TOPOLOGICAL CONCORDANCES

ERIK KJÆR PEDERSEN

In this paper we will prove that concordance implies isotopy for codimension ≥ 3, locally tame embeddings of locally triangulable spaces in topological manifolds, with the restriction that if the ambient manifold is 4-dimensional, we have to assume it simply-connected.

This theorem was first proved in the PL and Diff categories by J. F. P. Hudson [9]. Another method of proof has been developed by C. Rourke [17]. Our approach is to generalize the approach of C. Morlet [15] to the topological category. The theorem has also been proved by M. A. Armstrong [1] in case of a locally tame embedding of a complex in a PL-manifold, using PL-approximation [13] and Hudson’s theorem [9]. We have also generalized Morlet’s results on higher homotopy groups of concordance spaces, specifically the lemma of disjunction, to the topological case [3].

I would like to thank my thesis advisor professor R. Lashof for his help and encouragement in the preparation of this paper. Also I want to thank professor R. D. Edwards, who pointed out numerous improvements, using his stronger, relative version of the Miller-Connelly taming theorem [5]. In an earlier version of this paper we proved a slightly weaker theorem using only the non-relative PL-approximation theorem [4].

The results of this paper were announced in [16].

We use standard notation. D^{n} or B^{n} denotes the standard n-dimensional disc, S^{n} the n-sphere and R^{n} Euclidean space. By an embedding M ⊂ V of topological manifolds we mean a locally flat embedding, transversal to the boundary. If K is a locally triangulable space in the topological manifold V, the embedding is said to be locally tame or locally PL-able if for every point in K there is a neighborhood U in V such that K ∩ U and U have PL-structures such that K ∩ U ⊂ U is PL.

We prove the following theorem:

Theorem 8. Suppose V^{n} is a topological manifold with boundary (not necessarily compact) and (X, Y) is a compact locally triangulable pair of spaces, and suppose

ϕ : (X, Y) × I → (V, ∂V) × I

is a locally tame concordance (that is, ϕ^{−1}(V × I) = X × i for j = 0, 1, and

ϕ^{−1}(∂V × I) = Y × I,

and the embedding ϕ is locally PL). Suppose f|Y × I is a product concordance and dim(X − Y) ≤ n−3 and if n = 4, that π_{1}(V) = 0. Then there is an ambient isotopy H_{t} : V ×I → V ×I, t ∈ I, fixed on V × 0 ∪ ∂V × I and having compact support, such that

H_{1}ϕ = g × 1_{I}

where g denotes ϕ|X × 0.

For the moment we will restrict ourselves to the case n ≥ 5, and then we will discuss low dimensional difficulties later.

Proposition 1. Suppose V^{n} is an n-dimensional topological manifold, n ≥ 5, and let

ϕ : (D^{p}, ∂D^{p}) × (I, 0, 1) → (V, ∂V) × (I, 0, 1)

be a concordance such that ϕ|∂D^{p} × I = ϕ|∂D^{p} × 0 × 1_{I}. Denote ϕ|D^{p} × 0 by g and assume g(D^{p}) ⊆ V × 0 has a PL neighborhood U. Then there exists an ambient isotopy

F_{t} : V × I → V × I 0 ≤ t ≤ 1

with F_{t}|∂V × I a product isotopy, and there exists a concordance H : V × I → V × I, with H|∂V × I ∪ V × 0 = identity, such that the following images coincide:

H(g(D^{p}) × I) = g(D^{p}) × I

and

H(F_{1}ϕ(D^{p} × 1)) = F_{1}g(D^{p}) × I.

(Note: it is not asserted that F_{t}|V × 0 = identity, or that H is isotopic to the identity.)

The main work of this paper will be to prove Proposition 1. We shall need PL approximation theorems. We quote the Miller-Connelly-Edwards taming theorem [5]:

Theorem 2. Let M^{m} be a PL manifold, ϕ : K → M a locally tame embedding of a PL complex K of codimension ≥ 3 such that ϕ^{−1}(ϕ(K) ∩ ∂M) = L is a subcomplex of K and ϕ|L is PL. Then there is an ambient ε-isotopy H_{t} of M fixing ∂M such that H_{1} ∘ ϕ is a PL embedding.

To use Theorem 2 we need to have local PL structures.

Lemma 3. Let ϕ : D^{p} × I → V^{n} × I be a concordance as in Proposition 1. Then there is an extension

Φ : D^{p} × R^{n−p} × I → V^{n} × I,

where Φ is an embedding which is a product over the boundary.

