4-manifolds in 7-space

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1 Introduction

This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn theory of embeddings.

See general introduction on embeddings, notation and conventions.

2 Examples

The Hudson tori $\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$${{Stub}} == Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn theory of embeddings. See [[High_codimension_embeddings|general introduction on embeddings, notation and conventions]]. </wikitex> == Examples == <wikitex>; The Hudson tori $\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$ and $\Hud_{7,4,2}:S^1\times S^3\to\Rr^7$ are defined in [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Remark 3.5.d]]. For an orientable 4-manifold $N$, an embedding $f_0:N\to\Rr^7$ and a class $a\in H_2(N)$ one can construct an embedding $f_a:N\to\Rr^7$ by linked connected sum analogously to [[Embeddings_just_below_the_stable_range:_classification#Action_by_linked_embedded_connected_sum|embeddings into $\Rr^8$]]. If $N$ is simply-connected and CAT=PL, this gives a [[Embeddings_just_below_the_stable_range:_classification#A_generalization_to_highly-connected_manifolds|free transitive action]] of $H_2(N)$ on $E^7(N)$. We also have $W(f_u,f_0)=u$ for [[Embeddings_just_below_the_stable_range#The Whitney invariant|the Whitney invariant]]. Denote by $\eta:S^3\to S^2$ is the Hopf map. </wikitex> === The Lambrechts torus and the Hudson torus === <wikitex>; These two embeddings $\tau^1,\tau^2:S^1\times S^3\to\Rr^7$ are defined as compositions $S^1\times S^3\overset{pr_2\times t^i}\to S^3\times S^3\subset\Rr^7$, where $i=1,2$, $pr_2$ is the projection onto the second factor, $\subset$ is the standard inclusion and maps $t^i:S^1\times S^3\to S^3$ are defined below. We shall see that $t^i|_{S^1\times y}$ are embeddings for each $y\in S^3$, 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 $S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$. These examples appear in \cite{Skopenkov2006} but could be known earlier. Note that $\tau^2$ is PL isotopic to the Hudson torus $\Hud_{7,4,1}$ defined in [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Remark 3.5.d]]. Take the Hopf fibration $S^3\to S^7\overset{\eta_2}\to S^4$. Take the standard embeding $S^2\subset S^4$. Its complement has the homotopy type of $S^1$. Then $im\tau^1=\eta_2^{-1}(S^1)\cong S^1\times S^3\subset S^7$. This is the construction of Lambrechts (but could be known earlier). We have $$S^7-im\tau^1\sim \eta_2^{-1}(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-im f_0,$$ where $f_0:S^1\times S^3\to S^7$ is the standard embedding. </wikitex> === Embeddings of CP² into R⁷ === <wikitex>; We follow \cite{Boechat&Haefliger1970}, p. 164. Recall that $\Cc P^2_0$ is the mapping cylinder of $\eta$. 