Embeddings just below the stable range: classification

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{{beginthm|Theorem}}\label{hico} Let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Then [[Embeddings just below the stable range: classification#the Whitney invariant|the Whitney invariant]]
+
{{beginthm|Theorem}}\label{hico} Let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Then [[Embeddings just below the stable range: classification#The Whitney invariant|the Whitney invariant]]
$$W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})$$
$$W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})$$
is a bijection, provided $n\ge k+3$ or $n\ge2k+4$ in the PL or DIFF categories, respectively.
is a bijection, provided $n\ge k+3$ or $n\ge2k+4$ in the PL or DIFF categories, respectively.
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Because of the existence of knots the analogues of Theorem \ref{hico} for $n=k+2$ in the PL case, and for $n\le2k+3$ in the smooth case are false. So for the smooth category and $n\le2k+3$ a classification is much harder: for 40 years the ''only'' known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction \cite{Boechat1971}.
Because of the existence of knots the analogues of Theorem \ref{hico} for $n=k+2$ in the PL case, and for $n\le2k+3$ in the smooth case are false. So for the smooth category and $n\le2k+3$ a classification is much harder: for 40 years the ''only'' known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction \cite{Boechat1971}.
{{beginthm|Theorem}}\label{hicod} \cite{Skopenkov2008} Let $N$ be a closed homologically $(2k-2)$-connected $(4k-1)$-manifold. Then [[Embeddings just below the stable range: classification#the Whitney invariant|the Whitney invariant]]
+
{{beginthm|Theorem}}\label{hicod} \cite{Skopenkov2008} Let $N$ be a closed homologically $(2k-2)$-connected $(4k-1)$-manifold. Then [[Embeddings just below the stable range: classification#The Whitney invariant|the Whitney invariant]]
$$W:E^{6k}_D(N)\to H_{2k-1}(N)$$
$$W:E^{6k}_D(N)\to H_{2k-1}(N)$$
is surjective and for each $u\in H_{2k-1}(N)$ [[3-manifolds in 6-space#The Kreck invariant|the Kreck invariant]]
is surjective and for each $u\in H_{2k-1}(N)$ [[3-manifolds in 6-space#The Kreck invariant|the Kreck invariant]]

Revision as of 22:47, 19 March 2010

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

Contents

1 Introduction

Recall the unknotting theorem that if N is a connected manifold of dimension n>1, then there is just one isotopy class of embedding N \to\Rr^m if m \ge2n+1. In this page we summarise the situation for m=2n\ge6 and some more general situations.

For notation and conventions see high codimension embeddings.

2 Classification

Theorem 2.1. Let N be a closed connected n-manifold. The Whitney invariant

\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\  \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.

is bijective if either n\ge4 or n=3 and CAT=PL.

This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).

The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. See a description of E^{2n-k}(N) and of E^{2n-2k+1}(N) for a closed k-connected n-manifold N. Some estimations of E^{2n-k-1}(N) for a closed k-connected n-manifold N are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.

3 Examples

Together with the Haefliger knotted sphere, Hudson's examples were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction [Hudson1963] was not as explicit as the one below).

For m\ge n+2 define the standard embedding S^p\times S^{n-p}\to S^m as the composition S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m of standard embeddings.

3.1 Hudson tori 1

In this subsection we construct, for a\in\Zz and n\ge2, an embedding

\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.

Take the standard embeddings 2D^{n+1}\times S^{n-1}\subset\Rr^{2n} (where 2 means homothety with coefficient 2) and \partial D^2\subset\partial D^{n+1}. Fix a point x\in S^{n-1}. The Hudson torus \Hud_n(1) is the embedded connected sum of

\displaystyle 2\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.

(Unlike the unlinked embedded connected sum this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

For a\in\Zz instead of an embedded n-sphere 2\partial D^{n+1}\times x we can take a copies (1+\frac1k)\partial D^{n+1}\times x (k=1,\dots,a) of n-sphere outside D^{n+1}\times S^{n-1} `parallel' to \partial D^{n+1}\times x. Then we join these spheres by tubes so that the homotopy class of the resulting embedding

\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n  \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.

Let \Hud_n(a) be the connected sum of this embedding with the above standard embedding \partial D^2\times S^{n-1}\subset\Rr^{2n}.

Clearly, \Hud_n(0) is isotopic to the standard embedding.

Proposition 3.1. For n\ge3 odd \Hud_n(a) is isotopic to \Hud_n(a') if and only if a=a'.

For n\ge4 even \Hud_n(a) is isotopic to \Hud_n(a') if and only if a\equiv a'\mod2.

In particular, \Hud_n(1) is not isotopic to \Hud_n(0) for each n (this was the original motivation for Hudson).

Proposition 3.1 follows by the fifth part of Remark 4.1 and, for n even, by Theorem 2.1. It would be interesting to find an explicit construction of an isotopy between \Hud_{2k}(a) and \Hud_{2k}(a+2) (cf. [Vrabec1977], \S6) and to prove the analogue of Proposition 3.1 for n=2.

3.2 Hudson tori 2

In this subsection we give, for a\in\Zz and n\ge2, another construction of embeddings

\displaystyle \Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.

Define a map S^0\times S^{n-1}\to D^n to be the constant 0\in D^n on one component 1\times S^{n-1} and the standard embedding \varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n on the other component. This map gives an embedding

\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.

(See Figure 2.2 of [Skopenkov2006].) Each disk D^{n+1}\times x intersects the image of this embedding at two points lying in D^n\times x. Extend this embedding S^0\to D^n\times x for each x\in S^{n-1} to an embedding S^1\to D^{n+1}\times x. (See Figure 2.3 of [Skopenkov2006].) Thus we obtain the Hudson torus

\displaystyle \Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.

The embedding \Hud_n'(a) is obtained in the same way starting from a map \varphi of degree a.

The same proposition as above holds with \Hud_n replaced to \Hud_n'.

3.3 Remarks

We have \Hud_n(a) is PL isotopic to \Hud_n'(a) [Skopenkov2006a]. It would be interesting to prove the smooth analogue of this result.

For n=1 these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of S^1\times S^1.

Analogously one constructs the (left) Hudson torus \Hud_{4,2}(a):S^2\times S^2\to\Rr^7 for a\in\Zz or, more generally, \Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m for a\in\pi_n(S^{m-n+p-1}) and \Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m for a\in\pi_{n-p}(S^{m-n-1}).

3.4 An action of the first homology group on embeddings

In this subsection, for a\in H_1(N) and for a closed connected orientable n-manifold N with n\ge3, we construct an embedding f_a:N\to\Rr^{2n} from an embedding f_0:N\to\Rr^{2n}.

Represent a by an embedding a:S^1\to N. Since any orientable bundle over S^1 is trivial, \nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}. Identify \nu_{f_0}^{-1}a(S^1) with S^1\times S^{n-1}. It remains to make an embedded surgery of S^1\times*\subset S^1\times S^{n-1} to obtain an n-sphere \Sigma\subset C_{f_0}, and then we set f_a:=f_0\# \Sigma.

Take a vector field on S^1\times* normal to S^1\times S^{n-1}. Extend S^1\times* along this vector field to a smooth map \overline a:D^2\to S^{2n}. Since 2n>4 and n+2<2n, by general position we may assume that \overline a is an embedding and \overline a(Int D^2) misses f_0(N)\cup S^1\times S^{n-1}. Since n-1>1, we have \pi_1(V_{2n-2,n-1})=0. Hence the standard framing of S^1\times* in S^1\times S^{n-1} extends to an (n-1)-framing on \overline a(D^2) in \Rr^{2n}. Thus \overline a extends to an embedding

\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}
\displaystyle \mbox{Let}\qquad  \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.

This construction generalizes the construction of \Hud_n(a) (from \Hud_n(0)).

