4-manifolds in 7-space

Remark 1.1 (PL and piecewise smooth embeddings). Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure [Milnor&Stasheff1974, Complement, Theorems 10.5 and 10.6]. Since also any PL 4-manifold admits a unique smooth structure [Mandelbaum1980, $\S$1.2], we may consider a smooth 4-manifold as a PL 4-manifold.

A map of a smooth manifold is 'piecewise smooth (PS)' if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS.

For a smooth manifold $N$ let $E^m_{PS}(N)$ be the set of PS embeddings $N\to\R^m$ up to PS isotopy. The forgetful map $E^m_{PL}(N)\to E^m_{PS}(N)$ is 1-1 [Haefliger1967, 2.4]. So a description of $E^m_{PS}(N)$ is equivalent to a description of $E^m_{PL}(N)$.

Example 2.1 (Spinning construction). For an embedding $g:S^3\to D^6$ denote by $Sg$ the embedding

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Example 2.2. Two embeddings $\tau^1,\tau^2:S^1\times S^3\to S^7$ are defined as compositions

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where $k=1,2$ and maps $t^k:S^1\times S^3\to S^3$ are defined below. We shall see that $t^k|_{S^1\times y}$ is an embedding for any $y\in S^3$ and $k=1,2$, hence $\tau^1$ and $\tau^2$ are embeddings.

Define $t^1(s,y):=sy$, where $S^3$ is identified with the set of unit length quaternions and $S^1\subset S^3$ with the set of unit length complex numbers.

Define $t^2(e^{i\theta},y):=\eta(y)\cos\theta+\sin\theta$, where $\eta:S^3\to S^2$ is the Hopf fibration and $S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$.

Example 2.3. Define a map $\tau \colon \Z^2 \to E^7(S^1 \times S^3)$. Take a smooth map $\alpha:S^3\to V_{4,2}$. Assuming that $V_{4, 2}\subset (\R^4)^2$, we have $\alpha(x) = (\alpha_1(x), \alpha_2(x))$. Define the adjunction map $\R^2 \times S^3 \to \R^4$ by $((s, t), x) \mapsto \alpha_1(x)s + \alpha_2(x)t$. (Assuming that $V_{4, 2}\subset (\R^4)^{\R^2}$, this map is obtained from $\alpha$ by the exponential law.) Denote by $\overline\alpha:S^1\times S^3\to S^3$ the restriction of the adjunction map. We define the embedding $\tau_\alpha$ to be the composition

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We define the map $\tau$ by $\tau(l, b):=[\tau_{\alpha}]$, where $\alpha\colon S^3 \to V_{4, 2}$ represents $(l, b) \in \pi_3(V_{4, 2})$ (for the standard identification $\pi_3(V_{4, 2})=\Zz^2$).

Example 2.4 (The Lambrechts torus). There is an embedding $S^1\times S^3\to S^7$ whose complement is not homotopy equivalent to the complement of the standard embedding.

I learned this simple construction from P. Lambrechts. Take the Hopf fibration $S^3\to S^7\overset{\nu}\to S^4$. Take the Hopf linking $h:S^1\sqcup S^2\to S^4$ [Skopenkov2016h, Example 2.1]. Then

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Figure 1: Three intersecting disks spanning Borromean rings; a torus with a hole spanning one of the rings and disjoint from the spanning disks of the other two rings

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(I conjecture that $\nu^{-1}h(S^1)=\tau^1(S^1\times S^3)$.)

Example 2.5 (the Haefliger torus). There is a PL embedding $S^2\times S^2\to S^7$ which is not PS isotopic to a smooth embedding.

Take the Haefliger trefoil knot $S^3\to S^6$ [Skopenkov2016t, Example 2.1]. Extend it to a PL conical embedding $D^4\to D^7_-$. By [Haefliger1962, $\S$4.2] the trefoil knot also extends to a proper smooth embedding into $D^7_+$ of the punctured torus (or disk with handle), see Figure 1. These two extensions together form the required PL embedding $S^2\times S^2\to S^7$, see Figure 2 for $k=1$. By [Boechat&Haefliger1970, p.165] this PL embedding is not PS isotopic to a smooth embedding.

For a higher-dimensional generalization see [Boechat1971, 6.2].

Example 3.1 [Boechat&Haefliger1970, p.164]. There is an embedding $\Cc P^2\to\Rr^7$.