Proof. We need to deal specially with the case p = 0. For p = 0 choose an embedded path in V × 0, α, from ϕ(D^{0} × 0) to a point q such that ϕ(D^{0} × I) ∩ q × I = ∅ and another embedded path β in V × I from q × 1 to ϕ(D^{0} × 1) and such that the loop α ∪ q × I ∪ β ∪ ϕ is trivial in V × I. Doing general position patch by patch, this loop may be assumed to bound an embedded, locally flat disc in V × I, and since q × I does have a PL neighborhood, and we push ϕ(D^{0} × I) into this neighborhood by an ambient isotopy, we are done with this case. ϕ(D^{p} × 0) is by assumption contained in a PL neighborhood U and by Theorem 2 we may change the PL structure of U such that ϕ : D^{p} × 0 → U is a PL-embedding. A regular neighborhood of ϕ(D^{p} × 0), N,is a regular neighborhood of a point hence a disc, and by Zeeman’s unknotting theorem [21] we then get that ϕ(D^{p} × 0) is a standard subdisc hence an extension ϕ : D^{p} × 0 × R^{n−p} → V. If we extend ϕ : D^{p} × I → V × I to a collar of V × I by the product embedding we get

\overline{ϕ} : D^{p} × [0, 2) → V × [0, 2)

and it is now easy to find an ambient isotopy F_{t} of V × [0, 2) fixing V × 0 ∪ ∂V × [0, 2) such that F_{1}(ϕ(D^{p} × [0, 1])) ⊆ Φ(D^{p} × 0 × R^{n−p}) × [0, 2) so

F_{1}^{−1}(Φ(D^{p} × 0 × R^{n−p}) × [0, 2)) = W

is a PL neighborhood of ϕ(D^{p} × [0, 1]). Let S be a neighborhood of ϕ(D^{p} × 1) in V and ε ∈ R, both chosen so small that S×]1 − ε, 1 + ε[⊂ W. S×]1 − ε, 1 + ε[ is now an open subset of a PL manifold, hence inherits a PL structure [20] which is the product structure over the boundary, hence by the Kirby-Siebenmann product structure theorem [10] we may change the PL structure of W fixing it on the boundary to make S × {1} a PL submanifold. We therefore get a PL neighborhood of ϕ(D^{p} × [0, 1]) in V × I extending the neighborhood of ϕ(D^{p} × 0 ∪ ∂D^{p} × [0, 1]) in V × 0 ∪ ∂V × I we already had. By Theorem 2 we may change the PL structure, fixing it on the intersection with V × 0 ∪ ∂V × I to obtain that ϕ(D^{p} × I) is a subcomplex.

The proof is completed by taking a regular neighborhood of ϕ(D^{p} ×I) extending the given regular neighborhood of ϕ(D^{p} × 0 ∪ ∂D^{p} × I) and using Zeeman’s unknotting theorem twice [21]

□

We will now start constructing the isotopy of Proposition 1.

Lemma 4. Assumptions as in Proposition 1. Then there is an extension

Φ : D^{p} × R^{n−p} → V

of g : D^{p} → V, and an ambient isotope G_{1} of V × I with compact support such that a) G_{t}|∂V × I is a product isotopy.

b) G_{1}(ϕ(∂D^{p} × I)) ∩ g(D^{p}) × I = ∅.

c) G_{1} ∘ ϕ|(G ∘ ϕ)^{−1}(Φ(D^{p} × R^{n−p}) × I) is PL and intersects g(D^{p}) × I simplicially transversally in the sense of [19].

Proof. By Lemma 3 we may find an extension Φ : D^{p} × R^{n−p} → V of g and it is easy to find an isotopy that moves ϕ(∂D^{p} × I) off g(∂D^{p} × I) satisfying a). In the following we will then fix ∂V × I so that we only need c). By Theorem 2 we may tame a subcomplex of ϕ^{−1}(Φ(D^{p} × R^{n−p}) × I) by an ε-isotopy with compact support fixing Φ(∂D^{p} × R^{n−p}) and by shrinking the fibres of Φ we may then assume that ϕ|ϕ^{−1}(Φ(D^{p} × R^{n−p}) × I) is PL. Now since b) is satisfied we may find an ambient isotopy of Φ(D^{p} × R^{n−p}) × I with compact support, fixing Φ(∂D^{p} × R^{n−p}) × I that moves ϕ(D^{p} × I) to intersect g(D^{p}) × I simplicially transversally, and hence so that the intersection is a PL manifold, using [19]. □

The next step is to find isotopies so that we eventually move ϕ so that the intersection X = ϕ(D^{p} × I) ∩ g(D^{p}) × I becomes a product. For a locally tame submanifold A of V × I we use the notation: A^{i} = A ∩ V × i, dA^{i} = A ∩ ∂V × i (i = 0, 1) and dA = A ∩ ∂V × I. We say

τ(X) ≥ α

if X is obtained from X^{0} × I by adding handles of index j, with α ≤ j ≤ dim X − α, and the handles not touching d(X^{0}) × I. Notice that τ(X) > \frac{1}{2} dim X implies that X is a product. By abuse of notation we will denote the result of Lemma 4, G_{1} ∘ ϕ by ϕ. (Whereas g always refers to the embedding given in Prop. 1.)

Lemma 5. With notation as above let B = ϕ(D^{p} × I) ∪ g(D^{p}) × I. Then if τ(X) ≥ α − 1 we have

1) π_{i}(V × I − B, V × 0 − B^{0}) = 0 for i ≤ α + 2.