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$. Apriori this extension need not be unique (because it can be changed by a connected sum with an embedding $g:S^4\to D^6$). Surprisingly, it is unique, and is the only embedding $\Cc P^2\to\Rr^7$ (up to isotopy and a hyperplane reflection of $\Rr^7$). {{beginthm|Theorem}}\label{cp2} *There are exactly two smooth isotopy classes of smooth embeddings $\Cc P^2\to\Rr^7$ (differing by a hyperplane reflection of $\Rr^7$). *For each smooth embeddings $f:\Cc P^2\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is smoothly isotopic to $f$. *[[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|The Whitney invariant]] is a 1--1 correspondence $E^7_{PL}(\Cc P^2)\to\Z$. The inverse is given by [[Embeddings_just_below_the_stable_range:_classification#A_generalization_to_highly-connected_manifolds|linked connected sum]]. {{endthm}} This follows by \cite{Boechat&Haefliger1970}, \cite{Skopenkov2005}, Triviality Theorem (a) or by [[Embeddings_of_4-manifolds_in_7-space#Classification|general classification]]. </wikitex> === The Haefliger torus === <wikitex>; This is a PL embedding $S^2\times S^2\to\Rr^7$ which is (locally flat but) not PL isotopic to a smooth embedding \cite{Boechat&Haefliger1970}, p.165, \cite{Boechat1971}, 6.2. Take [[Knots,_i.e._embeddings_of_spheres#The_Haefliger_trefoil_knot|the Haefliger trefoil knot]] $S^3\to\Rr^6$. Extend it to a conical embedding $D^4\to\Rr^7_-$. By \cite{Haefliger1962}, the trefoil knot also extends to a smooth embedding $S^2\times S^2-Int D^4\to\Rr^7_+$ (see \cite{Skopenkov2006}, Figure 3.7.a). These two extensions together form the Haefliger torus (see \cite{Skopenkov2006}, Figure 3.7.b). </wikitex> == The Boechat-Haefliger invariant == <wikitex>; Let $N$ be a closed connected orientable 4-manifold and $f:N\to\Rr^7$ an embedding. Fix an orientation on $N$ and an orientation on $\Rr^7$. {{beginthm|Definition}} The composition $$ H_{s+1}(C_f,\partial C_f)\overset\partial\to H_s(\partial C_f)\overset{\nu_f}\to H_s(N) $$ of the boundary map $\partial$ and the projection $\nu_f$ is an isomorphism, cf. \cite{Skopenkov2008}, the Alexander Duality Lemma. The inverse $A_{f,s}$ to this composition is homology Alexander Duality isomorphism; it equals to the composition $H_s(N)\to H^{6-s}(C_f)\to H_{s+1}(C_f,\partial C_f)$ of the cohomology Alexander and Poincar\'e duality isomorphisms. {{endthm}} {{beginthm|Definition}}\label{dbh} A ''homology Seifert surface'' for $f$ is the image $A_{f,4}[N]\in H_5(C_f,\partial C_f)$ of the fundamental class $[N]$. Define $$ \varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N). $$ {{endthm}} {{beginthm|Remark}} (a) Take a small oriented disk $D^3_f\subset\Rr^7$ whose intersection with $fN$ consists of exactly one point of sign $+1$ and such that $\partial D^3_f\subset\partial C_f$. A ''meridian of $f$'' is $\partial D^3_f$. A homology Seifert surface $Y\in H_5(C_f,\partial)$ for $f$ is uniquely defined by the condition $Y\cap [\partial D^3_f]=1$. (b) We have $\varkappa(f)-\varkappa(f_0)=\pm2W(f,f_0)$ for [[Embeddings_just_below_the_stable_range#The Whitney invariant|the Whitney invariant]] $W(f,f_0)$. This is proved analogously to \cite{Skopenkov2006a}, \S2, The Boechat-Haefliger Invariant Lemma. (c) Definition \ref{dbh} is equivalent to the original one \cite{Boechat&Haefliger1970} by \cite{Crowley&Skopenkov2008}, Section Lemma 3.1. Hence $\varkappa(f)\mod2=PDw_2(N)$. (d) Earlier notation was $w_f$ \cite{Boechat&Haefliger1970}, $BH(f)$ \cite{Skopenkov2005} and $\aleph(f)$ \cite{Crowley&Skopenkov2008}. {{endthm}} </wikitex> == Classification == <wikitex>; See [[Embeddings_just_below_the_stable_range:_classification#Classification_just_below_the_stable_range|a classification]] of $E^7_{PL}(N)$ for a closed connected 4-manifold $N$ such that $H_1(N)=0$. Here we work in the smooth category. {{beginthm|Theorem}}\label{hae4} $E^7(S^4)\cong\Zz_{12}$. \cite{Haefliger1966}, \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008}. {{endthm}} {{beginthm|Theorem}}\label{clth4} \cite{Crowley&Skopenkov2008} Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$. There is the Boéchat-Haefliger invariant $$\varkappa:E^7_D(N)\to H_2(N)$$ whose image is $$im \varkappa=\{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$$ For each $u\in im \varkappa$ there is an injective invariant called the Kreck invariant, $$\eta_u:\varkappa^{-1}(u)\to\Zz_{\gcd(u,24)}$$ whose image is the subset of even elements. {{endthm}} Here $\gcd(u,24)$ is the maximal integer $k$ such that both $u\in H_2(N)$ and 24 are divisible by $k$. Thus $\eta_u$ is surjective if $u$ is not divisible by 2. Note that $u\in im \varkappa$ is divisible by 2 (for some $u$ or, equivalently, for each $u$) if and only if $N$ is spin. For the definition of the Kreck invariant see \cite{Crowley&Skopenkov2008}. {{beginthm|Corollary}}\label{coclth4} (a) There are exactly twelve isotopy classes of embeddings $N\to\Rr^7$ if $N$ is an integral homology 4-sphere (cf. Theorem \ref{hae4}). (b) Under assumptions of Theorem \ref{clth4} for each pair of embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$ $$\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).$$ (c) For each integer $u$ there are exactly $\gcd(u,12)$ isotopy classes of embeddings $f:S^2\times S^2\to\Rr^7$ with $\varkappa(f)=(2u,0)$, and the same holds for those with $\varkappa(f)=(0,2u)$. Other values of $\Zz^2$ are not in the image of $\varkappa$. (We take the standard basis in $H_2(S^2\times S^2)$.) (d) Take an integer $u$ and the Hudson torus $f_u:=\Hud_{7,4,2}(u):S^2\times S^2\to\Rr^7$ defined in [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Remark 3.5.d]]. If $u=6k\pm1$, then for each embedding $g:S^4\to\Rr^7$ the embedding $f_u\#g$ is isotopic to $f_u$. (For a general integer $u$ the number of isotopy classes of embeddings $f_u\#g$ is $\gcd(u,12)$.) <!-- The following can be obtained using \cite{Crowley&Skopenkov2008} (but not using \cite{Skopenkov2005}).--> (e) Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$ and the signature $\sigma(N)$ of $N$ is not divisible by the square of an integer $s\ge2$. Then for each embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is isotopic to $f$; in other words, $\varkappa$ is injective. (First proved in \cite{Skopenkov2005}.) (f) If $N$ is a closed connected 4-manifold such that $H_1(N)=0$ and $f(N)\subset\Rr^6$ for an embedding $f:N\to\Rr^7$, then for each embedding $g:S^4\to\Rr^7$ the embedding $f\#g$ is not isotopic to $f$. {{endthm}} Corollaries \ref{coclth4}.def exhibit examples of the effectiveness and the triviality of the embedded connected sum action of $E^7(S^4)$ on $E^7(N)$. </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$ and $\Hud_{7,4,2}:S^1\times S^3\to\Rr^7$$\Hud_{7,4,2}:S^1\times S^3\to\Rr^7$ are defined in Remark 3.5.d.

