Embeddings of manifolds with boundary: classification

Theorem 2.1. Assume that $N$ is a closed compact $n$-manifold. Then $N$ embeds into $\mathbb R^{2n}$.

Theorem 2.2. Assume that $N$ is a compact $n$-manifold with nonempty boundary. Then $N$ embeds into $\mathbb R^{2n-1}$.

Theorem 2.3. Assume that $N$ is a compact $n$-manifold and either

(a) $m \ge 2n+2$ or

(b) $N$ is connected and $m \ge 2n+1 \ge 5$.

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

Theorem 2.4. Assume that $N$ is a compact connected $n$-manifold with non-empty boundary and either

(a) $m \ge 2n$ or

(b) $N$ is $1$-connected, $m \ge 2n - 1\ge3$.

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

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 $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$

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 $\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

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 $f(N_0)$.

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

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.

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

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.

Proof of Lemma 4.4. The first bullet point follows because:

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.

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:

Proof of Lemma 4.6. Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get

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:

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.

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, $\S$5].

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.

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

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

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

is one-to-one.

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

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.

Theorem 6.1. Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n\geq2k+2$. Then $N$ embeds into $\mathbb R^{2n-k}$.

Theorem 6.2. Assume that $N$ is a compact $n$-manifold with nonempty boundary, $(N, \partial N)$ is $k$-connected and $n\ge2k+2$. Then $N$ embeds into $\mathbb R^{2n-k-1}$.

Theorem 6.3. Assume that $N$ is a closed $k$-connected $n$-manifold. Then for each $n\ge2k + 2$, $m \ge 2n - k + 1$ any two embeddings of $N$ into $\mathbb R^m$ are isotopic.

Theorem 6.4. Assume that $N$ is a $k$-connected $n$-manifold with non-empty boundary. Then for each $n\ge k+3$ and $m\ge2n-k$ any two embeddings of $N$ into $\mathbb R^m$ are isotopic.

Theorem 6.5. Assume $N$ is a closed orientable $k$-connected manifold embeddable into $\mathbb R^{2n-k-1}$. Then for each $k\ge1$ there exists a bijection

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

Theorem 2.1. Assume that $N$ is a closed compact $n$-manifold. Then $N$ embeds into $\mathbb R^{2n}$.

Theorem 2.2. Assume that $N$ is a compact $n$-manifold with nonempty boundary. Then $N$ embeds into $\mathbb R^{2n-1}$.

Theorem 2.3. Assume that $N$ is a compact $n$-manifold and either

(a) $m \ge 2n+2$ or

(b) $N$ is connected and $m \ge 2n+1 \ge 5$.

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

Theorem 2.4. Assume that $N$ is a compact connected $n$-manifold with non-empty boundary and either

(a) $m \ge 2n$ or

(b) $N$ is $1$-connected, $m \ge 2n - 1\ge3$.

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

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 $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$

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 $\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

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 $f(N_0)$.

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

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.

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

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.

Proof of Lemma 4.4. The first bullet point follows because:

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.

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:

Proof of Lemma 4.6. Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get

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:

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.

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, $\S$5].

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.

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

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

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

is one-to-one.

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

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.

Theorem 6.1. Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n\geq2k+2$. Then $N$ embeds into $\mathbb R^{2n-k}$.

Theorem 6.2. Assume that $N$ is a compact $n$-manifold with nonempty boundary, $(N, \partial N)$ is $k$-connected and $n\ge2k+2$. Then $N$ embeds into $\mathbb R^{2n-k-1}$.

Theorem 6.3. Assume that $N$ is a closed $k$-connected $n$-manifold. Then for each $n\ge2k + 2$, $m \ge 2n - k + 1$ any two embeddings of $N$ into $\mathbb R^m$ are isotopic.

Theorem 6.4. Assume that $N$ is a $k$-connected $n$-manifold with non-empty boundary. Then for each $n\ge k+3$ and $m\ge2n-k$ any two embeddings of $N$ into $\mathbb R^m$ are isotopic.

Theorem 6.5. Assume $N$ is a closed orientable $k$-connected manifold embeddable into $\mathbb R^{2n-k-1}$. Then for each $k\ge1$ there exists a bijection

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

'''Proof of Lemma \ref{lmm:L_well_def}.''' The first bullet point follows because: $$ \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned} $$ Here the second equality follows from Lemma \ref{lmm::saeki}. 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. {{endproof}} Lemma \ref{lmm:L_well_def} 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 $. Define the dual to [[Stiefel-Whitney_characteristic_classes|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. {{beginthm|Lemma}} \label{lmm::L_equality} 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: $$\rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$ {{endthm}} This Lemma was stated in a unpublished update of \cite{Tonkonog2010}, the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in \cite[Lemma 2.2]{Crowley&Skopenkov2016}.