2) π_{i}(ϕ(D^{p} × I) − X, ϕ(D^{p} × 0) − X^{0}) = 0 for i ≤ α + 1.
3) π_{i}(g(D^{p}) × I − X, g(D^{p}) − X^{0}) = 0 for i ≤ α + 1.

Proof. We remind the reader that p ≤ n − 3. First we prove 1. By general position (codimension 3) we have

π_{1}(V × I − B) ≅ π_{1}(V × I)

and

π_{1}(V × 0 − B^{0}) ≅ π_{1}(V).

Let Ṽ be the universal cover of V, and let B̃, B̃_{0} etc. be the preimages of B, B_{0}, etc. in Ṽ × I. Ṽ − B̃ and Ṽ × 0 − B̃^{0} are then simply connected by the above remarks, so it suffices to show that H_{k}(Ṽ × I − B̃, Ṽ × 0 − B̃^{0}) = 0 for k ≤ α + 3. By Poincaré-Lefschetz duality we have

H_{k}(Ṽ × I − B̃, Ṽ × 0 − B̃^{0}) ≅ H_{c}^{n+1−k}(Ṽ × I − B̃, Ṽ × 1 − B̃^{1})

where the c denotes cohomology with compact supports. We now have

H_{c}^{n+1−k}(Ṽ × I − B̃, Ṽ × 1 − B̃^{1}) ≅ lim{H^{n+1−k}(Ṽ × I − Ñ, Ṽ × 1 − Ñ^{1})}

where N is a closed neighborhood of B in V × I. Applying excision to each term, it follows that this is isomorphic to

lim{H_{c}^{n+1−k}(Ṽ × I, Ṽ × 1 ∪ Ñ)}

which is simply H_{c}^{n+1−k}(Ṽ × I, Ṽ × 1 ∪ B̃).

The exact sequence for the triple (Ṽ × I, Ṽ × 1 ∪ B̃, Ṽ × 1) shows that

H_{c}^{n+1−k}(Ṽ × I, Ṽ × 1 ∪ B̃) ≅ H_{c}^{n−k}(Ṽ × 1 ∪ B̃, Ṽ × 1)

which in turn by excision is isomorphic to

H_{c}^{n−k}(B̃, B̃^{1}).

The Mayer-Vietoris sequence applied to the pairs (ϕ_{0}(D^{p} × I, ϕ_{0}(D^{p} × 1)) and (ϕ_{1}(D^{p} × I, ϕ_{1}(D^{p} × 1)) shows that

H_{c}^{n−k}(B̃, B̃^{1}) ≅ H_{c}^{n−k−1}(X̃, X̃^{1}).

But since X is obtained from X^{1} by adjoining handles of dimension at most dim X −α+1 we have H_{c}^{n−k−1}(X̃, X̃^{1}) is 0 for n − k − 1 > dim X − α + 1 i. e. for k < (n − dim X) + α − 2, and then since n − dim X ≥ 5 we have that H_{c}^{n−k−1}(X̃, X̃^{1}) = 0 for k ≤ α + 2. This completes the proof of 1). The proof of 2) and 3) is completely analogous replacing V by D^{p} and B by ϕ^{−1}(X) respectively (g × 1)^{−1}(X). □

We are now ready for the main geometric construction. In the following lemma the picture is the following: We assume that ϕ has been moved by the isotopy given in Lemma 4, and the result we again denote be ϕ. We consider g(D^{p}) × I ⊂ V × I (g is the original g) with its extension Φ(D^{p} × R^{n−p}) × I defining a PL structure in a neighborhood of g(D^{p}) × I.

Lemma 6. Picture as above. Assume that τ(X) ≥ α − 1 (α ≥ 1). Then there is an ambient isotopy of H_{t} of V × I fixing ∂V × I such that H_{1} ∘ ϕ|(H_{1} ∘ ϕ)^{−1}(Φ(D^{p} × R^{n−p}) × I) is PL and intersects g(D^{p} × I) simplicially transversally. Moreover

τ(resulting X) ≥ α.

(Here of course resulting X = H_{1} ∘ ϕ(D^{p} × I) ∩ g(D^{p} × I.)

Proof. As above set B = g(D^{p}) × I ∪ ϕ(D^{p} × I).

Let μ_{s} : (D^{α−1}, S^{α−2}) → C be the cores of the α−1 handles mod X^{0}(X^{1}) in a handlebody decomposition of X with no handles of dimension less than α − 1 mod X^{0}(X^{1}). Since everything is PL near g(D^{p}) × I and g(D^{p}) × I intersects ϕ transversally, we can extend the embedding

μ_{s} : (D^{α−1}, S^{α−2}) → (X, X^{0})

to

F_{s} : (D^{α−1}, S^{α−2}) × I → (ϕ(D^{p} × I), ϕ(D^{p} × 0))

and

G_{s} : (D^{α−1}, S^{α−2}) × I → (g(D^{p}) × I, g(D^{p}) × 0)

such that

F_{s}|D^{α−1} × 0 = G_{s}|D^{α−1} × 0 = μ_{s},
F_{s}(D^{α−1} × I) ∩ g(D^{p}) × I = μ_{s}(D^{α−1})

and

G_{s}(D^{α−1} × I) ∩ ϕ(D^{p} × I) = μ_{s}(D^{α−1}).

