EMBEDDINGS OF TOPOLOGICAL MANIFOLDS

ERIK KJÆR PEDERSEN

It is the purpose of this paper to combine the methods of [8] and [9] to prove Haefliger type embedding theorems for topological manifolds. We prove the following embedding theorems:

Theorem 1. Let M be a topological manifold with boundary dim(M) = m, m ≥ 6, and let

α ∈ π_{r}(M, ∂M), r ≤ m − 3.

Assume that (M, ∂M) is 2r − m + 1 connected. Then α can be represented by a locally flat embedded disc

f : (D^{r}, S^{r−1}) → (M, ∂M).

The proof is modeled on the similar PL proof in [3] using the above mentioned modifications of the techniques.

Theorem 5. Let K be a finite k-dimensional complex, M^{n} a topological manifold m ≥ 6, m − k ≥ 3, and f a continuous map, f : K → M, so that

π_{i}(f) = 0 for i ≥ 2k − m + 1.

Then there is a complex K′ of dimension k′, k′ ≤ k, such that K′ is locally tamely embedded in M, and a simple homotopy equivalence h : K → K′ so that the diagram

\xymatrix{K \ar[rr]^{f} \ar[dr]_{h} &  & M \\  & K′ \ar@{^{(}->}[ur]}

is homotopy commutative.

Finally we obtain the “Haefliger type” theorem:

Theorem 7. Let f : M^{p} → V^{q} be a map of topological manifolds, q ≥ 6, q − p ≥ 3, M closed, and assume that

π_{i}(f) = 0 for i ≤ 2p − q − 1.

Then f is homotopic to a locally flat embedding.

Theorem 7 is proved as in the PL case using surgery and the normal theory due to Rourke and Sanderson classifying neighborhoods of topological manifolds [10] although we avoid the use of nonstable transversality. It generalizes the theorem of Lees [6]. Lees assumes the spaces are connected rather than the map.

For the proof of Theorem 1 we need a general position lemma.

Definition 2. Let P^{p} be a PL complex and M^{m} a topological manifold and f : (P, P′) → (M, ∂M) a continuous map, P′ a subcomplex of P. We say that f is in general position if K = im(f) has the structure of a complex such that f is PL, K ⊂ M is locally tamely embedded and the double point set S_{2}(f) ⊂ P is of dimension at most 2p − m.

Lemma 3. Let M be a topological manifold, dim(M) ≥ 6 and

f : (D^{r}, S^{r−1}) → (M, ∂M)

a continuous map, m − r ≥ 3. Then f is homotopic to a map in general position.

Proof. By the relative version of Lees’ immersion theorem (see [10]) the map

F : (D^{r}, S^{r−1}) × ℝ^{m−r} → (D^{r}, S^{r−1}) → (M, ∂M)

is homotopic to an immersion, since any bundle over D^{r} is trivial. Therefore we may assume that F is an immersion, so we can cover D^{r} by finitely many open sets U_{i} and shrink fibers so that F |U_{i} × ℝ^{m−r}, i = 1, 2, …, s is an embedding. Lert {Z_{i}} be a compact refinement of the covering {U_{i}} such that Z_{i} is a subcomplex of D^{r}. The proof will be by induction ob the following statement. There is an isotopy h_{i}^{t} of D^{r} × ℝ^{m−r} and subcomplexes Q_{i} of D^{r} such that ⋃_{j≤i} Z_{j} ⊂ int(Q_{i}) and F ∘ h_{i}^{1}|Q_{i} is in general position.

The induction starts trivially, so assume we have obtained the statement for i − 1. We may then as well assume ths true originally and denote F ∘ h_{i−1}^{1} by F and f ∘ h_{i−1}^{1} by f. Consider V = F (U_{i} × ℝ^{m−r}) ⊂ M. V has a PL structure induced by F. Let K_{i−1} = im(Q_{i}). Then by the induction hypothesis K_{i−1} is a PL complex locally tamely embedded in M, and of codimension at least 3. Therefore by [2] (see [9, Theorem 2] we can change the PL structure of V so that a compact regular neighborhood of K_{i−1} ∩ V is PL embedded. Hence if we shrink U_{i} and the fibers, we may assume K_{i−1} ∩ V ⊂ V is a PL embedding. We shrink U_{i} so little that U_{i} still contains Z_{i}. Since f : Q_{i−1} → K_{i−1} is PL the restriction of f to U_{i} ∩ Q_{i−1} is PL, so as before by shrinking U_{i} and Q_{i−1} we can find an isotopy of U_{i} × ℝ^{m−r} that fixes everything outside a compact set and a neighborhood of U_{i} ∩ Q_{i−1}, hence may be assumed to fix Q_{i−1} ∩ U_{i} × ℝ^{m−r}, and moves f|U_{i} to a PL embedding. The restriction of f to Q_{i−1} ∩ U_{i} is already in general position in the PL sense, so by PL general position we may move U_{i} as above so that we end up having f in general position in a closed neighborhood Q_{i} of ⋃_{j≤i} Z_{j}. This ends the induction step. □

