Embeddings of manifolds with boundary: classification

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Contents

1 Introduction

In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In \S4 we introduce an invariant of embedding of a n-manifold in (n-1)-space for even n. In \S6 which is independent from \S3, \S4 and \S5 we state generalisations of theorems from \S2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that N is a compact connected n-manifold.

(a) Then N embeds into \mathbb R^{2n}.

(b) If N has non-empty boundary, then N embeds into \mathbb R^{2n-1}.

Part (a) is well-known strong Whitney embedding theorem.

Proof of part (b). By strong strong Whitney immersion theorem there exist an immersion g\colon N\to\mathbb R^{2n-1}. Since N is connected and has non-empty boundary, it follows that N collapses to an (n-1)-dimensional subcomplex X\subset N of some triangulation of N. By general position we may assume that g|_{X} is an embedding, because 2(n-1) < 2n-1. Since g is an immersion, it follows that X has a sufficiently small regular neighbourhood M\supset X such that g|_{M} is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism h\colon N\to M. The composition g\circ h is an embedding of N.
\square
This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 6.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

Theorem 2.2. Assume that N is a compact connected n-manifold and either

(a) m \ge 2n+1 \ge 5 or

(b) N has non-empty boundary and m\geq 2n.

Then any two embeddings of N into \mathbb R^m are isotopic.

The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, \S 2, Theorems 2.1, 2.2].

Inequality in part (b) is sharp, see Proposition 3.1.

Part (b) in case n>2 can be found in [Edwards1968, \S 4, Corollary 5]. Case n=1 is clear. Both parts of this theorem are special cases of the Theorem 6.2. Case n=2 can be proved using the following ideas.

These basic results can be generalized to the highly-connected manifolds (see \S6). All stated theorems of \S2 and \S6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Example on non-isotopic embeddings

Denote by \mathrm{lk} the linking coefficient of two disjoint cycles with integer coefficient.

The following example is folklore.

Proposition 3.1. Let N=S^k\times [0, 1] be the cylinder over S^k. (a) Then there exist non-isotopic embeddings of N into \mathbb R^{2k+1}.

(b) Then for each a\in\mathbb Z there exist an embedding f\colon N\to\mathbb R^{2k+1} such that \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a.

Proof. Let h\colon S^k\to S^k be a map of degree a. To prove part (a) it is sufficient to take the identity map of S^k as a map of degree one and the constant map as a map of degree zero. Define g\colon S^k\times [0, 1] \to D^{k+1}\times S^k by the formula g(x, t) = (x, h(x)t), where 1_k:=(1,0,\ldots,0)\in S^k.

Let f=\mathrm ig, where \mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1} is the standard embedding.Thus \mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a

\square

4 Seifert linking form

Let N be a closed orientable connected n-manifold. By N_0 we denote the complement in N to an open n-ball. Thus \partial N_0 is the (n-1)-sphere. If the (co)homology coefficients are omitted, then we assume them to be \mathbb Z.

Denote by \mathrm{lk} the linking coefficient [Seifert&Threlfall1980, \S 77] of two disjoint cycles.

Example 4.1. For N=S^k\times S^1 and each k\ge2 there exists a bijection l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z given by \mathrm{lk}(S^k\times1_1, S^k\times-1_1).

The surjectivity of l is given by Proposition 3.1(b).

The following folklore result holds.

Lemma 4.2. Assume N is a closed orientable connected n-manifold, n is even and H_1(N) is torsion free. Then for each embedding f\colon N_0 \to \mathbb R^{2n-1} there exists a nowhere vanishing normal vector field to f(N_0).

Proof. There is an obstruction (Euler class) \bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N) to existence of a nowhere vanishing normal vector field to f(N_0).

A normal space to f(N_0) at any point of f(N_0) has dimension n-1. As n is even thus n-1 is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore \bar e=-\bar e. Since H_1(N) is torsion free, it follows that \bar e=0.

Since N_0 has non-empty boundary, we have that N_0 is homotopy equivalent to an (n-1)-complex. The dimension of this complex equals the dimension of normal space to f(N_0) at any point of f(N_0). Since \bar e=0, it follows that there exists a nowhere vanishing normal vector field to f(N_0).

\square

Denote by x, y two disjoint (n-1)-cycles in N_0 with integer coefficients.

Definition 4.3. For even n and every embedding f\colon N_0\to\mathbb R^{2n-1} denote

\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),

where s is a nowhere vanishing normal field to f(N_0) and s(x), s(y) are the results of the shift of f(x), f(y) by s.

Lemma 4.4 (L is well-defined). For even n and every embedding f\colon N_0\to\mathbb R^{2n-1} the integer L(f)(x, y):

  • is well-defined, i.e. does not change when s is replaced by s',
  • does not change when x or y are changed to homologous cycles and,
  • does not change when f is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

We will need the following supporting lemma.