Clearly, W(f_a) is a or a\mod2. Thus unless n=3 and CAT=DIFF

  • all isotopy classes of embedings N\to\Rr^{2n} can be obtained (from a certain given embedding f_0) by the above construction;
  • the above construction defines an action H_1(N;\Zz_{(n-1)})\to E^{2n}(N).

4 The Whitney invariant (for either n even or N orientable)

Fix orientations on \Rr^{2n} and, if n is odd, on N. Fix an embedding f_0:N\to\Rr^{2n}. For an embedding f:N\to\Rr^{2n} the restrictions of f and f_0 to N_0 are regular homotopic [Hirsch1959]. Since N_0 has an (n-1)-dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of f and assume that f=f_0 on N_0. Take a general position homotopy F:B^n\times I\to\Rr^{2n} relative to \partial B^n between the restrictions of f and g to B^n. Then f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I) (i.e. `the intersection of this homotopy with f(N-B^n)') is a 1-manifold (possibly non-compact) without boundary. Define W(f) to be the homology class of the closure of this 1-manifold:

\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n-1)})\cong H_1(N;\Zz_{(n-1)}).

The orientation on f\cap F is defined for N orientable as follows. (This orientation is defined for each n but used only for odd n.) For each point x\in f\cap F take a vector at x tangent to f\cap F. Complete this vector to a positive base tangent to N. Since n+2(n+1)>2\cdot2n, by general position there is a unique point y\in B^n\times I such that Fy=fx. The tangent vector at x thus gives a tangent vector at y to B^n\times I. Complete this vector to a positive base tangent to B^n\times I, where the orientation on B^n comes from N. The union of the images of the constructed two bases is a base at Fy=fx of \Rr^{2n}. If this base is positive, then call the initial vector of f\cap F positive. Since a change of the orientation on f\cap F forces a change of the orientation of the latter base of \Rr^{2n}, it follows that this condition indeed defines an orientation on f\cap F.

Remark 4.1.

  • The Whitney invariant is well-defined, i.e. independent of the choice of F and of the isotopy making f=f_0 outside B^n. This is so because the above definition is clearly equivalent to an alternative one. It is for being well-defined that we need \Zz_2-coefficients when n is even.
  • Clearly, W(f_0)=0. The definition of W depends on the choice of f_0, but we write W not W_{f_0} for brevity.
  • Since a change of the orientation on N forces a change of the orientation on B^n, the class W(f) is independent of the choice of the orientation on N. For the reflection \sigma:\Rr^{2n}\to\Rr^{2n} with respect to a hyperplane we have W(\sigma\circ f)=-W(f) (because we may assume that f=f_0=\sigma\circ f on N_0 and because a change of the orientation of \Rr^{2n} forces a change of the orientation of f\cap F).
  • The above definition makes sense for each n, not only for n\ge3.
  • Clearly, W(\Hud_n(a)) is a or a\mod2 for n\ge2 for the Hudson tori.
  • W(f\#g)=W(f) for each embeddings f:N\to\Rr^{2n} and g:S^n\to\Rr^{2n}.

5 A generalization to highly-connected manifolds

Examples are Hudson tori \Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m.

5.1 Classification

Theorem 5.1. Let N be a closed orientable homologically k-connected n-manifold, k\ge0. Then the Whitney invariant

\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})

is a bijection, provided n\ge k+3 or n\ge2k+4 in the PL or DIFF categories, respectively.

Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically k-connected manifolds. The proof works for homologically k-connected manifolds.

For k=0 this is covered by Theorem 2.1; for k\ge1 it is not. The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for k+3\le n\le2k+1.

E.g. by Theorem 5.1 the Whitney invariant W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)} is bijective for 1\le p\le q-2. It is in fact a group isomorphism; the generator is the Hudson torus.

Because of the existence of knots the analogues of Theorem 5.1 for n=k+2 in the PL case, and for n\le2k+3 in the smooth case are false. So for the smooth category and n\le2k+3 a classification is much harder: for 40 years the only known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

Theorem 5.2. [Skopenkov2008] Let N be a closed homologically (2k-2)-connected (4k-1)-manifold. Then the Whitney invariant

\displaystyle W:E^{6k}_D(N)\to H_{2k-1}(N)

is surjective and for each u\in H_{2k-1}(N) the Kreck invariant

\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}

is a 1-1 correspondence, where d(u) is the divisibility of the projection of u to the free part of H_1(N).

Recall that the divisibility of zero is zero and the divisibility of x\in G-\{0\} is \max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}.

E.g. by Theorem 5.2 the Whitney invariant W:E^{6k}_D(S^{2k-1}\times S^{2k})\to\Zz is surjective and for each u\in\Zz there is a 1-1 correspondence W^{-1}u\to\Zz_u.

Theorem 5.3. [Becker&Glover1971] Let N be a closed k-connected n-manifold embeddable into \Rr^m, m\ge2n-2k+1 and 2m\ge 3n+4. Then there is a 1-1 correspondence

\displaystyle E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.

For k=0,1 this is covered by Theorem 5.1; for k\ge2 it is not.

E.g. by Theorem 5.3 there is a 1-1 correspondence E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X), X:=V_{M,M+p+q-m+1}, for M>p+q, m\ge2q+3 and 2m\ge3q+3p+4. For a generalization see Knotted tori [Skopenkov2002].

5.2 The Whitney invariant

Let N be an n-manifold and f,f_0:N\to\Rr^m embeddings. Roughly speaking, W(f,f_0)=W_{f_0}(f) is defined as the homology class of the self-intersection set \Sigma(H) of a general position homotopy H between f and f_0. We present an accurate definition in the smooth category for m\ge n+2 when either m-n is even or N is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.

Fix orientations on N and on \Rr^m. Take embeddings f,f_0:N\to\Rr^m. Take a general position homotopy H:N\times I\to\Rr^m\times I between f_0 and f. By general position the closure Cl\Sigma(H) of the self-intersection set has codimension 2 singularities and so carries a homology class with \Zz_2 coefficients. (Note that Cl\Sigma(H) can be assumed to be a submanifold for 2m\ge3n+2.) For m-n odd it has a natural orientation and so carries a homology class with \Zz coefficients. Define the Whitney invariant

\displaystyle W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})

by W(f)=W_{f_0}(f):=[Cl\Sigma(H)]. Analogously to [Skopenkov2006], \S2.4, this is well-defined.

6 An orientation on the self-intersection set

Let f:N\to\Rr^m be a general position smooth map of an orientable n-manifold N. Assume that m\ge n+2 so that the closure Cl\Sigma(f) of the self-intersection set of f has codimension 2 singularities. Then

  • (1) \Sigma(f) has a natural orientation.
  • (2) the natural orientation on \Sigma(f) need not extend to Cl\Sigma(f).
  • (3) the natural orientation on \Sigma(f) extend to Cl\Sigma(f) if m-n is odd [Hudson1969], Lemma 11.4.
  • (4) f\Sigma(f) has a natural orientation if m-n is even.

Fix an orientation on N and on \Rr^m.

Let us prove (1). Take points x,y\in N outside singularities of \Sigma(f) and such that fx=fy. Then a (2n-m)-base \xi_x tangent to \Sigma(f) at x gives a (2n-m)-base \xi_y:=df_y^{-1}df_x(\xi_x) tangent to \Sigma(f) at y. Since N is orientable, we can take positive (m-n)-bases \eta_x and \eta_y at x and y normal to \xi_x and to \xi_y. If the base (df_x(\xi_x),df_x(\eta_x),df_y(\eta_y)) of \Rr^m is positive, then call the base \xi_x positive. This is well-defined because a change of the sign of \xi_x forces changes of the signs of \xi_y,\eta_x and \eta_y. (Note that a change of the orientation of N forces changes of the signs of \eta_x and \eta_y and so does not change the orientation of \Sigma(f).)