For an orientable 4-manifold $N$$N$, an embedding $f_0:N\to\Rr^7$$f_0:N\to\Rr^7$ and a class $a\in H_2(N)$$a\in H_2(N)$ one can construct an embedding $f_a:N\to\Rr^7$$f_a:N\to\Rr^7$ by linked connected sum analogously to embeddings into $\Rr^8$$\Rr^8$. If $N$$N$ is simply-connected and CAT=PL, this gives a free transitive action of $H_2(N)$$H_2(N)$ on $E^7(N)$$E^7(N)$. We also have $W(f_u,f_0)=u$$W(f_u,f_0)=u$ for the Whitney invariant.

Denote by $\eta:S^3\to S^2$$\eta:S^3\to S^2$ is the Hopf map.

2.1 The Lambrechts torus and the Hudson torus

These two embeddings $\tau^1,\tau^2:S^1\times S^3\to\Rr^7$$\tau^1,\tau^2:S^1\times S^3\to\Rr^7$ are defined as compositions $S^1\times S^3\overset{pr_2\times t^i}\to S^3\times S^3\subset\Rr^7$$S^1\times S^3\overset{pr_2\times t^i}\to S^3\times S^3\subset\Rr^7$, where $i=1,2$$i=1,2$, $pr_2$$pr_2$ is the projection onto the second factor, $\subset$$\subset$ is the standard inclusion and maps $t^i:S^1\times S^3\to S^3$$t^i:S^1\times S^3\to S^3$ are defined below. We shall see that $t^i|_{S^1\times y}$$t^i|_{S^1\times y}$ are embeddings for each $y\in S^3$$y\in S^3$, hence $\tau^1$$\tau^1$ and $\tau^2$$\tau^2$ are embeddings.

Define $t^1(s,y):=sy$$t^1(s,y):=sy$, where $S^3$$S^3$ is identified with the set of unit length quaternions and $S^1\subset S^3$$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$$t^2(e^{i\theta},y):=\eta(y)\cos\theta+\sin\theta$, where $S^2$$S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$$ai+bj+ck$.

These examples appear in [Skopenkov2006] but could be known earlier. Note that $\tau^2$$\tau^2$ is PL isotopic to the Hudson torus $\Hud_{7,4,1}$$\Hud_{7,4,1}$ defined in Remark 3.5.d.

Take the Hopf fibration $S^3\to S^7\overset{\eta_2}\to S^4$$S^3\to S^7\overset{\eta_2}\to S^4$. Take the standard embeding $S^2\subset S^4$$S^2\subset S^4$. Its complement has the homotopy type of $S^1$$S^1$. Then $im\tau^1=\eta_2^{-1}(S^1)\cong S^1\times S^3\subset S^7$$im\tau^1=\eta_2^{-1}(S^1)\cong S^1\times S^3\subset S^7$. This is the construction of Lambrechts (but could be known earlier). We have

$\displaystyle S^7-im\tau^1\sim \eta_2^{-1}(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-im f_0,$ $$\displaystyle S^7-im\tau^1\sim \eta_2^{-1}(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-im f_0,$$

where $f_0:S^1\times S^3\to S^7$$f_0:S^1\times S^3\to S^7$ is the standard embedding.

2.2 Embeddings of CP² into R⁷

We follow [Boechat&Haefliger1970], p. 164. Recall that $\Cc P^2_0$$\Cc P^2_0$ is the mapping cylinder of $\eta$$\eta$. Recall that $S^6=S^2*S^3$$S^6=S^2*S^3$. Define an embedding $f:\Cc P^2_0\to S^6$$f:\Cc P^2_0\to S^6$ by $f[(x,t)]:=[(x,\eta(x),t)]$$f[(x,t)]:=[(x,\eta(x),t)]$, where $x\in S^3$$x\in S^3$. In other words, the segment joining $x\in S^3$$x\in S^3$ and $\eta(x)\in S^2$$\eta(x)\in S^2$ is mapped onto the arc in $S^6$$S^6$ joining $x$$x$ to $\eta(x)$$\eta(x)$. Clearly, the boundary 3-sphere of $\Cc P^2_0$$\Cc P^2_0$ is standardly embedded into $S^6$$S^6$. Hence $f$$f$ extends to an embedding $\Cc P^2\to\Rr^7$$\Cc P^2\to\Rr^7$.

Apriori this extension need not be unique (because it can be changed by a connected sum with an embedding $g:S^4\to D^6$$g:S^4\to D^6$). Surprisingly, it is unique, and is the only embedding $\Cc P^2\to\Rr^7$$\Cc P^2\to\Rr^7$ (up to isotopy and a hyperplane reflection of $\Rr^7$$\Rr^7$).

Theorem 2.1.