F_{s}|D^{α−1} × 1 then represent an element of

π_{α−1}(ϕ(D^{p} × I) − X, ϕ(D^{p} × 0) − X^{0})

and G_{s}|D^{α−1} × 1 represents an element of

π_{α−1}(g(D^{p}) × I − X, g(D^{p}) × 0 − X^{0}).

These groups, however, are both zero by Lemma 5. Therefore since without loss 2α < p + 1, G_{s} and F_{s} extend to PL embeddings

f_{s} : (D^{α}, S_{+}^{α−1}, S_{−}^{α−1}) → (ϕ(D^{p} × I), X, ϕ(D^{p} × 0)),
g_{s} : (D^{α}, S_{+}^{α−1}, S_{−}^{α−1}) → (g(D^{p}) × I, X, g(D^{p}) × 0))

where S_{+}^{α−1}, S_{−}^{α−1} are respectively the upper and lower hemisphere of ∂D^{α}, and D^{α−1} × I is identified with a collar of S_{+}^{α−1} in D^{α}. The embeddings f_{s}, and g_{s} are made so that

f_{s}(D^{α}) ∩ g(D^{p}) × I = μ_{s}(D^{α−1}),
g_{s}(D^{α}) ∩ ϕ(D^{p} × I) = μ_{s}(D^{α−1}).

We now glue f_{s} and g_{s} together along S_{+}^{α−1} to obtain

κ_{s} = (f_{s} ∪_{μ_{s}} g_{s}) : D^{α} → B.

We now want to extend κ_{s} to

H_{s} : (D^{α}, S^{α−1}) × I → (V × I, V × 0)

that satisfies

H_{s}|D^{α} × 0 = κ_{s},
H_{s}(D^{α} × I) ∩ B = κ_{s}(D^{α})

and B ∪ H_{s}(D^{α} × I) is locally tamely embedded in V × I. This is done by first extending g_{s} to an embedding

(D^{α}, S_{+}^{α−1}, S_{−}^{α−1}) × I → (V × I, ϕ(D^{p} × I), V × 0)

using the PL structure in a neighborhood of g(D^{p}) × I and the transverse intersection of g(D^{p})×I and ϕ(D^{p} ×I) and then use the PL neighborhood of ϕ(D^{p} ×I) (Lemma 3) and the PL approximation theorem (Theorem 2) to extend this extension to the desired extension H_{s}. H_{s}|D^{α} × 1 now represents an element in π_{α}(V × I − B, V × 0 − B^{0}). This group is 0 by Lemma 5, so we may get a map

h_{s} : (D^{α}, S_{+}^{α−1}, S_{−}^{α−1}) → (V × I, V × 0, B)

extending H_{s} (we identify D^{α} × I with a collar of S_{−}^{α}) and such that

h_{s}D^{α+1} ∩ B = κ_{s}(D^{α}).

Since, without loss, 2(α + 1) < n + 1 we can change h_{s} relative to H_{s} by an arbitrarily small homotopy to make it a locally tame embedding. This is an elementary chart by chart argument using PL general position in each chart and Miller’s theorem (Theorem 2) to go from chart to chart.

We are now ready to find the desired isotopy. Firstly we may find an ambient isotopy of V × I fixing ∂V × I and carrying g(D^{p}) × I ∪ h_{s}(D^{α+1}) into the given PL neighborhood of g(D^{p}) × I fixing a smaller neighborhood of g(D^{p}) × I. This can be done since h_{s}(D^{α+1}) is locally flat and hence may inductively be pushed into an arbitrarily small neighborhood of g(D^{p}) × I. Having changed the picture by this isotopy we use Theorem 2 to find an isotopy that fixes a neighborhood of g(D^{p}) × I and such that Φ(D^{p} × R^{n−p}) × I ∩ B ∪ h_{s}(D^{α+1}) is PL embedded in Φ(D^{p} × R^{n−p}) × I.

Now we are entirely reduced to the PL situation and we now slide ϕ(D^{p} × I) across h_{s}(D^{α+1}) to remove a regular neighborhood of μ_{s}(D^{α+1}) in X as in the PL case, see [15] or [3]. Then τ(resulting X) ≥ α since we have removed a regular neighborhood of ⋃ μ_{s}(D^{α−1}) in X i. e. all α−1 handles mod. X^{0}(X^{1}), and at the end of the sliding we still have g(D^{p})×I simplicially transversal to ϕ(D^{p} × I). □

Proof of Proposition 1. First we assume n ≥ 6. By Lemma 4 and Lemma 6 we inductively obtain an isotopy F_{t} of V × I so that F_{t}|∂V × I is a product isotopy and F_{1}(ϕ(D^{p} × I)) ∩ g(D^{p}) × I = resulting X is a product, X^{0} × I, and what is left is to find the concordance H : V × I → V × I. In the following we will use the notationB_{0} = D(D^{p}) × I, B_{1} = F_{1}(ϕ(D^{p} × I)), B = B_{0} ∪ B_{1}, X = B_{0} ∩ B_{1},

