4-manifolds in 7-space

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1 Introduction

For notation and conventions see high codimension embeddings. Denote by $\eta:S^3\to S^2$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\eta:S^3\to S^2$ is the Hopf map.

2 Examples

The Hudson tori $\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$$\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$ and $\Hud_{7,4,2}:S^1\times S^3\to\Rr^7$$\Hud_{7,4,2}:S^1\times S^3\to\Rr^7$ are defined in Remark 3.5.

Analogously to the case $m=2n$$m=2n$ for an orientable 4-manifold $N$$N$, an embedding $f_0:N\to\Rr^7$$f_0:N\to\Rr^7$ and a class $a\in H_1(N)$$a\in H_1(N)$ one can construct an embedding $f_a:N\to\Rr^7$$f_a:N\to\Rr^7$. However, this embedding is no longer well-defined. We have $W(f_u,f_0)=u$$W(f_u,f_0)=u$ for the Whitney invariant, that is defined analogously to the case $m=2n$$m=2n$.

2.1 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 isotopy classes of 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 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$$f\#g$ is isotopic to $f$$f$.

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

2.2 The Lambrechts torus and the Hudson torus

These two embeddings $\tau^1,\tau^2:S^1\times S^3\to\Rr^7$$\tau^1,\tau^2:S^1\times S^3\to\Rr^7$ are defined [Skopenkov2006] as compositions $S^1\times S^3\overset{p_2\times t^i}\to S^3\times S^3\subset\Rr^7$$S^1\times S^3\overset{p_2\times t^i}\to S^3\times S^3\subset\Rr^7$, where $i=1,2$$i=1,2$, $p_2$$p_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$.

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.

Take the Hopf fibration $S^3\to S^7\overset\eta\to S^4$$S^3\to S^7\overset\eta\to S^4$. Take the standard embeding $S^2\subset S^4$$S^2\subset S^4$. Its complement has the homotopy type of $S^1$$S^1$. Then $im\tau^1=\eta^{-1}(S^1)\cong S^1\times S^3\subset S^7$$im\tau^1=\eta^{-1}(S^1)\cong S^1\times S^3\subset S^7$. This is the construction of Lambrechts motivated by the following property:

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

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.

2.3 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$$N$ be a closed connected orientable 4-manifold. Fix an orientation on $N$$N$ and an orientation on $\Rr^7$$\Rr^7$. A homology Seifert surface $A_f$$A_f$ for $f$$f$ is the image of the fundamental class $[N]$$[N]$ under the composition $H_4(N)\to H^2(C_f)\to H_5(C_f,\partial C_f)$$H_4(N)\to H^2(C_f)\to H_5(C_f,\partial C_f)$ of the Alexander and Poincar\'e duality isomorphisms. (This composition is an inverse to the composition $H_5(C_f,\partial C_f)\to H_4(\partial C_f)\to H_4(N)$$H_5(C_f,\partial C_f)\to H_4(\partial C_f)\to H_4(N)$ of the boundary map $\partial$$\partial$ and the projection $\nu_f$$\nu_f$ of the normal bundle, cf. [Skopenkov2008], the Alexander Duality Lemma; this justifies the name `homology Seifert surface'.)

Define $BH(f)$$BH(f)$ to be the image of $A_f^2=A_f\cap A_f$$A_f^2=A_f\cap A_f$ under the composition $H_3(C_f,\partial C_f)\to H^4(C_f)\to H_2(N)$$H_3(C_f,\partial C_f)\to H^4(C_f)\to H_2(N)$ of the Poincaré and Alexander duality isomorphisms. (This composition has a direct geometric definition $\nu\partial$$\nu\partial$ as above.)

This new definition is equivalent to the original one [Boechat&Haefliger1970] by [Crowley&Skopenkov2008], Section Lemma 3.1.

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

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

whose image is

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

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

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

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(BH)$$u\in im(BH)$ is divisible by 2 (for some $u$$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].

Theorem 4.2 implies that

• 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).
• 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$$f:S^2\times S^2\to\Rr^7$ with $BH(f)=(2u,0)$$BH(f)=(2u,0)$, and the same holds for those with $BH(f)=(0,2u)$$BH(f)=(0,2u)$. Other values of $\Zz^2$$\Zz^2$ are not in the image of $BH$$BH$. (We take the standard basis in $H_2(S^2\times S^2)$$H_2(S^2\times S^2)$.)

Theorem 4.2 implies the following examples (first proved in [Skopenkov2005]) of the triviality and the effectiveness of the connected sum action $E^7(S^4)\to E^7(N)$$E^7(S^4)\to E^7(N)$.

• 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$$f$ (in other words, $BH$$BH$ is injective).
• 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$.

The following can be obtained using [Crowley&Skopenkov2008] (but not using [Skopenkov2005]).

• 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. 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)$.)

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$

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

5 References

• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$$4$ dans $\Rr^7$$\Rr^7$, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
• [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
• [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013