[0,1) → E by the Collaring Theorem 2.5.9.
On the other hand, by Theorem 2.4.9, we have a homeomorphism f :M → M ×[0,1) such that f(A) ⊂ M × {0}. Then, g = k(h×id)f : M → E is the desired open embedding because g(A) is closed in the closed subspace k(h(M)× {0}) =h(M) ofE. ⊓⊔
Remark 2.6 Theorem 2.5.10 above is valid for anyE-manifoldM such that w(M) =w(E); equivalently,M has at mostw(E)-many components. Indeed, by Lemma 2.1.8,E has a discrete open collectionU such that cardU =w(E) and everyU ∈ U is homeomorphic toE itself.
Theorem 2.5.8 can be extended toE-manifolds as follows:
Theorem 2.5.11 (Unknotting) Let h:A →B be a homeomorphism be-tween E-deficient closed sets in anE-manifoldM andU be an open cover of M. If there is a homotopy f :A×I→M with f0 = id and f1 =h, then h can be extended to a homeomorphism ˜h:M →M that is ambiently invertibly U∗-isotopic to id, whereU∗={st(f({x} ×I),U)|x∈A}.
Proof. Without loss of generality, we may assume thatM is connected. Take V ∈cov(M) so that st3V ≺ U and letV[B] ={V ∈ V |V ∩B̸=∅}. First, we will construct an ambient invertible V[B]-isotopyψ :M×I→M such that ψ0 = id and A∩ψ1(B) = ∅. Let V′ be an open star-refinement of V. Since A∪BisE-deficient inM by Proposition 2.4.13, we apply Corollary 2.3.4(iv) to obtain a homeomorphismk:M ×E →M such that k(x,0) =xfor each
2.5 The Collaring and Unknotting Theorems 97 x∈A∪B and k isV′-close to prM. Takingv ∈E\ {0} and a Urysohn ma α:M →Iwithα(k−1(B)) ={1}andα(M\k−1(st(B,V′))) ={0}, we define an ambient invertible isotopyψ′:M×E×I→M×E by
ψ′(x, y, t) = (x, y+tα(x)v) for each (x, y, t)∈M ×E×I.
The isotopyψ can be defined byψt=kψ′tk−1 for eacht∈I.
To verify thatψis aV[B]-isotopy, for eachx∈M, letk−1(x) = (x′, y). If x′ ̸∈st(B,V′), thenα(x′) = 0, hence ψt(x) =kψ′(x′, y, t) =k(x′, y) =xfor every t∈I, that is,ψ({x} ×I) ={x}. Whenx′ ∈st(B,V′), we haveV′∈ V′ such thatx′∈V′ andV′∩B̸=∅. Since stV′≺ V, there is someV ∈ V such that st(V′,V′)⊂V, whereV∩B̸=∅, that is,V ∈ V[B]. BecausekisV′-close to prM, it follows that
ψ({x} ×I)⊂k({x′} ×E)⊂st(x′,V′)⊂st(V′,V′)⊂V.
It should be remarked thatψt−1=kψ′t−1k−1for eacht∈I, where ψt′−1(x, y) = (x, y−tα(x)v) for each (x, y, t)∈M×E×I.
Then, by the same argument as above, the isotopy defined byψt−1,t∈I, is a V[B]-isotopy.16
Letf′ :A×I→M be the homotopy defined by ft′=
{f2t if 06t61/2, ψ2t−1h if 1/26t61.
Sincef0′ = id and f1′ =ψ1h,f′|A× {0,1} is anE-deficient closed embedding intoM. Moreover,
f′({x} ×I)⊂st(f({x} ×I),V) for eachx∈A.
By the Strong Universality Theorem 2.3.15, there is an E-deficient closed embedding f′′ : A ×I → M such that f′′|A× {0} = f′|A× {0} = id, f′′|A× {1}=f′|A× {1}=ψ1handf′′isV-close tof′. Then,
f′′({x} ×I)⊂st(f′({x} ×I),V) for each x∈A.
On the other hand, we have anE-deficient closed embeddingj :A→E.
Then,j(A)×Iis anE-deficient closed set inE×R. Using the Open Embedding Theorem 2.5.10 and Klee’s Trick 2.5.8, we can construct an open embedding g : M → E×R such that g(f′′(x, t)) = (j(x), t) for (x, t)∈ A×I (cf. Fig.
2.13).
For eachx∈A, define
W(x) =g(st(f′({x} ×I),V))⊂g(M)
16Cf. Proposition 2.1.14.
98 2 Fundamental Results on Infinite-Dimensional Manifolds
E R
0 j(A)
g(M) M
1 g A
f′′(A×I)
{j(x)} ×I f′′({x} ×I)
W(x) j(A)×I
β
Fig. 2.13.The open embeddinggand the open collectionW
and letW={W(x)|x∈A}. Note that
{j(x)} ×I=gf′′({x} ×I)⊂W(x) for each x∈A.