• There are exactly two smooth isotopy classes of smooth embeddings $\Cc P^2\to\Rr^7$$\Cc P^2\to\Rr^7$ (differing by a hyperplane reflection of $\Rr^7$$\Rr^7$).
• For each smooth embeddings $f:\Cc P^2\to\Rr^7$$f:\Cc P^2\to\Rr^7$ and $g:S^4\to\Rr^7$$g:S^4\to\Rr^7$ the embedding $f\#g$$f\#g$ is smoothly isotopic to $f$$f$.
• The Whitney invariant is a 1--1 correspondence $E^7_{PL}(\Cc P^2)\to\Z$$E^7_{PL}(\Cc P^2)\to\Z$. The inverse is given by linked connected sum.

This follows by [Boechat&Haefliger1970], [Skopenkov2005], Triviality Theorem (a) or by general classification.

2.3 The Haefliger torus

This is a PL embedding $S^2\times S^2\to\Rr^7$$S^2\times S^2\to\Rr^7$ which is (locally flat but) not PL isotopic to a smooth embedding [Boechat&Haefliger1970], p.165, [Boechat1971], 6.2. Take the Haefliger trefoil knot $S^3\to\Rr^6$$S^3\to\Rr^6$. Extend it to a conical embedding $D^4\to\Rr^7_-$$D^4\to\Rr^7_-$. By [Haefliger1962], the trefoil knot also extends to a smooth embedding $S^2\times S^2-Int D^4\to\Rr^7_+$$S^2\times S^2-Int D^4\to\Rr^7_+$ (see [Skopenkov2006], Figure 3.7.a). These two extensions together form the Haefliger torus (see [Skopenkov2006], Figure 3.7.b).

3 The Boechat-Haefliger invariant

Let $N$$N$ be a closed connected orientable 4-manifold and $f:N\to\Rr^7$$f:N\to\Rr^7$ an embedding. Fix an orientation on $N$$N$ and an orientation on $\Rr^7$$\Rr^7$.

Definition 3.1. The composition

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

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

Definition 3.2. A homology Seifert surface for $f$$f$ is the image $A_{f,4}[N]\in H_5(C_f,\partial C_f)$$A_{f,4}[N]\in H_5(C_f,\partial C_f)$ of the fundamental class $[N]$$[N]$. Define

$\displaystyle \varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N).$ $$\displaystyle \varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N).$$

Remark 3.3. (a) Take a small oriented disk $D^3_f\subset\Rr^7$$D^3_f\subset\Rr^7$ whose intersection with $fN$$fN$ consists of exactly one point of sign $+1$$+1$ and such that $\partial D^3_f\subset\partial C_f$$\partial D^3_f\subset\partial C_f$. A meridian of $f$$f$ is $\partial D^3_f$$\partial D^3_f$. A homology Seifert surface $Y\in H_5(C_f,\partial)$$Y\in H_5(C_f,\partial)$ for $f$$f$ is uniquely defined by the condition $Y\cap [\partial D^3_f]=1$$Y\cap [\partial D^3_f]=1$.

(b) We have $\varkappa(f)-\varkappa(f_0)=\pm2W(f,f_0)$$\varkappa(f)-\varkappa(f_0)=\pm2W(f,f_0)$ for the Whitney invariant $W(f,f_0)$$W(f,f_0)$. This is proved analogously to [Skopenkov2006a], \S2, The Boechat-Haefliger Invariant Lemma.

(c) Definition 3.2 is equivalent to the original one [Boechat&Haefliger1970] by [Crowley&Skopenkov2008], Section Lemma 3.1. Hence $\varkappa(f)\mod2=PDw_2(N)$$\varkappa(f)\mod2=PDw_2(N)$.

(d) Earlier notation was $w_f$$w_f$ [Boechat&Haefliger1970], $BH(f)$$BH(f)$ [Skopenkov2005] and $\aleph(f)$$\aleph(f)$ [Crowley&Skopenkov2008].