We can see that (2) holds by considering the cone D^3\to\Rr^5 over a general position map S^2\to\Rr^4 having only one self-intersection point.

Let us prove (4). Take a (2n-m)-base \xi at a point x\in f\Sigma(f) outside singularities of f\Sigma(f). Since N is orientable, we can take a positive (m-n)-base \eta_+ normal to f\Sigma(f) in one sheet of f(N). Analogously construct an (m-n)-base \eta_- for the other sheet of f(N). If m-n is even, then the orientation of the base (\xi,\eta_+,\eta_-) of \Rr^m does not depend on choosing the first and the other sheet of f(N). If the base (\xi,\eta_+,\eta_-) is positive, then call the base \xi positive. This is well-defined because a change of the sign of \xi forces changes of the signs of \eta_+,\eta_- and so of (\xi,\eta_+,\eta_-). (Note that a change of the orientation of N forces changes of the signs of \eta_+,\eta_- and so does not change the orientation of f\Sigma(f).)

7 References

arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013

This page has not been refereed. The information given here might be incomplete or provisional.

\in D^n$ on one component N is a connected manifold of dimension n>1, then there is just one isotopy class of embedding N \to\Rr^m if m \ge2n+1. In this page we summarise the situation for m=2n\ge6 and some more general situations.

For notation and conventions see high codimension embeddings.

2 Classification

Theorem 2.1. Let N be a closed connected n-manifold. The Whitney invariant

\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\  \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.

is bijective if either n\ge4 or n=3 and CAT=PL.

This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).

The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. See a description of E^{2n-k}(N) and of E^{2n-2k+1}(N) for a closed k-connected n-manifold N. Some estimations of E^{2n-k-1}(N) for a closed k-connected n-manifold N are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.

3 Examples

Together with the Haefliger knotted sphere, Hudson's examples were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction [Hudson1963] was not as explicit as the one below).

For m\ge n+2 define the standard embedding S^p\times S^{n-p}\to S^m as the composition S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m of standard embeddings.

3.1 Hudson tori 1

In this subsection we construct, for a\in\Zz and n\ge2, an embedding

\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.

Take the standard embeddings 2D^{n+1}\times S^{n-1}\subset\Rr^{2n} (where 2 means homothety with coefficient 2) and \partial D^2\subset\partial D^{n+1}. Fix a point x\in S^{n-1}. The Hudson torus \Hud_n(1) is the embedded connected sum of

\displaystyle 2\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.

(Unlike the unlinked embedded connected sum this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

For a\in\Zz instead of an embedded n-sphere 2\partial D^{n+1}\times x we can take a copies (1+\frac1k)\partial D^{n+1}\times x (k=1,\dots,a) of n-sphere outside D^{n+1}\times S^{n-1} `parallel' to \partial D^{n+1}\times x. Then we join these spheres by tubes so that the homotopy class of the resulting embedding

\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n  \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.

Let \Hud_n(a) be the connected sum of this embedding with the above standard embedding \partial D^2\times S^{n-1}\subset\Rr^{2n}.

Clearly, \Hud_n(0) is isotopic to the standard embedding.

Proposition 3.1. For n\ge3 odd \Hud_n(a) is isotopic to \Hud_n(a') if and only if a=a'.

For n\ge4 even \Hud_n(a) is isotopic to \Hud_n(a') if and only if a\equiv a'\mod2.

In particular, \Hud_n(1) is not isotopic to \Hud_n(0) for each n (this was the original motivation for Hudson).

Proposition 3.1 follows by the fifth part of Remark 4.1 and, for n even, by Theorem 2.1. It would be interesting to find an explicit construction of an isotopy between \Hud_{2k}(a) and \Hud_{2k}(a+2) (cf. [Vrabec1977], \S6) and to prove the analogue of Proposition 3.1 for n=2.

3.2 Hudson tori 2

In this subsection we give, for a\in\Zz and n\ge2, another construction of embeddings

\displaystyle \Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.

Define a map S^0\times S^{n-1}\to D^n to be the constant 0\in D^n on one component 1\times S^{n-1} and the standard embedding \varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n on the other component. This map gives an embedding

\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.

(See Figure 2.2 of [Skopenkov2006].) Each disk D^{n+1}\times x intersects the image of this embedding at two points lying in D^n\times x. Extend this embedding S^0\to D^n\times x for each x\in S^{n-1} to an embedding S^1\to D^{n+1}\times x. (See Figure 2.3 of [Skopenkov2006].) Thus we obtain the Hudson torus

\displaystyle \Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.

The embedding \Hud_n'(a) is obtained in the same way starting from a map \varphi of degree a.

The same proposition as above holds with \Hud_n replaced to \Hud_n'.

3.3 Remarks

We have \Hud_n(a) is PL isotopic to \Hud_n'(a) [Skopenkov2006a]. It would be interesting to prove the smooth analogue of this result.

For n=1 these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of S^1\times S^1.

Analogously one constructs the (left) Hudson torus \Hud_{4,2}(a):S^2\times S^2\to\Rr^7 for a\in\Zz or, more generally, \Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m for a\in\pi_n(S^{m-n+p-1}) and \Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m for a\in\pi_{n-p}(S^{m-n-1}).

3.4 An action of the first homology group on embeddings

In this subsection, for a\in H_1(N) and for a closed connected orientable n-manifold N with n\ge3, we construct an embedding f_a:N\to\Rr^{2n} from an embedding f_0:N\to\Rr^{2n}.

Represent a by an embedding a:S^1\to N. Since any orientable bundle over S^1 is trivial, \nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}. Identify \nu_{f_0}^{-1}a(S^1) with S^1\times S^{n-1}. It remains to make an embedded surgery of S^1\times*\subset S^1\times S^{n-1} to obtain an n-sphere \Sigma\subset C_{f_0}, and then we set f_a:=f_0\# \Sigma.

Take a vector field on S^1\times* normal to S^1\times S^{n-1}. Extend S^1\times* along this vector field to a smooth map \overline a:D^2\to S^{2n}. Since 2n>4 and n+2<2n, by general position we may assume that \overline a is an embedding and \overline a(Int D^2) misses f_0(N)\cup S^1\times S^{n-1}. Since n-1>1, we have \pi_1(V_{2n-2,n-1})=0. Hence the standard framing of S^1\times* in S^1\times S^{n-1} extends to an (n-1)-framing on \overline a(D^2) in \Rr^{2n}. Thus \overline a extends to an embedding

\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}
\displaystyle \mbox{Let}\qquad  \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.

This construction generalizes the construction of \Hud_n(a) (from \Hud_n(0)).

Clearly, W(f_a) is a or a\mod2. Thus unless n=3 and CAT=DIFF

  • all isotopy classes of embedings N\to\Rr^{2n} can be obtained (from a certain given embedding f_0) by the above construction;
  • the above construction defines an action H_1(N;\Zz_{(n-1)})\to E^{2n}(N).

4 The Whitney invariant (for either n even or N orientable)

Fix orientations on \Rr^{2n} and, if n is odd, on N. Fix an embedding f_0:N\to\Rr^{2n}. For an embedding f:N\to\Rr^{2n} the restrictions of f and f_0 to N_0 are regular homotopic [Hirsch1959]. Since N_0 has an (n-1)-dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of f and assume that f=f_0 on N_0. Take a general position homotopy F:B^n\times I\to\Rr^{2n} relative to \partial B^n between the restrictions of f and g to B^n. Then f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I) (i.e. `the intersection of this homotopy with f(N-B^n)') is a 1-manifold (possibly non-compact) without boundary. Define W(f) to be the homology class of the closure of this 1-manifold:

\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n-1)})\cong H_1(N;\Zz_{(n-1)}).