Recall that $\Cc P^2_0$ is the mapping cylinder of the Hopf fibration $\eta:S^3\to S^2$. Recall that $S^6=S^2*S^3$. Define an embedding $f:\Cc P^2_0\to S^6$ by $f[(x,t)]:=[(x,\eta(x),t)]$, where $x\in S^3$. In other words, the segment joining $x\in S^3$ and $\eta(x)\in S^2$ is mapped onto the arc in $S^6$ joining $x$ to $\eta(x)$. Clearly, the boundary 3-sphere of $\Cc P^2_0$ is standardly embedded into $S^6$. Hence $f$ extends to an embedding $\Cc P^2\to\Rr^7$.

Alternatively, define an embedding $\Cc P^2\to\Rr^7$ by

$$\displaystyle (x:y:z)\mapsto(x\overline y, y\overline z, z\overline x,2|x|^2+|y|^2),\quad\text{where}\quad |x|^2+|y|^2+|z|^2=1.$$

Theorem 3.2. (a) There is only one embedding $\Cc P^2\to\Rr^7$ up to isotopy and a hyperplane reflection of $\Rr^7$. In other words, there are exactly two isotopy classes of embeddings $\Cc P^2\to\Rr^7$ (differing by composition with a hyperplane reflection of $\Rr^7$).

(b) For any pair of embeddings $f:\Cc P^2\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is isotopic to $f$.

(c) The Boechat-Haefliger invariant (defined below) is an injection $E^7_{PL}(\Cc P^2)\to H_2(\Cc P^2)\cong\Z$ whose image is the set of odd integeres. However, any PL embedding whose Boechat-Haefliger is different from $\pm1$ is not smoothable.

Definition 4.1. The composition

$$\displaystyle H_{s+1}(C_f,\partial)\overset\partial\to H_s(\partial C_f)\overset{\nu_f}\to H_s(N)$$

of the boundary map $\partial$ and the projection $\nu_f$ is an isomorphism. This is well-known, see [Skopenkov2008, $\S$2, the Alexander Duality Lemma]. The inverse $A_{f,s}$ to this composition is `the homology Alexander Duality isomorphism'; it equals to the composition $H_s(N)\to H^{6-s}(C_f)\to H_{s+1}(C_f,\partial)$ of the cohomology Alexander and Poincaré duality isomorphisms.

This is not to be confused with another well-known homology Alexander duality isomorphism $\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$ [Skopenkov2005, Alexander Duality Lemma 4.6].

Definition 4.2. A `homology Seifert surface' for $f$ is the image $A_{f,n}[N]\in H_{n+1}(C_f,\partial)$ of the fundamental class $[N]$.

Remark 4.3. Take a small oriented disk $D^{m-n}_f\subset\Rr^m$ whose intersection with $f(N)$ consists of exactly one point of sign $+1$ and such that $\partial D^{m-n}_f\subset\partial C_f$. A homology Seifert surface $Y\in H_{n+1}(C_f,\partial)$ for $f$ is uniquely defined by the condition $Y\cap [\partial D^{m-n}_f]=1$.

Definition 4.4. Define `the Boechat-Haefliger invariant' of $f$

$$\displaystyle \varkappa(f):=A_{f,2n+1-m}^{-1}\left(A_{f,n}[N]\cap A_{f,n}[N]\right)\in H_{2n+1-m}(N).$$

Clearly, a map $\varkappa:E^m(N)\to H_{2n+1-m}(N)$ is well-defined by $\varkappa([f]):=\varkappa(f)$.

Remark 4.5. (a) If $m=2n=6$, then $\varkappa(f)-\varkappa(f_0)=\pm2W(f,f_0)$ for any two embeddings $f,f_0:N\to\Rr^m$ [Skopenkov2008, $\S$2, The Boechat-Haefliger Invariant Lemma]. Here $W$ is the Whitney invariant [Skopenkov2016e, $\S$5], [Skopenkov2006, $\S$2]. We conjecture that this holds when $m-n$ is odd and that $\varkappa(f)=\varkappa(f_0)$ when $m-n$ is even.

(b) Definition 4.4 is equivalent to the original one for $m=2n-1=7$ [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1]. Earlier notation for $\varkappa(f)$ was $w_f$ [Boechat&Haefliger1970], $BH(f)$ [Skopenkov2005] and $\aleph(f)$ [Crowley&Skopenkov2008].

Theorem 5.1. (a) Any closed orientable 4-manifold embeds into $\Rr^7$.

(b) A closed 4-manifold $N$ embeds into $\Rr^7$ if and only if $\overline w_3(N)=0$.

Theorem 5.2. There is an isomorphism $E^7_D(S^4)\cong\Zz_{12}$.

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$$\displaystyle \varkappa:E^7_D(N)\to H_2(N)$$

$$\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$$

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$$\displaystyle \eta_u:\varkappa^{-1}(u)\to\Zz_{\gcd(u,24)}$$

whose image is the subset of even elements.