We now only need the following trivial lemma before the proof of Theorem 1.

Lemma 4. Let M be a manifold and assume (M × I, M × 0, M × 1) is written as the union of two triads V and W:

(M × I, M × 0, M × 1) = (V ⋃_{∂_{+}V =∂_{−}W} W, ∂_{−}V, ∂_{+}W).

Then if π_{j}(V, ∂_{+}V) = 0 for j ≤ r, r ≥ 2, then π_{j}(W, ∂_{+}W) = 0 for j ≤ r − 1.

Proof. Van Kampen’s Theorem applied to W and V shows that π_{1}(W) = π_{1}(M × I) since r ≥ 2; hence π_{1}(W) ≅ π_{1}(M × 1) = π_{1}(∂_{+}W). Let M̃ denote the universal covering space of M and denote inverse images in M̃ × I by . Then

H_{j+1}(Ṽ, ˜∂_{+}V) = H_{j+1}(M̃ × I, W̃) = H_{j}(W̃, { \atop ∂_{+}W})

by excision and the long exact sequence for the triple { \atop ∂_{+}W} ⊂ W̃ ⊂ M̃ × I. □

Proof of Theorem 1. The proof is by induction on the following statement:

F : (D^{r}, S^{r−1}) → (M, ∂M)

is homotopic to f_{i}, and ∂M has a collar in M that is decomposed as

(∂M × [0, 1], ∂M × 0, ∂M × 1) = (V ⋃_{∂_{+}V =∂_{−}W} W, ∂_{−}V, ∂_{+}W)

where V is obtained from ∂_{−}V = ∂M × 0 by adjoining handles of dimension less than 2r − m + 1 − i, f_{i}^{−1}(V) is a collar of ∂D^{r} in D^{r} and

f_{i}|\overline{D^{r} − f_{i}^{−1}(V)} : \overline{D^{r} − f_{0}^{−1}(V)} → \overline{M − V}

is a locally flat embedding.

We start the induction by homotoping f to f_{0} a map in general position as in Lemma 3. Let S_{2}(f_{0}) be the double point set of f_{0}. Then since the codimension of S_{2}(f_{0}) in D^{r} is at least 3 we can find a complex Z_{0} in D^{r} of dimension dim(S_{2}(f_{0})) + 1, i. e. of dimension less than 2r − m + 2 so that S_{2}(f_{0}) ⊂ Z_{0} and ∂D^{r} ∪ Z_{0} simplicially collapses to ∂D^{r} (see [3]). Let R_{0} be the image of Z_{0}. Then R_{0} is a subcomplex of im(f_{0}) of dimension at most 2r − m + 1, and by Newman’s engulfing theorem [7] we may assume that R_{0} is contained in some collar of ∂M, R_{0} ⊂ ∂M × [0, 1]. Let N_{0} be a regular neighborhood of R_{0} in ∂M × [0, 1] that intersects im(f_{0}) in a regular neighborhood of R_{0} in im(f_{0}) (see [9]). Then f_{0}^{−1}(N_{0}) is a regular neigborhood of Z_{0} so it collapses to ∂D^{r}. This picture is clearly homotopic to the picture where we have glued on a collar on ∂D^{r} and ∂M, so do that and put

V_{0} = ∂M × [0, 1] ∪ N_{0}, W_{0} = \overline{∂M × [−1, 1] − V_{0}}.