The orientation on f\cap F is defined for N orientable as follows. (This orientation is defined for each n but used only for odd n.) For each point x\in f\cap F take a vector at x tangent to f\cap F. Complete this vector to a positive base tangent to N. Since n+2(n+1)>2\cdot2n, by general position there is a unique point y\in B^n\times I such that Fy=fx. The tangent vector at x thus gives a tangent vector at y to B^n\times I. Complete this vector to a positive base tangent to B^n\times I, where the orientation on B^n comes from N. The union of the images of the constructed two bases is a base at Fy=fx of \Rr^{2n}. If this base is positive, then call the initial vector of f\cap F positive. Since a change of the orientation on f\cap F forces a change of the orientation of the latter base of \Rr^{2n}, it follows that this condition indeed defines an orientation on f\cap F.

Remark 4.1.

  • The Whitney invariant is well-defined, i.e. independent of the choice of F and of the isotopy making f=f_0 outside B^n. This is so because the above definition is clearly equivalent to an alternative one. It is for being well-defined that we need \Zz_2-coefficients when n is even.
  • Clearly, W(f_0)=0. The definition of W depends on the choice of f_0, but we write W not W_{f_0} for brevity.
  • Since a change of the orientation on N forces a change of the orientation on B^n, the class W(f) is independent of the choice of the orientation on N. For the reflection \sigma:\Rr^{2n}\to\Rr^{2n} with respect to a hyperplane we have W(\sigma\circ f)=-W(f) (because we may assume that f=f_0=\sigma\circ f on N_0 and because a change of the orientation of \Rr^{2n} forces a change of the orientation of f\cap F).
  • The above definition makes sense for each n, not only for n\ge3.
  • Clearly, W(\Hud_n(a)) is a or a\mod2 for n\ge2 for the Hudson tori.
  • W(f\#g)=W(f) for each embeddings f:N\to\Rr^{2n} and g:S^n\to\Rr^{2n}.

5 A generalization to highly-connected manifolds

Examples are Hudson tori \Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m.

5.1 Classification

Theorem 5.1. Let N be a closed orientable homologically k-connected n-manifold, k\ge0. Then the Whitney invariant

\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})

is a bijection, provided n\ge k+3 or n\ge2k+4 in the PL or DIFF categories, respectively.

Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically k-connected manifolds. The proof works for homologically k-connected manifolds.

For k=0 this is covered by Theorem 2.1; for k\ge1 it is not. The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for k+3\le n\le2k+1.

E.g. by Theorem 5.1 the Whitney invariant W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)} is bijective for 1\le p\le q-2. It is in fact a group isomorphism; the generator is the Hudson torus.

Because of the existence of knots the analogues of Theorem 5.1 for n=k+2 in the PL case, and for n\le2k+3 in the smooth case are false. So for the smooth category and n\le2k+3 a classification is much harder: for 40 years the only known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

Theorem 5.2. [Skopenkov2008] Let N be a closed homologically (2k-2)-connected (4k-1)-manifold. Then the Whitney invariant

\displaystyle W:E^{6k}_D(N)\to H_{2k-1}(N)

is surjective and for each u\in H_{2k-1}(N) the Kreck invariant

\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}

is a 1-1 correspondence, where d(u) is the divisibility of the projection of u to the free part of H_1(N).

Recall that the divisibility of zero is zero and the divisibility of x\in G-\{0\} is \max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}.

E.g. by Theorem 5.2 the Whitney invariant W:E^{6k}_D(S^{2k-1}\times S^{2k})\to\Zz is surjective and for each u\in\Zz there is a 1-1 correspondence W^{-1}u\to\Zz_u.

Theorem 5.3. [Becker&Glover1971] Let N be a closed k-connected n-manifold embeddable into \Rr^m, m\ge2n-2k+1 and 2m\ge 3n+4. Then there is a 1-1 correspondence

\displaystyle E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.

For k=0,1 this is covered by Theorem 5.1; for k\ge2 it is not.

E.g. by Theorem 5.3 there is a 1-1 correspondence E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X), X:=V_{M,M+p+q-m+1}, for M>p+q, m\ge2q+3 and 2m\ge3q+3p+4. For a generalization see Knotted tori [Skopenkov2002].

5.2 The Whitney invariant

Let N be an n-manifold and f,f_0:N\to\Rr^m embeddings. Roughly speaking, W(f,f_0)=W_{f_0}(f) is defined as the homology class of the self-intersection set \Sigma(H) of a general position homotopy H between f and f_0. We present an accurate definition in the smooth category for m\ge n+2 when either m-n is even or N is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.

Fix orientations on N and on \Rr^m. Take embeddings f,f_0:N\to\Rr^m. Take a general position homotopy H:N\times I\to\Rr^m\times I between f_0 and f. By general position the closure Cl\Sigma(H) of the self-intersection set has codimension 2 singularities and so carries a homology class with \Zz_2 coefficients. (Note that Cl\Sigma(H) can be assumed to be a submanifold for 2m\ge3n+2.) For m-n odd it has a natural orientation and so carries a homology class with \Zz coefficients. Define the Whitney invariant

\displaystyle W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})

by W(f)=W_{f_0}(f):=[Cl\Sigma(H)]. Analogously to [Skopenkov2006], \S2.4, this is well-defined.

6 An orientation on the self-intersection set

Let f:N\to\Rr^m be a general position smooth map of an orientable n-manifold N. Assume that m\ge n+2 so that the closure Cl\Sigma(f) of the self-intersection set of f has codimension 2 singularities. Then

  • (1) \Sigma(f) has a natural orientation.
  • (2) the natural orientation on \Sigma(f) need not extend to Cl\Sigma(f).
  • (3) the natural orientation on \Sigma(f) extend to Cl\Sigma(f) if m-n is odd [Hudson1969], Lemma 11.4.
  • (4) f\Sigma(f) has a natural orientation if m-n is even.

Fix an orientation on N and on \Rr^m.

Let us prove (1). Take points x,y\in N outside singularities of \Sigma(f) and such that fx=fy. Then a (2n-m)-base \xi_x tangent to \Sigma(f) at x gives a (2n-m)-base \xi_y:=df_y^{-1}df_x(\xi_x) tangent to \Sigma(f) at y. Since N is orientable, we can take positive (m-n)-bases \eta_x and \eta_y at x and y normal to \xi_x and to \xi_y. If the base (df_x(\xi_x),df_x(\eta_x),df_y(\eta_y)) of \Rr^m is positive, then call the base \xi_x positive. This is well-defined because a change of the sign of \xi_x forces changes of the signs of \xi_y,\eta_x and \eta_y. (Note that a change of the orientation of N forces changes of the signs of \eta_x and \eta_y and so does not change the orientation of \Sigma(f).)

We can see that (2) holds by considering the cone D^3\to\Rr^5 over a general position map S^2\to\Rr^4 having only one self-intersection point.

Let us prove (4). Take a (2n-m)-base \xi at a point x\in f\Sigma(f) outside singularities of f\Sigma(f). Since N is orientable, we can take a positive (m-n)-base \eta_+ normal to f\Sigma(f) in one sheet of f(N). Analogously construct an (m-n)-base \eta_- for the other sheet of f(N). If m-n is even, then the orientation of the base (\xi,\eta_+,\eta_-) of \Rr^m does not depend on choosing the first and the other sheet of f(N). If the base (\xi,\eta_+,\eta_-) is positive, then call the base \xi positive. This is well-defined because a change of the sign of \xi forces changes of the signs of \eta_+,\eta_- and so of (\xi,\eta_+,\eta_-). (Note that a change of the orientation of N forces changes of the signs of \eta_+,\eta_- and so does not change the orientation of f\Sigma(f).)