A_{0} = Φ(D^{p} × B^{n−p} × 0) × I,
A_{1} = F_{1}Φ(D^{p} × B^{n−p} × I)

where Φ is the extension of ϕ given by Lemma 3. Since F_{1} ∘ ϕ|F_{1} ∘ ϕ)^{−1}(int A_{0}) is PL we can find a regular PL neighborhood N of X in A_{0} such that

a) Z_{i} = N ∩ B_{i} is a regular neighborhood of X in B_{i}, i = 0, 1.

b) N ⊂ int(A_{0} ∩ A_{1}).

c) N ∩ ∂V × I = (N^{0} ∩ ∂V × 0) × I ⊂ ∂V × I.

d) N is PL homeomorphic to N^{0} × I via a homeomorphism h that restricts to a homeomorphism (h|Z_{i}) : Z_{i} → Z_{i}^{0} × I and is the identity on N ∩ ∂V × I = (N^{0} ∩ ∂V × 0) × I.

To see this we triangulate a neighborhood U of X in A_{0} such that B ∩ U and X and boundaries are triangulated as full subcomplexes. We then take N to be a derived neighborhood of X, so a) is fulfilled, and so is b) if we take N small enough. To obtain c) we just need to arrange that the triangulation on A_{0} ∩ ∂V × I = (A_{0}^{0} ∩ ∂V × 0) × I is the product triangulation. To find h : N → N^{0} × I we note that X is PL homeomorphic to X^{0} × I by a homeomorphism which is the identity on dX = X ∩ ∂V × I. We denote this homeomorphism by h. h can by Hudson’s theorem [9] and uniqueness of regular neighborhoods, be extended to a product structure on Z_{i} agreeing with the identity on Z_{i} ∩ ∂V × I. Repeating this argument h can be extended to a product structure on N agreeing with the identity on (N ∩ ∂V × 0) × I, so d) is proved.

We next consider

Y_{i} = \overline{B_{i} − Z_{i}}, i = 0, 1.

Y_{i} is contained in C_{i} = \overline{A_{i} − N}. We claim:

e) Y_{i} is PL homeomorphic to Y_{i}^{0} × I via a homeomorphism which agrees with h on Y_{i} ∩ Z_{i} and with the identity on Y_{i} ∩ ∂V × I.

f) C_{0} is PL homeomorphic to C_{0}^{0} × I via a homeomorphism agreeing with h on C_{0} ∩ N and with the identity on

C_{0} ∩ ∂V × I = (C_{0}^{0} ∩ ∂V × 0) × I.

g) C_{1} is homeomorphic to C_{1}^{0} × I via a homeomorphism k that agrees with h on C_{1} ∩ N and with the identity on C_{1} ∩ ∂V × I = (C_{1}^{0} ∩ ∂V × 0) × I.

h) C_{1}^{0} can be given a PL structure such that there is an ambient isotopy l^{t} of C_{1}^{0} × I fixing ∂C_{1}^{0} × I ∪ C_{1}^{0} × 0 and such that

Y_{1} → C_{1} ⟶^{k} C_{1}^{0} × I ⟶^{μ} C_{1}^{0} × I

is PL.

e) follows from the same argument that proved that Z_{i} is homeomorphic to Z_{i}^{0} × I. Since that homeomorphism was realized by a PL ambient isotopy, the complement, Y_{i}, will also be a product.

f) follows from the PL s-cobordism theorem. N is PL embedded in A_{0} so C_{0} is a PL manifold. (C_{0}, C_{0}^{0}, C_{0}^{1}) is an s-cobordism since π_{1}(C_{0}^{1}) ≅ π_{1}(C_{0}) ≅ π_{1}(A_{0}), by general position, and the Mayer-Vietoris sequence applied to the universal covering space shows that H_{∗}(C̃_{0}^{i}) ≅ H_{∗}(C̃_{0}), i = 0, 1. The torsion is zero, since torsion is additive over unions. The s-cobordism is already given a product structure over ∂C_{0}^{0}, by gluing the product structures on the boundary of A_{0} and N together. We now extend this product structure to C_{0} using the PL s-cobordism theorem. To prove g) we proceed as follows: C_{1}^{0} = \overline{A_{a}^{0} − N^{0}} and N^{0} is a codimension 0 submanifold of A_{1}^{0}, meeting the boundary transversally. Therefore if we let Q = \overline{∂N^{0} − ∂V × 0} Q is a codimension 1 submanifold of C_{1}^{0} as well as of N^{0}. Taking a collar neighborhood in each we obtain an open neighborhood Q × R ⊂ A_{1}^{0} of Q in A_{1}^{0}. The induced PL structure on Q × R can be moved to a product structure near Q × 0 i. e. one can change the PL structure of A_{1}^{0} so that Q inherits a structure as PL submanifold, see [11] or [12]. (Note this is the only point in this paper where the handle straightening does not have a starting point on the boundary, so we have to straighten 0-handles, thus we use the Annulus conjecture and hence surgery theory.) This then give a PL structure to C_{1}^{0} since ∂C_{1}^{0}∩(∂V ×I ∪∂N) is a product we may extend the PL structure to C_{1}^{0}∪∂C_{1}^{0}∩(∂V ×I ∪∂N) and then further to C_{1} using triangulation theory since C_{1}^{0} is a deformation retract of C_{1}, and we have ensured that the structure on ∂C_{1}^{0} ∩ (∂V × I ∪ ∂N) is the product structure. Y_{1}^{0} is locally tamely embedded in C_{1}^{0} so by Theorem 2, we can move the PL structure of C_{1}^{0} to make Y_{1}^{0} a PL-manifold on the boundary. Y_{1} is PL-embedded since it is a product embedding, so we can use Theorem 2 to find the desired isotopy l^{t} fixing C_{1}^{0} ∪ ∂C_{1}^{0} × I. We may now proceed as in g) to get h).

