3-manifolds in 6-space

This page has been accepted for publication in the Bulletin of the Manifold Atlas.

For notation and conventions see high codimension embeddings.

1 The Haefliger trefoil knot

Let us construct a smooth embedding $t:S^3\to\Rr^6$$For notation and conventions see [[High_codimension_embeddings|high codimension embeddings]]. == The Haefliger trefoil knot == <wikitex>; Let us construct a smooth embedding $t:S^3\to\Rr^6$ (which is a generator of $E^6(S^3)\cong\Zz$) \cite{Haefliger1962}, 4.1. A miraculous property of this embedding is that it is not $smoothly$ isotopic to the standard embedding, but is $piecewise$ $smoothly$ isotopic to the standard embedding. Denote coordinates in $\Rr^6$ by $(x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2)$. The higher-dimensional trefoil knot $t$ is obtained by joining with two tubes the higher-dimensional $Borromean$ $rings$, i.e. the three spheres given by the following three systems of equations: $$\left\{\begin{array}{c} x=0\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\ |x|^2+2|y|^2=1 \end{array}\right..$$ See Figures 3.5 and 3.6 of \cite{Skopenkov2006}. </wikitex> == The Hopf construction of an embedding <wikitex> $\Rr P^3\to S^5$ </wikitex> == <wikitex>; Represent $\Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1.$ Define $$Ho:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad Ho[(x,y)]=(x^2,2xy,y^2).$$ It is easy to check that $Ho$ is an embedding. (The image of this embedding is given by the equations $b^2=4ac$, $|a|^2+|b|^2+|c|^2=1$.) It would be interesting to obtain an explicit construction of an embedding $f:\Rr P^3\to S^6$ which is not isotopic to the composition of the Hopf embedding with the standard inclusion $S^5\subset\Rr^6$. (Such an embedding $f$ is unique up to PL isotopy by [[Classification_just_below_the_stable_range|classification just below the stable range]].) </wikitex> == Algebraic embeddings from the theory of integrable systems == <wikitex>; Some 3-manifolds appear in the theory of integrable systems together with their embeddings into $\Rr^6$ (given by a system of algebraic equations) \cite{Bolsinov&Fomenko2004}, Chapter 14. E.g. the following system of equations corresponds to the Euler integrability case: $$R_1^2+R_2^2+R_3^2=c_1,\quad R_1S_1+R_2S_2+R_3S_3=c_2,\quad \frac{S_1^2}{A_1}+\frac{S_2^2}{A_2}+\frac{S_3^2}{A_3}=c_3,$$ where $R_i$ and $S_i$ are variables while $A_1<A_2<A_3$ and $c_i$ are constants. This defines embeddings of $S^3$, $S^1\times S^2$ or $\Rr P^3$ into $\Rr^6$. </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}t:S^3\to\Rr^6$ (which is a generator of $E^6(S^3)\cong\Zz$$E^6(S^3)\cong\Zz$) [Haefliger1962], 4.1. A miraculous property of this embedding is that it is not $smoothly$$smoothly$ isotopic to the standard embedding, but is $piecewise$$piecewise$ $smoothly$$smoothly$ isotopic to the standard embedding.

Denote coordinates in $\Rr^6$$\Rr^6$ by $(x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2)$$(x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2)$. The higher-dimensional trefoil knot $t$$t$ is obtained by joining with two tubes the higher-dimensional $Borromean$$Borromean$ $rings$$rings$, i.e. the three spheres given by the following three systems of equations:

$\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1 \end{array}\right..$ $$\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1 \end{array}\right..$$

See Figures 3.5 and 3.6 of [Skopenkov2006].

2 The Hopf construction of an embedding $\Rr P^3\to S^5$$\Rr P^3\to S^5$

Represent $\Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1.$$\Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1.$ Define

$\displaystyle Ho:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad Ho[(x,y)]=(x^2,2xy,y^2).$ $$\displaystyle Ho:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad Ho[(x,y)]=(x^2,2xy,y^2).$$

It is easy to check that $Ho$$Ho$ is an embedding. (The image of this embedding is given by the equations $b^2=4ac$$b^2=4ac$, $|a|^2+|b|^2+|c|^2=1$$|a|^2+|b|^2+|c|^2=1$.)

It would be interesting to obtain an explicit construction of an embedding $f:\Rr P^3\to S^6$$f:\Rr P^3\to S^6$ which is not isotopic to the composition of the Hopf embedding with the standard inclusion $S^5\subset\Rr^6$$S^5\subset\Rr^6$. (Such an embedding $f$$f$ is unique up to PL isotopy by classification just below the stable range.)

3 Algebraic embeddings from the theory of integrable systems

Some 3-manifolds appear in the theory of integrable systems together with their embeddings into $\Rr^6$$\Rr^6$ (given by a system of algebraic equations) [Bolsinov&Fomenko2004], Chapter 14. E.g. the following system of equations corresponds to the Euler integrability case:

$\displaystyle R_1^2+R_2^2+R_3^2=c_1,\quad R_1S_1+R_2S_2+R_3S_3=c_2,\quad \frac{S_1^2}{A_1}+\frac{S_2^2}{A_2}+\frac{S_3^2}{A_3}=c_3,$ $$\displaystyle R_1^2+R_2^2+R_3^2=c_1,\quad R_1S_1+R_2S_2+R_3S_3=c_2,\quad \frac{S_1^2}{A_1}+\frac{S_2^2}{A_2}+\frac{S_3^2}{A_3}=c_3,$$

where $R_i$$R_i$ and $S_i$$S_i$ are variables while $A_1<A_2<A_3$$A_1<A_2<A_3$ and $c_i$$c_i$ are constants. This defines embeddings of $S^3$$S^3$, $S^1\times S^2$$S^1\times S^2$ or $\Rr P^3$$\Rr P^3$ into $\Rr^6$$\Rr^6$.

4 References

• [Bolsinov&Fomenko2004] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems, Chapman \& Hall/CRC, Boca Raton, FL, 2004. MR2036760 (2004j:37106) Zbl 1056.37075
• [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
• [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.

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