Lemma 4.5. Let f:N_0\to \mathbb R^{2n-1} be an embedding. Let s,s' be two nowhere vanishing normal vector fields to f(N_0). Then

\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y

where s(y) is the result of the shift of f(y) by s, and d(s,s')\in H_2(N_0) is (Poincare dual to) the first obstruction to s,s' being homotopic in the class of the nowhere vanishing vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for n=3, but the proof is valid in all dimensions.

Proof of Lemma 4.4. The first bullet point follows because:
\displaystyle  \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}

Here the second equality follows from Lemma 4.5.

For each two homologous (n-1)-cycles x, x' in N_0, the image of the homology between x and x' is a n-chain X of f(N_0) such that \partial X = f(x) - f(x'). Since s is a nowhere vanishing normal field to f(N_0), this implies that the supports of s(y) and X are disjoint. Hence \mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y)).

Since isotopy of f is a map from \mathbb R^{2n-1}\times [0, 1] to \mathbb R^{2n-1}\times [0, 1], it follows that this isotopy gives an isotopy of the link f(x)\sqcup s(y). Now the third bullet point follows because the linking coefficient is preserved under isotopy.

\square

Lemma 4.4 implies that L(f) generates a bilinear form H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z denoted by the same letter.

Denote by \rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2) the reduction modulo 2.

Define the dual to Stiefel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2) to be the class of the cycle on which two general position normal fields to f(N_0) are linearly dependent.

Lemma 4.6. Let f:N_0\to \mathbb R^{2n-1} be an embedding. Then for every X, Y \in H_{n-1}(N_0) the following equality holds:

\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 4.6. Let -s be the normal field to f(N_0) opposite to s. We get
\displaystyle  \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}

The first congruence is clear.

The second equality holds because if we shift the link s(X)\sqcup f(Y) by -s, we get the link f(X)\sqcup -s(Y) and the linking coefficient will not change after this shift.

The third equality follows from Lemma 4.5.

Thus it is sufficient to show that \rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0). Denote by s' a general perturbation of s. We get:

\displaystyle  \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).

The first equality holds because s' and s are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between s' and -s degenerates only at those points x where s'(x)=s(x). These points x are exactly points where s'(x) and s(x) are linearly dependent. All those point x form a 2-cycle modulo two in N_0. The homotopy class of this 2-cycle is \mathrm{PD}\bar w_{n-2}(N_0) by the definition of Stiefel-Whitney class.

\square

5 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let N be a closed orientable connected n-manifold. By N_0 we denote the complement in N to an open n-ball. Thus \partial N_0 is the (n-1)-sphere. Denote \mathrm{Emb}^mN the set of all embeddings f\colon N\to\mathbb R^m up to isotopy. For a free Abelian group A, let B_n^∗A be the group of bilinear forms \phi \colon A \times A \to \mathbb Z such that \phi(x, y) = (−1)^n \phi(y, x) and \phi(x, x) is even for each x (the second condition automatically holds for n odd).

Definition 5.1. For each even n define an invariant W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2). For each embedding f\colon N_0\to\mathbb R^{2n-1} construct any PL embedding g\colon N\to\mathbb R^{2n} by adding a cone over f(\partial N_0). Now let W\Lambda([f]) = W(g), where W is Whitney invariant, [Skopenkov2016e, \S5].

Lemma 5.2. The invariant W\Lambda is well-defined for n\ge4.

Proof. Note that Unknotting Spheres Theorem implies that \partial N_0 unknots in \mathbb R^{2n}. Thus f|_{\partial N_0} can be extended to embedding of an n-ball B^n into \mathbb R^{2n}. Unknotting Spheres Theorem implies that n-sphere unknots in \mathbb R^{2n}. Thus all extensions of f are isotopic in PL category. Note also that if f and g are isotopic then their extensions are isotopic as well. And Whitney invariant W is invariant for PL embeddings.

\square

Definition 5.3 of G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N) if n is even and H_1(N) is torsion-free. Take a collection \{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0 such that W\Lambda(f_z)=z. For each f such that W\Lambda(f)=z define

\displaystyle  G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)

where x,y\in H_{n-1}(N_0).

Note also that G depends on choice of collection \{f_z\}. The following Theorems hold for any choice of \{f_z\}.

Theorem 5.4. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n even. The map

\displaystyle  G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),

is one-to-one.

Lemma 5.5. For each even n\in H_{n-1}(N) and each x the following equality holds: W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right).

An equivalemt statement of Theorem 5.4:

Theorem 5.6. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n even. Then

(a) The map L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N) is an injection.

(b) The image of L consists of all symmetric bilinear forms \phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z such that \rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y). Here \bar w_2(N_0) is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

6 A generalization to highly-connected manifolds

In this paragraph we consider only punctured manifolds to simplify statements of the theorems, but the results seem to hold for manifolds with arbitrary non-empty boundary.

By N_0 denote the complement in N to an open n-ball. Thus \partial N_0 is the (n-1)-sphere.

Theorem 6.1. Assume that N is a closed compact k-connected n-manifold.

(a) If n\geq 2n+3, then N embeds into \mathbb R^{2n-k}.

(b) Then N_0 embeds into \mathbb R^{2n-k-1}.