Using PL regular neighborhood theory and Lemma 3 we may find regular neighborhoods N_{i} of Y_{i} in C_{i} and homeomorphisms

h_{i} : N_{i} → N_{i}^{0} × I

extending the homeomorphisms of Y_{i} to Y_{i}^{0} × I given by e), and so that h_{i} agrees with the identity on N_{i} ∩ ∂V × I, and with h on N ∩ N_{i}. Further we choose N_{i} so small that N_{0} ∩ N_{1} = ∅. N_{0} ∪ N ∪ N_{1} is now a (topological) regular neighborhood of B in V × I, so we can argue as before that D = V × I − N_{0} ∪ N ∪ N_{1} is a topological s-cobordism with a given product structure on the boundary.

Therefore the homeomorphism

h_{0} ∪ h ∪ h_{1} : N_{0} ∪ N ∪ N_{1} → (N_{0}^{0} ∪ N^{0} ∪ N_{1}^{0}) × I

extends by [18] to a homeomorphism.

H : V × I → V × I

which is the identity on ∂V × I ∪ V × 0, and since it is an extension of h_{0} ∪ h ∪ h_{1} it satisfies H(B) = B^{0} × I.

For n = 5 we note that X is at most 0-dimensional, so if τ(X) > 0 X is empty. We can therefore find extensions Φ_{0} and Φ_{1} that are disjoint, so there are disjoint regular neighborhoods N_{0} and N_{1}, and we may proceed as above.

□

We may now prove:

Theorem 7. Suppose V^{n} is an n-dimensional topological manifold, n ≥ 5, and let

ϕ : (D^{p}, ∂D^{p}) × (I, 0, 1) → (V, ∂V) × (I, 0, 1)

be a concordance such that ϕ|∂D^{p} × I = (ϕ|∂D^{p} × 0) × 1_{I}. Denote ϕ|D^{p} × 0 by g and assume g(D^{p}) ⊆ V × 0 has a PL neighborhood U. Then there is a ambient isotopy with compact support

K_{t} : V × I → V × I 0 ≤ t ≤ 1

with K_{t}|∂V × I ∪ V × 0 the identity and

K_{1} ∘ ϕ = g × 1_{I}

Proof. Since we have the same assumptions as in Proposition 1, the proof of that provides us with an ambient isotopy F_{t} and a concordance H. We denote F_{t}|V × 0 by F_{t}^{0}, and as a first approximation we define

L_{t} = F_{2t} for 0 ≤ t ≤ \frac{1}{2},
L_{t} = H^{−1} ∘ (F_{2−2t}^{0} × 1_{I}) ∘ (F_{1}^{0} × 1_{I})^{−1} ∘ H ∘ F_{1} for \frac{1}{2} ≤ t ≤ 1

then L_{t} is another solution to Proposition 1, so we do have

H(L_{1}(ϕ(D^{p} × I)) = L_{1}(g(D^{p}))) × I

and

H(g(D^{p} × I)) = g(D^{p}) × I

but we now further have

L_{1}(g(D^{p})) × I = L_{1}^{0}(g(D^{p})) × I = g(D^{p}) × I

and hence

L_{1}(ϕ(D^{p} × I)) = g(D^{p}) × I

as images. However, L_{t}|V × 0 ∪ ∂V × I is not the identity. We note, however, that L_{t}|V × 0 ∪ ∂V × I is F_{2t}, for t ∈ [0, \frac{1}{2}] and F_{2t−1} for t ∈ [\frac{1}{2}, 1]. We first modify L_{t} to get T_{t} with T_{0} = L_{o}, T_{1} = L_{1} and T_{t}|∂V × I the identity as follows:

Let ∂V × [0, 2] ⊂ V be a collar of the boundary [2]. and denote

V (t) = V − ∂V × [0, t) ⊂ V

and

k^{t} : V → V (t)

the homeomorphism defined to be the identity for x ∉ ∂V × [0, 2) and

k^{t}(y, s) = (y, t + \frac{1}{2}(2 − t)s) for (y, s) ∈ ∂V × [0, 2).