This then starts the induction. Assume the hypothesis for i. Denote

f_{i}(\overline{D^{r} − f_{i}^{−1}(V_{i}))},

i. e. the image of D^{r} with a collar of the boundary deleted, by B^{r}. Then B^{r} is locally flatly embedded,

B^{r} ⊂ \overline{M − V_{i}},

and therefore extends to an embedding

B^{r} × ℝ^{n−r} ⊂ \overline{M − V_{i}}

by [9, Lemma 3]. By Lemma 4 and [8, Theorem 1], W_{i} has a strong deformation retract, ∂W_{i} ∪ K_{i}, where K_{i} is a locally tamely embedded complex of dimension at most max(2, 2r − m − i + 1). Using [2] we may change the PL structure on B^{r} × ℝ^{m−r} and shrink the fibers so that

K_{i} ∩ B^{r} × ℝ^{m−r} ⊂ B^{r} × ℝ^{m−r}

is a PL embedding. We then isotop B^{r} so that B^{r} is PL embedded in B^{r} × ℝ^{m−r} and finally such that K_{i} and B^{r} are in general position. This isotoping B^{r} can obviously be done as a homotopy of f_{i}. Let C_{i} = K_{i} ∩ B^{r}. Then, since B^{r} is of codimension at least 3, C_{i} is of dimnesion at most 2r − m − i − 2 so we can find a complex Z_{i} in B^{r} of dimension one higher so that Z_{i} ∪ ∂B^{r} simplicially collapses to ∂B^{r} and Z_{i} contains C_{i}. Consider

K_{i} ∪ Z_{i} ⊂ K_{i} ∪ B^{r} ⊂ \overline{M − V_{i}}.

By Newman’s engulfing theorem K_{i} ∪Z_{i} ∪V is contained in a collar ∂M ×I of the boundary, so we let N_{i} be a regular neighborhood of K_{i} ∪ Z_{i} that intersects K_{i} ∪ B^{r} in a regular neighborhood of K_{i} ∪ Z_{i} in K_{i} ∪ B^{r} (see [9]), hence intersects B^{r} in a regular neighborhood of Z_{i}. Define

V_{i+1} = V_{i} ∪ N_{i}, W_{i+1} = \overline{∂M × [0, 1] − V_{i+1}},

Then V_{i+1} is obtained from M by adjoining handles of dimension at most 2r − m − i − 1 since that is the dimension of Z_{i}. f_{i+1}^{−1}(V_{i+1}) is a collar neighborhood of D^{r} since f_{i+1}^{−1}(N_{i}) is a regular neighborhood of Z_{i} and Z_{i} simplicially collapses to ∂B^{r}.

Eventually we get the dimension of K_{i} to be 2, so when we put K_{i} in general position to B^{r} the intersection becomes empty and therefore when we have completed that step we have obtained that f is homotopic to a map f_{i} such that there is a collar of ∂M in M with f_{i}^{−1}(∂M ×I) a collar of ∂D^{r} in D^{r} and f_{i}|f_{i}^{−1}(M −∂M ×[0, 1]) is a locally flat embedding. It is now easy to homotop f so as to pinch off the collar where f is not yet an embedding. □

We now show how this can be used to embed complexes in topological manifolds up to homotopy type.

Theorem 5. Let K^{k} be a finite complex, M^{m} a topological manifold, m ≥ 6, m − k ≥ 3, and f a continuous map, f : K → M, such that π_{i}(f) = 0 for i ≤ 2k − m + 1. Then there is a complex K′ of dimension k′, k′ ≤ k, such that K′ is locally tamely embedded in M, and a simple homotopy equivalence h : K → K′ such that the diagram

\xymatrix{K \ar[rr]^{f} \ar[dr]_{h} &  & M \\  & K′ \ar@{^{(}->}[ur]}

is homotopy commutative.

Theorem 5 follows immediately from Theorem 1 in [8] and the following proposition.

Proposition 6. Let M^{m} be a topological manifold of dimension m ≥ 6, and let K^{k} be a PL complex, m − k ≥ 3, and f a continuous map f : K → M such that π_{i}(f) = 0 for i ≤ 2k − m + 1. Then there is a codimension 0 submanifold N of M and a simple homotopy equivalence h : K → N such that the diagram

\xymatrix{K \ar[rr]^{f} \ar[dr]_{h} &  & M \\  & N \ar@{^{(}->}[ur]}

os homotopy commutative.