The Diff case of part (a) is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].

The Diff case of part (b) is in [Hirsch1961a, Corollary 4.2]. For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].

Theorem 6.2. Assume that N is a closed k-connected n-manifold.

(a) If n\ge2k + 2, m \ge 2n - k + 1, then any two embeddings of N into \mathbb R^m are isotopic.

(b) If m \ge 2n - k, then any two embeddings of N_0 into \mathbb R^m are isotopic.

For part (a) see Theorem 2.4 of the survey [Skopenkov2016c, \S 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

For the PL case of part (b) see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

For k>1 part (b) is corollary of Theorem 6.3 below (but can be proved in a straightforward way). For k=0 part (b) coincides with Theorem 2.2b.

Proof of Theorem 6.2(b) for k=1. By Theorem 7.6 below every two immersions of N into \mathbb R^{2n-1} are regulary homotopic. Hence for every two embeddings f,g\colon N\to\mathbb R^{2n-1} there exist an immersion F\colon N\times[0,1]\to\mathbb R^{2n-1}\times[0,1] such that F(x, 0) = (f(x), 0) and F(x, 1)=(g(x), 1) for each x\in N. Since N is 1-connected and has non-empty boundary, it follows that N collapses to an (n-2)-dimensional subcomplex X\subset N of some triangulation of N. By general position we may assume that F|_{X\times[0,1]} is an embedding, because 2(n-1) < 2n. Since F is an immersion, it follows that X has a sufficiently small regular neighbourhood M\supset X such that F|_{M\times[0,1]} is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism h\colon N\to M. It is clear that f is isotopic to f\circ h and g is isotopic to g\circ h. Thus the restriction F|_{M\times[0,1]} is a concordance of f\circ h and g\circ h. By concordance implies isotopy Theorem f and g are isotopic.
\square

Denote by \mathrm{Emb}^{m}N_0 the set embeddings of N_0 into \mathbb R^{m} up to isotopy.

Theorem 6.3. Assume N is a closed k-connected n-manifold. Then for each k\ge1 there exists a bijection

\displaystyle  W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),

where \mathbb Z_{(s)} denote \mathbb Z for s even and \mathbb Z_2 for s odd.

For definition of W_0' and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(W_0')]. See also [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 5.6, which describes \mathrm{Emb}^{2n-1}(N_0) and differs from the general case.

7 Comments

Theorem 7.1.[Smale-Hirsch] The space of immersions of a manifold in \mathbb R^{m} is homotopically equivalent to the space of linear monomorphisms from TM to \mathrm R^{m}.

See [Hirsch1959] and [Haefliger&Poenaru1964].

Theorem 7.2. If N is immersible in \mathbb R^{m+r} with a transversal r-field then it is immersible in \mathbb R^{m}.

This is [Hirsch1959, Theorem 6.4].

Theorem 7.3. Every n-manifold N with non-empty boundary is immersible in \mathbb R^{2n-1}.

Theorem 7.4.[Whitney] Every n-manifold N is immersible in \mathbb R^{2n-1}.

See [Hirsch1961a, Theorem 6.6].

Theorem 7.5. Every k-connected n-manifold N with non-empty boudary is immersible in \mathbb R^{m} for each m\geq2n-k-1.

Proof. It suffices to show that exists an immersion of N in \mathbb R^{2n-k-1}. It suffices to show that exists a linear monomorphism from TM to \mathbb R^{2n-k-1}. Lets cunstruct such linear monomorphist on each r-skeleton of N. It is clear that linear monomorphism exists on 0-skeleton of N.

The obstruction to continue the linear monomorphism from (r-1)-skeleton to r-skeleton lies in H_{n-r}(N; \pi_{r-1}(V_{2n-k-1, n})), where V_{2n-k, n} is Stiefel manifold of n-frames in \mathbb R^{2n-k}.

For r=1,\ldots,n-k-1 we know \pi_{r-1}(V_{2n-k-1, n}) = 0.

For r=n-k,\ldots, n we have H_{n-r}(N; *) = 0 since N is k-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

Other variant. By theorem 7.2 it suffies to show that that there exists an immersion of N into \mathbb R^{2n} with k tranversal linearly independent fields. It is true because N is k-connected.

\square

Theorem 7.6. Suppose N is a k-connected n-manifold with non-empty boudary and m\geq2n-k. Then every two immersions of N in \mathbb R^m are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from TM to \mathbb R^{2n-k}. Lets cunstruct such homotopy on each r-skeleton of N. It is clear that homotopy exists on 0-skeleton of N.

The obstruction to continue the homotopy from (r-1)-skeleton to r-skeleton lies in H_{n-r}(N; \pi_r(V_{2n-k, n})), where V_{2n-k, n} is Stiefel manifold of n-frames in \mathbb R^{2n-k}.

For r=1,\ldots,n-k-1 we know \pi_r(V_{2n-k, n}) = 0.

For r=n-k,\ldots, n we have H_{n-r}(N; *) = 0 since N is k-connected and has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

\square


8 References

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