7 References

arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013

This page has not been refereed. The information given here might be incomplete or provisional.

\times S^{n-1}$ and the standard embedding $\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an ''embedding'' $$S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ (See Figure 2.2 of \cite{Skopenkov2006}.) Each disk $D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$. Extend this embedding $S^0\to D^n\times x$ for each $x\in S^{n-1}$ to an embedding $S^1\to D^{n+1}\times x$. (See Figure 2.3 of \cite{Skopenkov2006}.) Thus we obtain ''the Hudson torus'' $$\Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ The embedding $\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi$ of degree $a$. The same proposition as above holds with $\Hud_n$ replaced to $\Hud_n'$. === Remarks === ; We have $\Hud_n(a)$ is PL isotopic to $\Hud_n'(a)$ \cite{Skopenkov2006a}. It would be interesting to prove the smooth analogue of this result. For $n=1$ these construction give what we call the ''left'' Hudson torus. The ''right'' Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$. Analogously one constructs the (left) Hudson torus $\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$ for $a\in\Zz$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$ and $\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_{n-p}(S^{m-n-1})$. === An action of the first homology group on embeddings === ; In this subsection, for $a\in H_1(N)$ and for a closed connected orientable $n$-manifold $N$ with $n\ge3$, we construct an embedding $f_a:N\to\Rr^{2n}$ from an embedding $f_0:N\to\Rr^{2n}$. Represent $a$ by an embedding $a:S^1\to N$. Since any orientable bundle over $S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Identify $\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$. It remains to make an embedded surgery of $S^1\times*\subset S^1\times S^{n-1}$ to obtain an $n$-sphere $\Sigma\subset C_{f_0}$, and then we set $f_a:=f_0\# \Sigma$. Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$. Extend $S^1\times*$ along this vector field to a smooth map $\overline a:D^2\to S^{2n}$. Since n>4$ and $n+2<2n$, by general position we may assume that $\overline a$ is an embedding and $\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times*$ in $S^1\times S^{n-1}$ extends to an $(n-1)$-framing on $\overline a(D^2)$ in $\Rr^{2n}$. Thus $\overline a$ extends to an embedding $$\widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}$$ $$\mbox{Let}\qquad \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.$$ This construction generalizes the construction of $\Hud_n(a)$ (from $\Hud_n(0)$). Clearly, $W(f_a)$ is $a$ or $a\mod2$. Thus unless $n=3$ and CAT=DIFF * all isotopy classes of embedings $N\to\Rr^{2n}$ can be obtained (from a certain given embedding $f_0$) by the above construction; * the above construction defines an action $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$. == The Whitney invariant (for either n even or N orientable) == ; Fix orientations on $\Rr^{2n}$ and, if $n$ is odd, on $N$. Fix an embedding $f_0:N\to\Rr^{2n}$. For an embedding $f:N\to\Rr^{2n}$ the restrictions of $f$ and $f_0$ to $N_0$ are regular homotopic \cite{Hirsch1959}. Since $N_0$ has an $(n-1)$-dimensional spine, it follows that these restrictions are isotopic, cf. \cite{Haefliger&Hirsch1963}, 3.1.b, \cite{Takase2006}, Lemma 2.2. So we can make an isotopy of $f$ and assume that $f=f_0$ on $N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^{2n}$ relative to $\partial B^n$ between the restrictions of $f$ and $g$ to $B^n$. Then $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (i.e. `the intersection of this homotopy with $f(N-B^n)$') is a 1-manifold (possibly non-compact) without boundary. Define $W(f)$ to be the homology class of the closure of this 1-manifold: $$W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n-1)})\cong H_1(N;\Zz_{(n-1)}).$$ The orientation on $f\cap F$ is defined for $N$ orientable as follows. (This orientation is defined for each $n$ but used only for odd $n$.) For each point $x\in f\cap F$ take a vector at $x$ tangent to $f\cap F$. Complete this vector to a positive base tangent to $N$. Since $n+2(n+1)>2\cdot2n$, by general position there is a unique point $y\in B^n\times I$ such that $Fy=fx$. The tangent vector at $x$ thus gives a tangent vector at $y$ to $B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$, where the orientation on $B^n$ comes from $N$. The union of the images of the constructed two bases is a base at $Fy=fx$ of $\Rr^{2n}$. If this base is positive, then call the initial vector of $f\cap F$ positive. Since a change of the orientation on $f\cap F$ forces a change of the orientation of the latter base of $\Rr^{2n}$, it follows that this condition indeed defines an orientation on $f\cap F$. {{beginthm|Remark}}\label{re5} *The Whitney invariant is well-defined, i.e. independent of the choice of $F$ and of the isotopy making $f=f_0$ outside $B^n$. This is so because the above definition is clearly equivalent to [[Embeddings just below the stable range: classification#The Whitney invariant|an alternative one]]. It is for being well-defined that we need $\Zz_2$-coefficients when $n$ is even. *Clearly, $W(f_0)=0$. The definition of $W$ depends on the choice of $f_0$, but we write $W$ not $W_{f_0}$ for brevity. *Since a change of the orientation on $N$ forces a change of the orientation on $B^n$, the class $W(f)$ is independent of the choice of the orientation on $N$. For the reflection $\sigma:\Rr^{2n}\to\Rr^{2n}$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$ (because we may assume that $f=f_0=\sigma\circ f$ on $N_0$ and because a change of the orientation of $\Rr^{2n}$ forces a change of the orientation of $f\cap F$). *The above definition makes sense for each $n$, not only for $n\ge3$. *Clearly, $W(\Hud_n(a))$ is $a$ or $a\mod2$ for $n\ge2$ for the [[Embeddings just below the stable range: classification#Hudson tori 1|Hudson tori]]. *$W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$. {{endthm}} == A generalization to highly-connected manifolds == ; Examples are [[Embeddings just below the stable range: classification#Remarks|Hudson tori]] $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$. === Classification === ; {{beginthm|Theorem}}\label{hico} Let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Then [[Embeddings just below the stable range: classification#The Whitney invariant|the Whitney invariant]] $$W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})$$ is a bijection, provided $n\ge k+3$ or $n\ge2k+4$ in the PL or DIFF categories, respectively. {{endthm}} Theorem \ref{hico} was proved in \cite{Haefliger&Hirsch1963}, \cite{Hudson1969}, \S11, \cite{Boechat&Haefliger1970}, \cite{Boechat1971}, \cite{Vrabec1977} ''homotopically'' $k$-connected manifolds. The proof works for ''homologically'' $k$-connected manifolds. For $k=0$ this is covered by Theorem \ref{th4}; for $k\ge1$ it is not. The PL case of Theorem \ref{hico} gives nothing but [[High codimension embeddings: classification#Unknotting theorems|the Unknotting Spheres Theorem]] for $k+3\le n\le2k+1$. E.g. by Theorem \ref{hico} the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for N is a connected manifold of dimension n>1, then there is just one isotopy class of embedding N \to\Rr^m if m \ge2n+1. In this page we summarise the situation for m=2n\ge6 and some more general situations.

For notation and conventions see high codimension embeddings.

2 Classification

Theorem 2.1. Let N be a closed connected n-manifold. The Whitney invariant

\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\  \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.

is bijective if either n\ge4 or n=3 and CAT=PL.

This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).

The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. See a description of E^{2n-k}(N) and of E^{2n-2k+1}(N) for a closed k-connected n-manifold N. Some estimations of E^{2n-k-1}(N) for a closed k-connected n-manifold N are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.

3 Examples

Together with the Haefliger knotted sphere, Hudson's examples were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction [Hudson1963] was not as explicit as the one below).

For m\ge n+2 define the standard embedding S^p\times S^{n-p}\to S^m as the composition S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m of standard embeddings.