Proof. Filter K by simpleces of nondecreasing dimension

∅ = K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ K_{s} = K

where K_{i+1} is obtained from K_{i} by adjoining a simplex. The proof will be by induction on the following statement: There is a codimension 0 submanifold N_{j} of M and a homotopy of f to f_{j} such that f_{j}(K_{j}) ⊂ N_{j} and f_{j}|K_{j} : K_{j} → N_{j} is a simple homotopy equivalence.

It is trivial to start the induction so assume the statement for j. K_{j+1} = K_{j} ⋃_{S^{r−1}} D^{r} for some r, and f_{j}|D^{r} determines an element in π_{r}(M, N_{j}). We want to prove that the map

π_{r}(\overline{M − N_{j}}, ∂N_{j}) → π_{r}(M, N_{j})

is onto. For r ≤ m − r − 1 this is trivial since N_{j} is obtained from ∂N_{j} by adjoining handles of dimension ≥ m − r, so

π_{s}(∂N_{j}) → π_{s}(N_{j})

as well as

π_{s}(\overline{M − N_{j}}) → π_{s}(M)

is an isomorphism for s ≤ m − r − 2 and onto for s = m − r − 1. For r = m − r ≥ 3 we have as above

π_{s}(N_{j}, ∂N_{j}) = 0 for s ≤ r − 1

and

π_{s}(\overline{M − N_{j}}, ∂N_{j}) = 0 for s ≤ r − 1

and since r − 1 ≥ 2 this implies by Blakers-Massey [1] that

π_{s}(\overline{M − N_{j}}, ∂N_{j}) → π_{s}(M, N_{j})

is an isomorphism for s ≤ 2r − 3 and onto for s = 2r − 2.

In case r > m − r we have, since r ≤ k, that 2k − m + 1 ≥ 2 so π_{1}(K) ≅ π_{1}(M); since r ≥ 3, π_{1}(K_{j}) ≅ π_{1}(K) and

π_{1}(N_{j}) ≅ π_{1}(K) ≅ π_{1}(N_{j}) ≅ π_{1}(\overline{M − N_{j}})

so all fundamental groups are the same. We denote universal covering spaces by and we have

H_{s}(M̃, Ñ_{j}) ≅ H_{s}((\overline{M − N_{j}}) , ˜∂N_{j})

by excision, but

H_{s}(M̃, Ñ_{j}) = 0 for s ≤ min(r − 1, 2k − m + 1)

so

π_{s}(\overline{M − N_{j}}, ∂N_{j}) = 0 for s ≤ min(r − 1, 2k − m + 1)

and as before π_{s}(N_{j}, ∂N_{j}) = 0 for s ≤ m − r − 1 so by Blakers-Massey [1]

π_{s}(\overline{M − N_{j}}, ∂N_{j}) → π_{s}(M, N_{j})

is onto for s ≤ min(r − 1, 2k − m + 1) + m − r − 1 = 2k − r. Hence we can choose a map

α : (D^{r}, S^{r−1}) → (\overline{M − N_{j}}, ∂N_{j})

representing f_{j}|D^{r}. Since π_{s}(\overline{M − N_{j}}, N_{j}) = 0 for s ≤ min(r − 1, 2k − m + 1), α can, by Proposition 6 be chosen to be a locally flat embedding which can be extended to an embedding

A : (D^{r}S^{r−1}) × ℝ^{m−r} → (\overline{M − N_{j}}, ∂N_{j})

(see [9, Lemma 3]). We now define N_{j+1} = N_{j} ∪ A(D^{r} × B^{m−r}) where B^{m−r} is the unit ball in ℝ^{m−r}. Sinve K_{j} ⊂ K_{j+1} ⊂ K are cofibrations we can homotop f_{j} to f_{j+1} such that we get a Mayor-Vietoris diagram

\xymatrix{ & K_{j+1} &  &  & N_{j+1} \\ K_{j} \ar@{^{(}->}[ur] &  & D^{r} \ar@{^{(}->}[ul] \ar[r]^{f_{j+1}} & N_{j} \ar@{^{(}->}[ur] &  & A(D^{r} × B^{m−r}) \ar@{^{(}->}[ul] \\  & S^{r−1} \ar@{^{(}->}[ul] \ar@{^{(}->}[ur] &  &  & A(S^{r−1} × B^{m−r}) \ar@{^{(}->}[ul] \ar@{^{(}->}[ur]}

where f_{j+1} is a simple homotopy equivalence on the 3 small terms hence on K_{j+1}. This ends the induction step □ Theorem 7. Let f : M^{p} → V^{q} be a map of topological manifolds, q ≤ 6, q − p ≥ 3, M closed, and assume that π_{i}(f) = for i ≤ 2p − q + 1. Then f is homotopic to a locally flat embedding.