We define T_{t} by

T_{t}(x) = \begin{cases}k^{1−|1−t|} × 1_{I} ∘ L_{t} ∘ (k^{1−|1−t|})^{−1} × 1_{I} for x ∈ V (1 − |1 − t|) \\ (L_{s}^{0}(y), r) for x = (y, s, r) ∈ ∂V × [0, 1 − |1 − t|] × I. \end{cases}

In a similar fashion we modify T_{t} to get K_{t} with K_{0} = L_{0}, K_{1} = L_{1} but now K_{t} also fixes V × 0. Specifically we let ρ_{s} : [0, 1] → [s, 1] be defined by ρ_{s}(t) = s + (1 − s)t and define

K_{t}(x) = \begin{cases}(1_{v} × ρ_{\frac{1}{2} −| \frac{1}{2} −t|} ∘ T_{t} ∘ 1_{V} × ρ_{\frac{1}{2} −| \frac{1}{2} −t|}^{−1})(x) for x ∈ V × [\frac{1}{2} − |\frac{1}{2} − t|, 1] \\ (T_{s}^{0}(y), s) for x = (y, s) ∈ V × (0, \frac{1}{2} − |\frac{1}{2} − t|). \end{cases}

On easily checks that K_{t}|∂V × I ∪ V × 0 is the identity (since T_{0}^{0} = 1_{V}) and we thus have found an isotopy with the required properties, except we only have K_{1}ϕ(D^{p}×I)) = g(D^{p})×I as sets, but this may now be straightened by an Alexander isotopy. We finally note that since F_{t} by construction has compact support so does K_{t}. □

We are now ready to state and prove our main theorem, Theorem 8. Suppose V^{n} is a paracompact topological manifold and (X, Y) is a compact locally triangulable pair of spaces, and suppose

ϕ : (X, Y) × (I, 0, 1) → (V, ∂V) × I

is locally tame concordance (that is ϕ^{−1}(∂V × I) = Y × I and ϕ is locally PL with varying structures both in X and V. Suppose f|Y ×I is a product concordance and dim(X−Y) ≤ n−3 and if n = 4, that π_{1}(V) = 0. Then there is an ambient isotopy H_{t} : V × I → V × I, t ∈ I, fixed on V × 0 ∪ ∂V × I and having compact support, such that if we denote g = ϕ|X × 0

H_{1}ϕ = g × 1_{I}.

We first consider the case where dim(V) ≥ 5.

The proof will be in a number of steps. First we straighten ϕ near Y. Step 1. There is an ambient isotopy H_{t} fixing V × 0 ∪ ∂V × I such that H_{1} ∘ ϕ is a product embedding in a neighborhood of Y.

Proof. Write Y = C_{1} ∪ ⋯ ∪ C_{n} where each C_{i} is a closed subset of Y having an open neighborhood of U_{i} in V (we consider g an inclusion) such that both U_{i} and U_{i} ∩ X have PL structures. By the taming theorem we may change the PL-structure of U_{i} such that (X, Y) ∩ U_{i} is a subpolyhedron of U_{i}. We inductively suppose that f has been straightened

over some neighborhood V_{j−1} of ⋃_{i=1}^{j−1} C_{i} in X. Let E be a compact PL neighborhood of C_{j} in

∂U_{j} ∩ Y (where ∂U_{j} = U_{j} ∩ ∂V), and let F be a compact PL neighborhood of (⋃_{i=1}^{j−1} C_{i}) ∩ E

in ∂V_{j} ∩Y, both chosen so small that F ⊂ V_{j−1}. Let N be a PL regular neighborhood (which meets the boundary regularly) of F in U_{j} such that N ∩ X ⊂ V_{j−1}. Let V_{0} = V − int_{V} N and X_{0} = X − int_{V} N and Y_{0} = X_{0} ∩ ∂V_{0} where int_{V} denotes the interior of N with respect to V.

In a neighborhood of E_{0} × I we now have PL structures on both (X_{0}, Y_{0}) and on a neighborhood of H_{0} ×I in V ×I so we may apply the taming theorem (Theorem 2) to change the situation to a PL situation and then proceed as in the PL-case, using the Haefliger trick (see [17, Lemma 7]) to straighten the concordance of X_{0} × I in V_{0} × I near E_{0} × I keeping fixed V_{0} × 0 ∪ ∂V_{0} × I. We extend this straightening isotopy over N × I via the identity and note that by making support of the straightening isotopy small enough we have not

disturbed a neighborhood of ⋃_{i=1}^{j−1} C_{i}, thus by induction, since the induction step also proves

step 1 by putting C_{0} = ∅ we have finished step 1 of the proof of Theorem 8.

Step 2. Assumptions as in Theorem 8 and further that ϕ is a product concordance over a neighborhood of some closed subset C of X with C ⊃ Y. Suppose D is a simplex in some triangulated open subset of X such that D ∩ C = ∂D. Then there is an ambient isotopy F_{t}, fixed on V × 0 ∪ ∂V × I, having compact support, such that F_{1}f is a product over a neighborhood of C ∪ D in X.