4 Classification

See a classification of $E^7_{PL}(N)$$E^7_{PL}(N)$ for a closed connected 4-manifold $N$$N$ such that $H_1(N)=0$$H_1(N)=0$. Here we work in the smooth category.

Theorem 4.1. $E^7(S^4)\cong\Zz_{12}$$E^7(S^4)\cong\Zz_{12}$. [Haefliger1966], [Skopenkov2005], [Crowley&Skopenkov2008].

Theorem 4.2. [Crowley&Skopenkov2008] Let $N$$N$ be a closed connected 4-manifold such that $H_1(N)=0$$H_1(N)=0$. There is the Boéchat-Haefliger invariant

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

whose image is

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

For each $u\in im \varkappa$$u\in im \varkappa$ there is an injective invariant called the Kreck invariant,

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

whose image is the subset of even elements.

Here $\gcd(u,24)$$\gcd(u,24)$ is the maximal integer $k$$k$ such that both $u\in H_2(N)$$u\in H_2(N)$ and 24 are divisible by $k$$k$. Thus $\eta_u$$\eta_u$ is surjective if $u$$u$ is not divisible by 2. Note that $u\in im \varkappa$$u\in im \varkappa$ is divisible by 2 (for some $u$$u$ or, equivalently, for each $u$$u$) if and only if $N$$N$ is spin.

For the definition of the Kreck invariant see [Crowley&Skopenkov2008].

Corollary 4.3. (a) There are exactly twelve isotopy classes of embeddings $N\to\Rr^7$$N\to\Rr^7$ if $N$$N$ is an integral homology 4-sphere (cf. Theorem 4.1).

(b) Under assumptions of Theorem 4.2 for each pair of embeddings $f:N\to\Rr^7$$f:N\to\Rr^7$ and $g:S^4\to\Rr^7$$g:S^4\to\Rr^7$

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

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

(d) Take an integer $u$$u$ and the Hudson torus $f_u:=\Hud_{7,4,2}(u):S^2\times S^2\to\Rr^7$$f_u:=\Hud_{7,4,2}(u):S^2\times S^2\to\Rr^7$ defined in Remark 3.5.d. If $u=6k\pm1$$u=6k\pm1$, then for each embedding $g:S^4\to\Rr^7$$g:S^4\to\Rr^7$ the embedding $f_u\#g$$f_u\#g$ is isotopic to $f_u$$f_u$. (For a general integer $u$$u$ the number of isotopy classes of embeddings $f_u\#g$$f_u\#g$ is $\gcd(u,12)$$\gcd(u,12)$.)

(e) Let $N$$N$ be a closed connected 4-manifold such that $H_1(N)=0$$H_1(N)=0$ and the signature $\sigma(N)$$\sigma(N)$ of $N$$N$ is not divisible by the square of an integer $s\ge2$$s\ge2$. Then for each embeddings $f:N\to\Rr^7$$f:N\to\Rr^7$ and $g:S^4\to\Rr^7$$g:S^4\to\Rr^7$ the embedding $f\#g$$f\#g$ is isotopic to $f$$f$; in other words, $\varkappa$$\varkappa$ is injective. (First proved in [Skopenkov2005].)

(f) If $N$$N$ is a closed connected 4-manifold such that $H_1(N)=0$$H_1(N)=0$ and $f(N)\subset\Rr^6$$f(N)\subset\Rr^6$ for an embedding $f:N\to\Rr^7$$f:N\to\Rr^7$, then for each embedding $g:S^4\to\Rr^7$$g:S^4\to\Rr^7$ the embedding $f\#g$$f\#g$ is not isotopic to $f$$f$.

Corollaries 4.3.def exhibit examples of the effectiveness and the triviality of the embedded connected sum action of $E^7(S^4)$$E^7(S^4)$ on $E^7(N)$$E^7(N)$.

5 References

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• [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension $4k$$4k$ dans $\Rr^{6k+1}$$\Rr^{6k+1}$, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
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