3.1 Hudson tori 1

In this subsection we construct, for a\in\Zz and n\ge2, an embedding

\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.

Take the standard embeddings 2D^{n+1}\times S^{n-1}\subset\Rr^{2n} (where 2 means homothety with coefficient 2) and \partial D^2\subset\partial D^{n+1}. Fix a point x\in S^{n-1}. The Hudson torus \Hud_n(1) is the embedded connected sum of

\displaystyle 2\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.

(Unlike the unlinked embedded connected sum this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

For a\in\Zz instead of an embedded n-sphere 2\partial D^{n+1}\times x we can take a copies (1+\frac1k)\partial D^{n+1}\times x (k=1,\dots,a) of n-sphere outside D^{n+1}\times S^{n-1} `parallel' to \partial D^{n+1}\times x. Then we join these spheres by tubes so that the homotopy class of the resulting embedding

\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n  \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.

Let \Hud_n(a) be the connected sum of this embedding with the above standard embedding \partial D^2\times S^{n-1}\subset\Rr^{2n}.

Clearly, \Hud_n(0) is isotopic to the standard embedding.

Proposition 3.1. For n\ge3 odd \Hud_n(a) is isotopic to \Hud_n(a') if and only if a=a'.

For n\ge4 even \Hud_n(a) is isotopic to \Hud_n(a') if and only if a\equiv a'\mod2.

In particular, \Hud_n(1) is not isotopic to \Hud_n(0) for each n (this was the original motivation for Hudson).

Proposition 3.1 follows by the fifth part of Remark 4.1 and, for n even, by Theorem 2.1. It would be interesting to find an explicit construction of an isotopy between \Hud_{2k}(a) and \Hud_{2k}(a+2) (cf. [Vrabec1977], \S6) and to prove the analogue of Proposition 3.1 for n=2.

3.2 Hudson tori 2

In this subsection we give, for a\in\Zz and n\ge2, another construction of embeddings

\displaystyle \Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.

Define a map S^0\times S^{n-1}\to D^n to be the constant 0\in D^n on one component 1\times S^{n-1} and the standard embedding \varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n on the other component. This map gives an embedding

\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.

(See Figure 2.2 of [Skopenkov2006].) Each disk D^{n+1}\times x intersects the image of this embedding at two points lying in D^n\times x. Extend this embedding S^0\to D^n\times x for each x\in S^{n-1} to an embedding S^1\to D^{n+1}\times x. (See Figure 2.3 of [Skopenkov2006].) Thus we obtain the Hudson torus

\displaystyle \Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.

The embedding \Hud_n'(a) is obtained in the same way starting from a map \varphi of degree a.

The same proposition as above holds with \Hud_n replaced to \Hud_n'.

3.3 Remarks

We have \Hud_n(a) is PL isotopic to \Hud_n'(a) [Skopenkov2006a]. It would be interesting to prove the smooth analogue of this result.

For n=1 these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of S^1\times S^1.

Analogously one constructs the (left) Hudson torus \Hud_{4,2}(a):S^2\times S^2\to\Rr^7 for a\in\Zz or, more generally, \Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m for a\in\pi_n(S^{m-n+p-1}) and \Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m for a\in\pi_{n-p}(S^{m-n-1}).

3.4 An action of the first homology group on embeddings

In this subsection, for a\in H_1(N) and for a closed connected orientable n-manifold N with n\ge3, we construct an embedding f_a:N\to\Rr^{2n} from an embedding f_0:N\to\Rr^{2n}.

Represent a by an embedding a:S^1\to N. Since any orientable bundle over S^1 is trivial, \nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}. Identify \nu_{f_0}^{-1}a(S^1) with S^1\times S^{n-1}. It remains to make an embedded surgery of S^1\times*\subset S^1\times S^{n-1} to obtain an n-sphere \Sigma\subset C_{f_0}, and then we set f_a:=f_0\# \Sigma.

Take a vector field on S^1\times* normal to S^1\times S^{n-1}. Extend S^1\times* along this vector field to a smooth map \overline a:D^2\to S^{2n}. Since 2n>4 and n+2<2n, by general position we may assume that \overline a is an embedding and \overline a(Int D^2) misses f_0(N)\cup S^1\times S^{n-1}. Since n-1>1, we have \pi_1(V_{2n-2,n-1})=0. Hence the standard framing of S^1\times* in S^1\times S^{n-1} extends to an (n-1)-framing on \overline a(D^2) in \Rr^{2n}. Thus \overline a extends to an embedding

\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}
\displaystyle \mbox{Let}\qquad  \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.

This construction generalizes the construction of \Hud_n(a) (from \Hud_n(0)).

Clearly, W(f_a) is a or a\mod2. Thus unless n=3 and CAT=DIFF

  • all isotopy classes of embedings N\to\Rr^{2n} can be obtained (from a certain given embedding f_0) by the above construction;
  • the above construction defines an action H_1(N;\Zz_{(n-1)})\to E^{2n}(N).

4 The Whitney invariant (for either n even or N orientable)

Fix orientations on \Rr^{2n} and, if n is odd, on N. Fix an embedding f_0:N\to\Rr^{2n}. For an embedding f:N\to\Rr^{2n} the restrictions of f and f_0 to N_0 are regular homotopic [Hirsch1959]. Since N_0 has an (n-1)-dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of f and assume that f=f_0 on N_0. Take a general position homotopy F:B^n\times I\to\Rr^{2n} relative to \partial B^n between the restrictions of f and g to B^n. Then f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I) (i.e. `the intersection of this homotopy with f(N-B^n)') is a 1-manifold (possibly non-compact) without boundary. Define W(f) to be the homology class of the closure of this 1-manifold:

\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n-1)})\cong H_1(N;\Zz_{(n-1)}).

The orientation on f\cap F is defined for N orientable as follows. (This orientation is defined for each n but used only for odd n.) For each point x\in f\cap F take a vector at x tangent to f\cap F. Complete this vector to a positive base tangent to N. Since n+2(n+1)>2\cdot2n, by general position there is a unique point y\in B^n\times I such that Fy=fx. The tangent vector at x thus gives a tangent vector at y to B^n\times I. Complete this vector to a positive base tangent to B^n\times I, where the orientation on B^n comes from N. The union of the images of the constructed two bases is a base at Fy=fx of \Rr^{2n}. If this base is positive, then call the initial vector of f\cap F positive. Since a change of the orientation on f\cap F forces a change of the orientation of the latter base of \Rr^{2n}, it follows that this condition indeed defines an orientation on f\cap F.

Remark 4.1.

  • The Whitney invariant is well-defined, i.e. independent of the choice of F and of the isotopy making f=f_0 outside B^n. This is so because the above definition is clearly equivalent to an alternative one. It is for being well-defined that we need \Zz_2-coefficients when n is even.
  • Clearly, W(f_0)=0. The definition of W depends on the choice of f_0, but we write W not W_{f_0} for brevity.
  • Since a change of the orientation on N forces a change of the orientation on B^n, the class W(f) is independent of the choice of the orientation on N. For the reflection \sigma:\Rr^{2n}\to\Rr^{2n} with respect to a hyperplane we have W(\sigma\circ f)=-W(f) (because we may assume that f=f_0=\sigma\circ f on N_0 and because a change of the orientation of \Rr^{2n} forces a change of the orientation of f\cap F).
  • The above definition makes sense for each n, not only for n\ge3.
  • Clearly, W(\Hud_n(a)) is a or a\mod2 for n\ge2 for the Hudson tori.
  • W(f\#g)=W(f) for each embeddings f:N\to\Rr^{2n} and g:S^n\to\Rr^{2n}.

5 A generalization to highly-connected manifolds

Examples are Hudson tori \Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m.