Corollary 5.4 ([Crowley&Skopenkov2008, Corollary 1.2]). (a) There are exactly twelve isotopy classes of embeddings $N\to\Rr^7$ if $N$ is an integral homology 4-sphere (cf. Theorem 5.2).

(b) Identify $H_2(S^2\times S^2) = \Z^2$ using the standard basis. For any integer $u$ there are exactly $\gcd(u,12)$ isotopy classes of embeddings $f:S^2\times S^2\to\Rr^7$ with $\varkappa(f)=(2u,0)$, and the same holds for those with $\varkappa(f)=(0,2u)$. Other values of $\Zz^2$ are not in the image of $\varkappa$.

Addendum 5.5 ([Crowley&Skopenkov2008, Addendum 1.3]). If $H_1(N)=0$ and $f:N\to\Rr^7$, $g:S^4\to\Rr^7$ are embeddings, then

$$\displaystyle \varkappa(f\#g)=\varkappa(f)\quad\text{and}\quad\eta_{\varkappa(f)}[f\#g]\equiv\eta_{\varkappa(f)}[f]+\eta_0[g]\mod\gcd(\varkappa(f),24).$$

Corollary 5.6 ([Crowley&Skopenkov2008, Corollary 1.4]). (a) Take an integer $u$ and the Hudson torus $f_u:=\Hud_{7,4,2}(u):S^2\times S^2\to\Rr^7$ defined in Remark 3.5.d of [Skopenkov2016e], [Skopenkov2006, Example 2.10]. If $u=6k\pm1$, then for any embedding $g:S^4\to\Rr^7$ the embedding $f_u\#g$ is isotopic to $f_u$. Moreover, for any integer $u$ the number of isotopy classes of embeddings $f_u\#g$ is $\gcd(u,12)$.

(b) If $H_1(N)=0$ and $\sigma(N)$ is not divisible by the square of an integer $s\ge2$. Then for any pair of embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is isotopic to $f$; in other words, $\varkappa$ is injective.

(c) If $H_1(N)=0$ and $f(N)\subset\Rr^6$ for an embedding $f:N\to\Rr^7$, then for every embedding $g:S^4\to\Rr^7$ the embedding $f\#g$ is not isotopic to $f$.

Remark 1.1 (PL and piecewise smooth embeddings). Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure [Milnor&Stasheff1974, Complement, Theorems 10.5 and 10.6]. Since also any PL 4-manifold admits a unique smooth structure [Mandelbaum1980, $\S$1.2], we may consider a smooth 4-manifold as a PL 4-manifold.

A map of a smooth manifold is 'piecewise smooth (PS)' if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS.

For a smooth manifold $N$ let $E^m_{PS}(N)$ be the set of PS embeddings $N\to\R^m$ up to PS isotopy. The forgetful map $E^m_{PL}(N)\to E^m_{PS}(N)$ is 1-1 [Haefliger1967, 2.4]. So a description of $E^m_{PS}(N)$ is equivalent to a description of $E^m_{PL}(N)$.

Example 2.1 (Spinning construction). For an embedding $g:S^3\to D^6$ denote by $Sg$ the embedding

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Example 2.2. Two embeddings $\tau^1,\tau^2:S^1\times S^3\to S^7$ are defined as compositions

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where $k=1,2$ and maps $t^k:S^1\times S^3\to S^3$ are defined below. We shall see that $t^k|_{S^1\times y}$ is an embedding for any $y\in S^3$ and $k=1,2$, hence $\tau^1$ and $\tau^2$ are embeddings.

Define $t^1(s,y):=sy$, where $S^3$ is identified with the set of unit length quaternions and $S^1\subset S^3$ with the set of unit length complex numbers.

Define $t^2(e^{i\theta},y):=\eta(y)\cos\theta+\sin\theta$, where $\eta:S^3\to S^2$ is the Hopf fibration and $S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$.

Example 2.3. Define a map $\tau \colon \Z^2 \to E^7(S^1 \times S^3)$. Take a smooth map $\alpha:S^3\to V_{4,2}$. Assuming that $V_{4, 2}\subset (\R^4)^2$, we have $\alpha(x) = (\alpha_1(x), \alpha_2(x))$. Define the adjunction map $\R^2 \times S^3 \to \R^4$ by $((s, t), x) \mapsto \alpha_1(x)s + \alpha_2(x)t$. (Assuming that $V_{4, 2}\subset (\R^4)^{\R^2}$, this map is obtained from $\alpha$ by the exponential law.) Denote by $\overline\alpha:S^1\times S^3\to S^3$ the restriction of the adjunction map. We define the embedding $\tau_\alpha$ to be the composition

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We define the map $\tau$ by $\tau(l, b):=[\tau_{\alpha}]$, where $\alpha\colon S^3 \to V_{4, 2}$ represents $(l, b) \in \pi_3(V_{4, 2})$ (for the standard identification $\pi_3(V_{4, 2})=\Zz^2$).