Proof. We can find a PL neighborhood U of int D in int V since D is locally flatly embedded in V. By taming X ∩U and possibly shrinking U we may assume that X ∩U is a subpolyhedron of U. Let V be a neighborhood of C in X over which ϕ is straight. Let D_{0} ⊂ int D be a slightly smaller PL copy of D such that D − V ⊂ int D and let N be a PL regular neighborhood of ∂D_{0} in U, such that N intersects X and D_{0} in regular neighborhoods and N ∩ X ⊂ V.

Define B = D_{0} − int N which is a ball, and V_{0} = V − (C ∪ int N). Since B does have a PL neighborhood in V_{0}, we may straighten ϕ|B × I in V_{o}, by an ambient isotopy with compact support and then apply the taming theorem and the Haefliger trick [17, Lemma 7] to straighten ϕ over a neighborhood of B in X − int N. We may now extend this isotopy to all of V by the identity, since it fixes the boundary and has compact support, so we have now straightened ϕ over a neighborhood of C ∪ D in X.

Step 3. Assume that ϕ is straight over a neighborhood of C, C_{0} some closed subset of X containing Y. This we may assume by step 1. Write \overline{X − C_{0}} as a union of closed subsets C_{1}, …, C_{n} such that each C_{i} has a neighborhood U_{i} in the interior of V with a PL structure in which U_{i} ∩X is a subpolyhedron (we consider g an inclusion and use the taming theorem). By induction we may assume that f has been straightened over a neighborhood of V_{j−1} of

⋃_{i=0}^{j−1} C_{i} in X. Let E be a compact PL neighborhood of C_{j} in U_{j} ∩ X and let F be a compact

PL neighborhood of (⋃_{i=0}^{j−1} C_{i}) ∩ E in U_{j} ∩ X, both chosen so small that F ⊂ V_{j−1}. Define C = (⋃_{i=0}^{j−1} C_{i}) ∪ F and G = E − F, so that C ∪ G = ⋃_{i=0}^{j} C_{i}. Now C ∩ G is a subpolyhedron of G and ϕ is straight over a neighborhood of C so step 2 can be applied repeatedly to the simplices {D} of G−C, in order of increasing dimension, for some triangulation of (G, G∩C) to ultimately straighten ϕ on a neighborhood of C ∪ G = ⋃_{i=0}^{j} C_{i}. This completes the proof for n ≥ 5. We now consider the remaining cases. For n < 3 there is nothing to prove.

For n = 3 we may use the theorem of Homma-Gluck that homotopy (and hence concordance)implies isotopy for 1-complexes in 4-manifolds see [7] and [6]. For n = 4 and V 1-connected we use the following trick due to Morlet [14]

First consider

ϕ : D^{0} × I → V^{4} × I

as in the proof of Lemma 4 we extend ϕ to an embedding

G : D^{1} × I → V^{4} × I

so that G|∂D^{1} × I is the disjoint union of ϕ and a trivial concordance. It is easy to find a PL neighborhood of G(D^{1} × I) (as in Lemma 4) and having that it is easy to isotope ϕ to a product embedding, not fixing V × 0, but that isotopy may be changed to fix V × 0 as in proof of Proposition 1. Finally consider

ϕ : D^{1} × I → V^{4} × I

ϕ|D^{1} × 1 is homotopic hence by [6] isotopic to ϕ|D^{1} × 0. So we may change ϕ by an isotopy to make ϕ|D^{1} × 1 = ϕ|D^{1} × 0. We may further, since the isotopy given by [6] is in the homotopy class of the given homotopy relative endpoints, assume that ϕ is homotopic to g × 1_{I} relative to ϕ|∂(D^{1} × I) = g × 1_{I}|∂(D^{1} × I). This homotopy may be realized as an embedding

Φ : D^{1} × I × I → V^{4} × I × I

since we may first general position Φ patch by patch and finally delete the 0-dimensional double points using the standard Whitney trick and the fact that V is 1-connected. We thus have ϕ concordant to g × 1_{I} and hence isotopic relative to V × 0 ∪ ∂V × I, to g × 1_{I} by the 5 dimensional case.

Remark. One could get rid of the 1-connected assumption if one could prove that it is possible to isotop ϕ so that the following map is an embedding

ϕ : D^{1} × I → V × I → V.

If this composition is in general position this may be done using sunny collapsing a la the PL proof due to Hudson. The difficulty seems to be to isotop ϕ so that the composition is in general position since at least the naive approach requires fibrewise PL approximation which is only known in codimension ≥ 4. Of course it may be interesting to notice that what goes wrong in the present line of proof is only the final application of the s-cobordism theorem since it is easy to isotop ϕ : D^{p} × V × I off (ϕ|D^{p} × 0) × 1_{I}, by a patch by patch argument.

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Matematisk Institut, Odense Universitet, Niels Bohrs Allé, DK-5000 Odense, Denmark