5.1 Classification

Theorem 5.1. Let N be a closed orientable homologically k-connected n-manifold, k\ge0. Then the Whitney invariant

\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})

is a bijection, provided n\ge k+3 or n\ge2k+4 in the PL or DIFF categories, respectively.

Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically k-connected manifolds. The proof works for homologically k-connected manifolds.

For k=0 this is covered by Theorem 2.1; for k\ge1 it is not. The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for k+3\le n\le2k+1.

E.g. by Theorem 5.1 the Whitney invariant W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)} is bijective for 1\le p\le q-2. It is in fact a group isomorphism; the generator is the Hudson torus.

Because of the existence of knots the analogues of Theorem 5.1 for n=k+2 in the PL case, and for n\le2k+3 in the smooth case are false. So for the smooth category and n\le2k+3 a classification is much harder: for 40 years the only known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

Theorem 5.2. [Skopenkov2008] Let N be a closed homologically (2k-2)-connected (4k-1)-manifold. Then the Whitney invariant

\displaystyle W:E^{6k}_D(N)\to H_{2k-1}(N)

is surjective and for each u\in H_{2k-1}(N) the Kreck invariant

\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}

is a 1-1 correspondence, where d(u) is the divisibility of the projection of u to the free part of H_1(N).

Recall that the divisibility of zero is zero and the divisibility of x\in G-\{0\} is \max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}.

E.g. by Theorem 5.2 the Whitney invariant W:E^{6k}_D(S^{2k-1}\times S^{2k})\to\Zz is surjective and for each u\in\Zz there is a 1-1 correspondence W^{-1}u\to\Zz_u.

Theorem 5.3. [Becker&Glover1971] Let N be a closed k-connected n-manifold embeddable into \Rr^m, m\ge2n-2k+1 and 2m\ge 3n+4. Then there is a 1-1 correspondence

\displaystyle E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.

For k=0,1 this is covered by Theorem 5.1; for k\ge2 it is not.

E.g. by Theorem 5.3 there is a 1-1 correspondence E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X), X:=V_{M,M+p+q-m+1}, for M>p+q, m\ge2q+3 and 2m\ge3q+3p+4. For a generalization see Knotted tori [Skopenkov2002].

5.2 The Whitney invariant

Let N be an n-manifold and f,f_0:N\to\Rr^m embeddings. Roughly speaking, W(f,f_0)=W_{f_0}(f) is defined as the homology class of the self-intersection set \Sigma(H) of a general position homotopy H between f and f_0. We present an accurate definition in the smooth category for m\ge n+2 when either m-n is even or N is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.

Fix orientations on N and on \Rr^m. Take embeddings f,f_0:N\to\Rr^m. Take a general position homotopy H:N\times I\to\Rr^m\times I between f_0 and f. By general position the closure Cl\Sigma(H) of the self-intersection set has codimension 2 singularities and so carries a homology class with \Zz_2 coefficients. (Note that Cl\Sigma(H) can be assumed to be a submanifold for 2m\ge3n+2.) For m-n odd it has a natural orientation and so carries a homology class with \Zz coefficients. Define the Whitney invariant

\displaystyle W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})

by W(f)=W_{f_0}(f):=[Cl\Sigma(H)]. Analogously to [Skopenkov2006], \S2.4, this is well-defined.

6 An orientation on the self-intersection set

Let f:N\to\Rr^m be a general position smooth map of an orientable n-manifold N. Assume that m\ge n+2 so that the closure Cl\Sigma(f) of the self-intersection set of f has codimension 2 singularities. Then

  • (1) \Sigma(f) has a natural orientation.
  • (2) the natural orientation on \Sigma(f) need not extend to Cl\Sigma(f).
  • (3) the natural orientation on \Sigma(f) extend to Cl\Sigma(f) if m-n is odd [Hudson1969], Lemma 11.4.
  • (4) f\Sigma(f) has a natural orientation if m-n is even.

Fix an orientation on N and on \Rr^m.

Let us prove (1). Take points x,y\in N outside singularities of \Sigma(f) and such that fx=fy. Then a (2n-m)-base \xi_x tangent to \Sigma(f) at x gives a (2n-m)-base \xi_y:=df_y^{-1}df_x(\xi_x) tangent to \Sigma(f) at y. Since N is orientable, we can take positive (m-n)-bases \eta_x and \eta_y at x and y normal to \xi_x and to \xi_y. If the base (df_x(\xi_x),df_x(\eta_x),df_y(\eta_y)) of \Rr^m is positive, then call the base \xi_x positive. This is well-defined because a change of the sign of \xi_x forces changes of the signs of \xi_y,\eta_x and \eta_y. (Note that a change of the orientation of N forces changes of the signs of \eta_x and \eta_y and so does not change the orientation of \Sigma(f).)

We can see that (2) holds by considering the cone D^3\to\Rr^5 over a general position map S^2\to\Rr^4 having only one self-intersection point.

Let us prove (4). Take a (2n-m)-base \xi at a point x\in f\Sigma(f) outside singularities of f\Sigma(f). Since N is orientable, we can take a positive (m-n)-base \eta_+ normal to f\Sigma(f) in one sheet of f(N). Analogously construct an (m-n)-base \eta_- for the other sheet of f(N). If m-n is even, then the orientation of the base (\xi,\eta_+,\eta_-) of \Rr^m does not depend on choosing the first and the other sheet of f(N). If the base (\xi,\eta_+,\eta_-) is positive, then call the base \xi positive. This is well-defined because a change of the sign of \xi forces changes of the signs of \eta_+,\eta_- and so of (\xi,\eta_+,\eta_-). (Note that a change of the orientation of N forces changes of the signs of \eta_+,\eta_- and so does not change the orientation of f\Sigma(f).)

7 References

arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013

This page has not been refereed. The information given here might be incomplete or provisional.