To prove Theorem 7 we need some lemmas.

Lemma 8. With the assumptions of Theorem 7 there is a codimension 0 submanifolds N of V such that π_{1}(∂N) ≅ π_{1}(N) and a simple homotopy equivalence h : M → N such that the diagram

\xymatrix{M \ar[rr] \ar[dr] &  & V \\  & N \ar@{^{(}->}[ur]}

is homotopy commutative.

Proof. There is a p-dimensional complex K which is simple homotopy equivalent to M. K is obtained as follows. Let C be the total space of the normal discbundle of M. Then C is a PL manifold and determines by definition the simple homotopy type of M. We can then let K be a PL spine of D. The codimension 0 submanifold N of V is now constructed by Proposition 6. □

The proof of Theorem 7 is completed by the following lemma.

Lemma 9. Let f : M^{p} → V^{q} be a simple homotopy equivalence, M and V topological manifolds, M closed, q ≥ 6, q − p ≥ 3, and π_{1}(∂V) → π_{1}(V) an isomorphism. Then f is homotopic to a locally flat embedding.

Proof. It is proved in [10] that there is a classifying space B Top_{r} for topological neighborhoods of codimension r. The classification goes via microbundles: to a topological neighborhood P ⊂ Q one assigns the stable microbundle pair (τ_{Q}, i^{∗}τ_{Q}) which is then classified by B Top_{r}. There is a map B Top_{r} → BG_{r} assigning to the microbundle pair the corresponding spherical fibration pair. Since spherical fibration pairs split uniquely BG_{r} is identified with BG_{r} the classifying space for spherical fibrations of dimension r.

If M ⊂ W is a locally flat inclusion of topological manifolds of codimension ≥ 3 we can take a regular neighborhood N of M in W (see [4]). By [11] the map ∂N ⊂ N → M is equivalent to a spherical fibration ν, and ν_{M} = ν ⊕ i^{∗}ν_{W} as spherical fiber spaces. Therefore the map B Top_{r} → BG_{r} is described as follows: Take a closed regular neighborhood N, and turn ∂N → M into an r-spherical fibration ν. The classifying map for ν is then M → B Top_{r} BG_{r}.

We now return to our problem. Consider r : V → M, a homotopy inverse to f, and the restriction of r to ∂V → M. This map is by Spivak [11] equivalent to a spherical fibration ξ : E → M. Let \overline{E}(ξ) be the mapping cone of ξ; then (\overline{E}(ξ), E(ξ)) is naturally homotopy equivalenbt to (V, ∂V). Assume that the map ξ → BG_{r} lifts to B Top_{r}:

\xymatrix{ & B Top_{r} \ar[d] \\ M \ar[ur] \ar[r] & BG_{r}}

We then have M locally flatly embedded in a manifold and if we take a regular neighborhood W of M we get a fiber homotopy equivalence

[figure: (W, ∂W); ≃; (V, ∂V); r; M]

and if the homotopy equivalence is homotopic to a homeomorphism we have proved that f is homotopic to an embedding. By Sullivan theory this is determined by the normal obstruction in [W, G/ Top] = [M, G/ Top] but we have freedom in the choice of the lifting of ξ to B Top_{r} corresponding to G_{r}/ Top_{r}. But since r ≥ 3 this space is homotopy equivalent to G/ Top by [10]. Hence we can choose our lifting so as to make the normal obstruction 0. This then ends the proof once we show the existence of one lifting. However ξ ⊕ f^{∗}ν_{V} = ν_{M} as spherical fibrations, so stably ξ = ν_{M} ⊕ (f^{∗}ν_{V})^{−1}, hence

M ⟶^{ξ} BG_{r} ⟶ BG

lifts to B Top. and since G_{r}/ Top_{r} = G/ Top, ξ : M → BG_{r}, lifts to B Top_{r}. □

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Odense University, Odense, Denmark