By Lemma 2.5.1, we have mapsβ, γ:E→Rsuch that (1) β(x)>0 forx∈E,
(2) β(x) = 1 andγ(x)>0 ifx∈j(A), (3) β(x) =γ(x) = 0 if (x,0)̸∈∪
x∈AW(x), (4) γ(x)>0 ifβ(x)>0,
(5) {{x} ×[−γ(x), β(x) +γ(x)]|(x,0)∈∪
x∈AW(x)} ≺ W.
In the right-side picture of Fig. 2.13, we have an ambient invertibleW-isotopy φ′:E×R×I→E×Rsliding points vertically so as to move{0} ×E to the graph of β. The following is the precise definition ofφ′:
φ′t(x, s) =
(
x, tβ(x) +((1−t)β(x) +γ(x))s β(x) +γ(x)
)
if 06s6β(x) +γ(x), (
x, s+tβ(x)(s+γ(x)) γ(x)
)
if −γ(x)6s60,
(x, s) otherwise.
Then,φ′0= id andφ′1g|A=gψ1h. Indeed, for every x∈A, φ′1g(x) =φ′1gf′(x,0) =φ′1gf′′(x,0) =φ′1(j(x),0)
= (j(x),1) =gf′′(x,1) =gf′(x,1) =gψ1h(x).
Sinceφ′t(g(M)) =g(M) for eacht∈I, we have an ambient invertibleg−1 (W)-isotopyφ:M ×I→M defined byφt=g−1φ′tg. Then,φ0= id andφ1|A= g−1φ′1g|A=ψ1h. Thus,his extended to a homeomorphism ˜h=ψ−11 φ1:M → M. Moreover, we can define an ambient invertible isotopyη:M×I→M by
ηt=
{φ2t if 06t61/2, ψ−12t−1φ1 if 1/26t61.
2.5 The Collaring and Unknotting Theorems 99 Then,η0= id andη1=ψ−11 φ1= ˜h.
To verify that η is a U∗-isotopy, let x ∈ M. When φ({x} ×I) is not contained ing−1(W(y)) for anyy∈A, sinceφis ag−1(W)-isotopy, it follows that φ({x} ×I) ={x}. Then,
x̸∈ ∪
y∈A
g−1(W(y)) = st(f′(A×I),V),
hencex̸∈ st(f1/2′ (A),V) = st(B,V). In this case, for eacht ∈ I, ψt(x) = x, that is,ψt−1(x) =x. Thus, we haveη({x} ×I) ={x}.
When φ({x} ×I) is contained in g−1(W(y)) for some y ∈ A, since the isotopy defined byψ−1t ,t∈I, is aV[B]-isotopy, it follows that
η({x} ×I)⊂st(g−1(W(y)),V) = st(st(f′({y} ×I),V),V)
⊂st(f({y} ×I),stV)⊂st(f({y} ×I),U)∈ U∗. Therefore,η is aU∗-isotopy. ⊓⊔
The Open Embedding Theorem 2.5.10 can be generalized as follows:
Theorem 2.5.12 (Open Embedding Approximation) Let M and N be connected E-manifolds,Abe an E-deficient closed set in M andf :M →N be a map such thatf|Ais anE-deficient closed embedding intoN. Then, for each open coverU ofN, there exists an open embeddingg:M →N such that g|A=f|A andg isU-homotopic tof.
Proof. LetV ∈cov(N) such that stV ≺ U. RecallM ∈ FE (Theorem 2.1.10).
By the Strong Universality Theorem 2.3.15, we have an E-deficient closed embeddingh:M →N such that h|A=f|Aand hisV-homotopic tof. By Corollary 2.3.5, we have a homeomorphism φ : M ×[0,1) → M such that φ(x,0) =xfor each x∈A and φis h−1(V)-homotopic to prM. Since h(M) is collared inN by the Collaring Theorem 2.5.9, we have an open embedding k:M×[0,1)→N such thatk(x,0) =h(x) for eachx∈M. By Lemma 2.5.2, we can requirekto satisfy the condition that
{k({x} ×[0,1))x∈M}
≺ V,
which implies that kisV-homotopic toh◦prM. Since h◦prM isV-homotopic tohφandhisV-homotopic tof,kis stV-homotopic, soU-homotopic tof φ.
Thus, we have an open embeddingg=k◦φ−1:M →N which isU-homotopic tof. Moreover,
g(x) =kφ−1(x) =k(x,0) =h(x) =f(x) for eachx∈A, that is,g|A=f|A. ⊓⊔
100 2 Fundamental Results on Infinite-Dimensional Manifolds