4-manifolds in 7-space

Revision as of 12:02, 7 March 2010

1 Introduction

For notation and conventions see high codimension embeddings.

2 Examples

There is The Hudson torus $== Introduction == <wikitex>; For notation and conventions see [[High_codimension_embeddings#Notation and conventions|high codimension embeddings]]. </wikitex> == Examples == <wikitex>; There is [[Embeddings_just_below_the_stable_range#Examples#Remarks|The Hudson torus]] $\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$. Analogously to [[Embeddings_just_below_the_stable_range#Examples#An action of the first homology group on embeddings|the case $m=2n$]] for an orientable 4-manifold $N$, an embedding $f_0:N\to\Rr^7$ and a class $a\in H_1(N)$ one can construct an embedding $f_a:N\to\Rr^7$. However, this embedding is no longer well-defined. We have $W_(f_u,f_0)=u$ for the Whitney invariant (which is defined analogously to [[Embeddings_just_below_the_stable_range#The Whitney invariant (for either n odd or N orientable)|The Whitney invariant for $m=2n$]]. </wikitex> === An embedding of CP2 into R7 === <wikitex>; \cite{Boechat&Haefliger1970}, p. 164. It suffices to construct {\it an embedding $f:\Cc P^2_0\to S^6$ such that the boundary 3-sphere is the standard one.} Recall that $\Cc P^2_0$ is the mapping cylinder of the Hopf map $\eta:S^3\to S^2$. Recall that $S^6=S^2*S^3$. Define $f[(x,t)]:=[(x,\eta(x),t)]$, 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 $\eta(x)$. </wikitex> === The Lambrechts torus and the Hudson torus === <wikitex>; \cite{Skopenkov2006}. These two embeddings $\tau^1,\tau^2:S^1\times S^3\to\Rr^7$ are defined as compositions $S^1\times S^3\overset{p_2\times t^i}\to\rightarrow S^3\times S^3\subset\Rr^7$, where $i=1,2$, $p_2$ 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 $t^2(e^{i\theta},y):=H(y)\cos\theta+\sin\theta$, where $H:S^3\to S^2$ is the Hopf map and $S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$. Note that $\tau^2$ is PL isotopic to [[Embeddings_just_below_the_stable_range#Examples#Remarks|The Hudson torus]] $\Hud_{7,4,1}$. Take the Hopf fibration $S^3\to S^7\overset\eta\to S^4$. Take the standard embeding $S^2\subset S^4$. Its complement has the homotopy type of $S^1$. Then $im\tau^1=\eta^{-1}(S^1)\cong S^1\times S^3\subset S^7$. This is the construction of Lambrechts motivated by the following property: $$S^7-im\tau^1\simeq \eta^{-1}(S^2)\cong S^2\times S^3\not\simeq S^2\vee S^3\vee S^5\simeq S^7-im f_0,$$ where $f_0:S^1\times S^3\to S^7$ is the standard embedding. </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{Haefliger1962}, \cite{Boechat&Haefliger1970}, p.165, \cite{Boechat1971}, 6.2. Take the Haefliger trefoil knot $S^3\to\Rr^6$. Extend this knot 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}, 3.7.b). </wikitex> == The Boechat-Haefliger invariant == <wikitex>; </wikitex> == Classification == <wikitex>; </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$.

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

We have $W_(f_u,f_0)=u$ for the Whitney invariant (which is defined analogously to The Whitney invariant for $m=2n$.

2.1 An embedding of CP2 into R7

[Boechat&Haefliger1970], p. 164. It suffices to construct {\it an embedding $f:\Cc P^2_0\to S^6$ such that the boundary 3-sphere is the standard one.} Recall that $\Cc P^2_0$ is the mapping cylinder of the Hopf map $\eta:S^3\to S^2$. Recall that $S^6=S^2*S^3$. Define $f[(x,t)]:=[(x,\eta(x),t)]$, where $x\in S^3$. In other words, the segment joining $x\in S^3$ and $\eta(x)\in S^2$ is mapped

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2.2 The Lambrechts torus and the Hudson torus

[Skopenkov2006]. These two embeddings $\tau^1,\tau^2:S^1\times S^3\to\Rr^7$ are defined as compositions $S^1\times S^3\overset{p_2\times t^i}\to\rightarrow S^3\times S^3\subset\Rr^7$, where $i=1,2$, $p_2$ 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 $t^2(e^{i\theta},y):=H(y)\cos\theta+\sin\theta$, where $H:S^3\to S^2$ is the Hopf map and $S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$.

Note that $\tau^2$ is PL isotopic to The Hudson torus $\Hud_{7,4,1}$.

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

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

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

2.3 The Haefliger torus

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

3 The Boechat-Haefliger invariant

4 Classification

5 References

• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$ dans $\Rr^7$, (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 $4k$ dans $\Rr^{6k+1}$, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$-spheres in $6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [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.