Example 2.4 (The Lambrechts torus). There is an embedding $S^1\times S^3\to S^7$ whose complement is not homotopy equivalent to the complement of the standard embedding.

I learned this simple construction from P. Lambrechts. Take the Hopf fibration $S^3\to S^7\overset{\nu}\to S^4$. Take the Hopf linking $h:S^1\sqcup S^2\to S^4$ [Skopenkov2016h, Example 2.1]. Then

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Figure 1: Three intersecting disks spanning Borromean rings; a torus with a hole spanning one of the rings and disjoint from the spanning disks of the other two rings

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(I conjecture that $\nu^{-1}h(S^1)=\tau^1(S^1\times S^3)$.)

Example 2.5 (the Haefliger torus). There is a PL embedding $S^2\times S^2\to S^7$ which is not PS isotopic to a smooth embedding.

Take the Haefliger trefoil knot $S^3\to S^6$ [Skopenkov2016t, Example 2.1]. Extend it to a PL conical embedding $D^4\to D^7_-$. By [Haefliger1962, $\S$4.2] the trefoil knot also extends to a proper smooth embedding into $D^7_+$ of the punctured torus (or disk with handle), see Figure 1. These two extensions together form the required PL embedding $S^2\times S^2\to S^7$, see Figure 2 for $k=1$. By [Boechat&Haefliger1970, p.165] this PL embedding is not PS isotopic to a smooth embedding.

For a higher-dimensional generalization see [Boechat1971, 6.2].

Example 3.1 [Boechat&Haefliger1970, p.164]. There is an embedding $\Cc P^2\to\Rr^7$.

Recall that $\Cc P^2_0$ is the mapping cylinder of the Hopf fibration $\eta:S^3\to S^2$. Recall that $S^6=S^2*S^3$. Define an embedding $f:\Cc P^2_0\to S^6$ by $f[(x,t)]:=[(x,\eta(x),t)]$, where $x\in S^3$. In other words, the segment joining $x\in S^3$ and $\eta(x)\in S^2$ is mapped onto the arc in $S^6$ joining $x$ to $\eta(x)$. Clearly, the boundary 3-sphere of $\Cc P^2_0$ is standardly embedded into $S^6$. Hence $f$ extends to an embedding $\Cc P^2\to\Rr^7$.

Alternatively, define an embedding $\Cc P^2\to\Rr^7$ by

$$\displaystyle (x:y:z)\mapsto(x\overline y, y\overline z, z\overline x,2|x|^2+|y|^2),\quad\text{where}\quad |x|^2+|y|^2+|z|^2=1.$$

Theorem 3.2. (a) There is only one embedding $\Cc P^2\to\Rr^7$ up to isotopy and a hyperplane reflection of $\Rr^7$. In other words, there are exactly two isotopy classes of embeddings $\Cc P^2\to\Rr^7$ (differing by composition with a hyperplane reflection of $\Rr^7$).

(b) For any pair of embeddings $f:\Cc P^2\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is isotopic to $f$.

(c) The Boechat-Haefliger invariant (defined below) is an injection $E^7_{PL}(\Cc P^2)\to H_2(\Cc P^2)\cong\Z$ whose image is the set of odd integeres. However, any PL embedding whose Boechat-Haefliger is different from $\pm1$ is not smoothable.

Definition 4.1. The composition

$$\displaystyle H_{s+1}(C_f,\partial)\overset\partial\to H_s(\partial C_f)\overset{\nu_f}\to H_s(N)$$

of the boundary map $\partial$ and the projection $\nu_f$ is an isomorphism. This is well-known, see [Skopenkov2008, $\S$2, the Alexander Duality Lemma]. The inverse $A_{f,s}$ to this composition is `the homology Alexander Duality isomorphism'; it equals to the composition $H_s(N)\to H^{6-s}(C_f)\to H_{s+1}(C_f,\partial)$ of the cohomology Alexander and Poincaré duality isomorphisms.

This is not to be confused with another well-known homology Alexander duality isomorphism $\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$ [Skopenkov2005, Alexander Duality Lemma 4.6].

Definition 4.2. A `homology Seifert surface' for $f$ is the image $A_{f,n}[N]\in H_{n+1}(C_f,\partial)$ of the fundamental class $[N]$.

Remark 4.3. Take a small oriented disk $D^{m-n}_f\subset\Rr^m$ whose intersection with $f(N)$ consists of exactly one point of sign $+1$ and such that $\partial D^{m-n}_f\subset\partial C_f$. A homology Seifert surface $Y\in H_{n+1}(C_f,\partial)$ for $f$ is uniquely defined by the condition $Y\cap [\partial D^{m-n}_f]=1$.

Definition 4.4. Define `the Boechat-Haefliger invariant' of $f$

$$\displaystyle \varkappa(f):=A_{f,2n+1-m}^{-1}\left(A_{f,n}[N]\cap A_{f,n}[N]\right)\in H_{2n+1-m}(N).$$

Clearly, a map $\varkappa:E^m(N)\to H_{2n+1-m}(N)$ is well-defined by $\varkappa([f]):=\varkappa(f)$.

Remark 4.5. (a) If $m=2n=6$, then $\varkappa(f)-\varkappa(f_0)=\pm2W(f,f_0)$ for any two embeddings $f,f_0:N\to\Rr^m$ [Skopenkov2008, $\S$2, The Boechat-Haefliger Invariant Lemma]. Here $W$ is the Whitney invariant [Skopenkov2016e, $\S$5], [Skopenkov2006, $\S$2]. We conjecture that this holds when $m-n$ is odd and that $\varkappa(f)=\varkappa(f_0)$ when $m-n$ is even.

(b) Definition 4.4 is equivalent to the original one for $m=2n-1=7$ [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1]. Earlier notation for $\varkappa(f)$ was $w_f$ [Boechat&Haefliger1970], $BH(f)$ [Skopenkov2005] and $\aleph(f)$ [Crowley&Skopenkov2008].

Theorem 5.1. (a) Any closed orientable 4-manifold embeds into $\Rr^7$.

(b) A closed 4-manifold $N$ embeds into $\Rr^7$ if and only if $\overline w_3(N)=0$.

Theorem 5.2. There is an isomorphism $E^7_D(S^4)\cong\Zz_{12}$.

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$$\displaystyle \varkappa:E^7_D(N)\to H_2(N)$$

$$\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$$

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$$\displaystyle \eta_u:\varkappa^{-1}(u)\to\Zz_{\gcd(u,24)}$$

whose image is the subset of even elements.

Corollary 5.4 ([Crowley&Skopenkov2008, Corollary 1.2]). (a) There are exactly twelve isotopy classes of embeddings $N\to\Rr^7$ if $N$ is an integral homology 4-sphere (cf. Theorem 5.2).

(b) Identify $H_2(S^2\times S^2) = \Z^2$ using the standard basis. For any integer $u$ there are exactly $\gcd(u,12)$ isotopy classes of embeddings $f:S^2\times S^2\to\Rr^7$ with $\varkappa(f)=(2u,0)$, and the same holds for those with $\varkappa(f)=(0,2u)$. Other values of $\Zz^2$ are not in the image of $\varkappa$.

Addendum 5.5 ([Crowley&Skopenkov2008, Addendum 1.3]). If $H_1(N)=0$ and $f:N\to\Rr^7$, $g:S^4\to\Rr^7$ are embeddings, then

$$\displaystyle \varkappa(f\#g)=\varkappa(f)\quad\text{and}\quad\eta_{\varkappa(f)}[f\#g]\equiv\eta_{\varkappa(f)}[f]+\eta_0[g]\mod\gcd(\varkappa(f),24).$$

Corollary 5.6 ([Crowley&Skopenkov2008, Corollary 1.4]). (a) Take an integer $u$ and the Hudson torus $f_u:=\Hud_{7,4,2}(u):S^2\times S^2\to\Rr^7$ defined in Remark 3.5.d of [Skopenkov2016e], [Skopenkov2006, Example 2.10]. If $u=6k\pm1$, then for any embedding $g:S^4\to\Rr^7$ the embedding $f_u\#g$ is isotopic to $f_u$. Moreover, for any integer $u$ the number of isotopy classes of embeddings $f_u\#g$ is $\gcd(u,12)$.

(b) If $H_1(N)=0$ and $\sigma(N)$ is not divisible by the square of an integer $s\ge2$. Then for any pair of embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is isotopic to $f$; in other words, $\varkappa$ is injective.

(c) If $H_1(N)=0$ and $f(N)\subset\Rr^6$ for an embedding $f:N\to\Rr^7$, then for every embedding $g:S^4\to\Rr^7$ the embedding $f\#g$ is not isotopic to $f$.