\le p\le q-2$. It is in fact a group isomorphism; the generator is [[Embeddings just below the stable range: classification#Hudson tori 1|the Hudson torus]]. Because of the existence of knots the analogues of Theorem \ref{hico} for $n=k+2$ in the PL case, and for $n\le2k+3$ in the smooth case are false. So for the smooth category and $n\le2k+3$ a classification is much harder: for 40 years the ''only'' known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction \cite{Boechat1971}. {{beginthm|Theorem}}\label{hicod} \cite{Skopenkov2008} Let $N$ be a closed homologically $(2k-2)$-connected $(4k-1)$-manifold. Then [[Embeddings just below the stable range: classification#The Whitney invariant|the Whitney invariant]] $$W:E^{6k}_D(N)\to H_{2k-1}(N)$$ is surjective and for each $u\in H_{2k-1}(N)$ [[3-manifolds in 6-space#The Kreck invariant|the Kreck invariant]] $$\eta_u:W^{-1}u\to\Zz_{d(u)}$$ is a 1-1 correspondence, where $d(u)$ is the divisibility of the projection of $u$ to the free part of $H_1(N)$. {{endthm}} Recall that the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$. E.g. by Theorem \ref{hicod} the Whitney invariant $W:E^{6k}_D(S^{2k-1}\times S^{2k})\to\Zz$ is surjective and for each $u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$. {{beginthm|Theorem}}\label{begl} \cite{Becker&Glover1971} Let $N$ be a closed $k$-connected $n$-manifold embeddable into $\Rr^m$, $m\ge2n-2k+1$ and m\ge 3n+4$. Then there is a 1-1 correspondence $$E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.$$ {{endthm}} For $k=0,1$ this is covered by Theorem \ref{hico}; for $k\ge2$ it is not. E.g. by Theorem \ref{begl} there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X)$, $X:=V_{M,M+p+q-m+1}$, for $M>p+q$, $m\ge2q+3$ and m\ge3q+3p+4$. For a generalization see [[Knotted tori|Knotted tori]] \cite{Skopenkov2002}.
=== The Whitney invariant === ; Let $N$ be an $n$-manifold and $f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ of a general position homotopy $H$ between $f$ and $f_0$. We present an accurate definition in the smooth category for $m\ge n+2$ when either $m-n$ is even or $N$ is orientable \cite{Skopenkov2010}. The definition in the PL category is analogous \cite{Hudson1969}, \S12, \cite{Vrabec1977}, p. 145, \cite{Skopenkov2006}, \S2.4. Fix orientations on $N$ and on $\Rr^m$. Take embeddings $f,f_0:N\to\Rr^m$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$. By general position the closure $Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities and so carries a homology class with $\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$ can be assumed to be a submanifold for m\ge3n+2$.) For $m-n$ odd it has a [[Embeddings just below the stable range: classification#An orientation on the self-intersection set|natural orientation]] and so carries a homology class with $\Zz$ coefficients. Define the Whitney invariant $$W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})$$ by $W(f)=W_{f_0}(f):=[Cl\Sigma(H)].$ Analogously to \cite{Skopenkov2006}, \S2.4, this is well-defined. == An orientation on the self-intersection set == ; Let $f:N\to\Rr^m$ be a general position smooth map of an orientable $n$-manifold $N$. Assume that $m\ge n+2$ so that the closure $Cl\Sigma(f)$ of the self-intersection set of $f$ has codimension 2 singularities. Then * (1) $\Sigma(f)$ has a natural orientation. * (2) the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$. * (3) the natural orientation on $\Sigma(f)$ extend to $Cl\Sigma(f)$ if $m-n$ is odd \cite{Hudson1969}, Lemma 11.4. * (4) $f\Sigma(f)$ has a natural orientation if $m-n$ is even. Fix an orientation on $N$ and on $\Rr^m$. Let us prove (1). Take points $x,y\in N$ outside singularities of $\Sigma(f)$ and such that $fx=fy$. Then a $(2n-m)$-base $\xi_x$ tangent to $\Sigma(f)$ at $x$ gives a $(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)$ at $y$. Since $N$ is orientable, we can take positive $(m-n)$-bases $\eta_x$ and $\eta_y$ at $x$ and $y$ normal to $\xi_x$ and to $\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$ is positive, then call the base $\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$ forces changes of the signs of $\xi_y,\eta_x$ and $\eta_y$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_x$ and $\eta_y$ and so does not change the orientation of $\Sigma(f)$.) We can see that (2) holds by considering the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point. Let us prove (4). Take a $(2n-m)$-base $\xi$ at a point $x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$. Since $N$ is orientable, we can take a positive $(m-n)$-base $\eta_+$ normal to $f\Sigma(f)$ in one sheet of $f(N)$. Analogously construct an $(m-n)$-base $\eta_-$ for the other sheet of $f(N)$. If $m-n$ is even, then the orientation of the base $(\xi,\eta_+,\eta_-)$ of $\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$. If the base $(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$ positive. This is well-defined because a change of the sign of $\xi$ forces changes of the signs of $\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)$.) == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}N is a connected manifold of dimension n>1, then there is just one isotopy class of embedding N \to\Rr^m if m \ge2n+1. In this page we summarise the situation for m=2n\ge6 and some more general situations.

For notation and conventions see high codimension embeddings.

2 Classification

Theorem 2.1. Let N be a closed connected n-manifold. The Whitney invariant

\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\  \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.

is bijective if either n\ge4 or n=3 and CAT=PL.

This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).

The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. See a description of E^{2n-k}(N) and of E^{2n-2k+1}(N) for a closed k-connected n-manifold N. Some estimations of E^{2n-k-1}(N) for a closed k-connected n-manifold N are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.

3 Examples

Together with the Haefliger knotted sphere, Hudson's examples were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction [Hudson1963] was not as explicit as the one below).

For m\ge n+2 define the standard embedding S^p\times S^{n-p}\to S^m as the composition S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m of standard embeddings.

3.1 Hudson tori 1

In this subsection we construct, for a\in\Zz and n\ge2, an embedding

\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.

Take the standard embeddings 2D^{n+1}\times S^{n-1}\subset\Rr^{2n} (where 2 means homothety with coefficient 2) and \partial D^2\subset\partial D^{n+1}. Fix a point x\in S^{n-1}. The Hudson torus \Hud_n(1) is the embedded connected sum of

\displaystyle 2\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.

(Unlike the unlinked embedded connected sum this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

For a\in\Zz instead of an embedded n-sphere 2\partial D^{n+1}\times x we can take a copies (1+\frac1k)\partial D^{n+1}\times x (k=1,\dots,a) of n-sphere outside D^{n+1}\times S^{n-1} `parallel' to \partial D^{n+1}\times x. Then we join these spheres by tubes so that the homotopy class of the resulting embedding

\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n  \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.

Let \Hud_n(a) be the connected sum of this embedding with the above standard embedding \partial D^2\times S^{n-1}\subset\Rr^{2n}.

Clearly, \Hud_n(0) is isotopic to the standard embedding.

Proposition 3.1. For n\ge3 odd \Hud_n(a) is isotopic to \Hud_n(a') if and only if a=a'.

For n\ge4 even \Hud_n(a) is isotopic to \Hud_n(a') if and only if a\equiv a'\mod2.

In particular, \Hud_n(1) is not isotopic to \Hud_n(0) for each n (this was the original motivation for Hudson).

Proposition 3.1 follows by the fifth part of Remark 4.1 and, for n even, by Theorem 2.1. It would be interesting to find an explicit construction of an isotopy between \Hud_{2k}(a) and \Hud_{2k}(a+2) (cf. [Vrabec1977], \S6) and to prove the analogue of Proposition 3.1 for n=2.

3.2 Hudson tori 2

In this subsection we give, for a\in\Zz and n\ge2, another construction of embeddings

\displaystyle \Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.

Define a map S^0\times S^{n-1}\to D^n to be the constant 0\in D^n on one component 1\times S^{n-1} and the standard embedding \varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n on the other component. This map gives an embedding

\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.

(See Figure 2.2 of [Skopenkov2006].) Each disk D^{n+1}\times x intersects the image of this embedding at two points lying in D^n\times x. Extend this embedding S^0\to D^n\times x for each x\in S^{n-1} to an embedding S^1\to D^{n+1}\times x. (See Figure 2.3 of [Skopenkov2006].) Thus we obtain the Hudson torus

\displaystyle \Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.

The embedding \Hud_n'(a) is obtained in the same way starting from a map \varphi of degree a.

The same proposition as above holds with \Hud_n replaced to \Hud_n'.

3.3 Remarks

We have \Hud_n(a) is PL isotopic to \Hud_n'(a) [Skopenkov2006a]. It would be interesting to prove the smooth analogue of this result.

For n=1 these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of S^1\times S^1.

Analogously one constructs the (left) Hudson torus \Hud_{4,2}(a):S^2\times S^2\to\Rr^7 for a\in\Zz or, more generally, \Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m for a\in\pi_n(S^{m-n+p-1}) and \Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m for a\in\pi_{n-p}(S^{m-n-1}).

3.4 An action of the first homology group on embeddings

In this subsection, for a\in H_1(N) and for a closed connected orientable n-manifold N with n\ge3, we construct an embedding f_a:N\to\Rr^{2n} from an embedding f_0:N\to\Rr^{2n}.

Represent a by an embedding a:S^1\to N. Since any orientable bundle over S^1 is trivial, \nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}. Identify \nu_{f_0}^{-1}a(S^1) with S^1\times S^{n-1}. It remains to make an embedded surgery of S^1\times*\subset S^1\times S^{n-1} to obtain an n-sphere \Sigma\subset C_{f_0}, and then we set f_a:=f_0\# \Sigma.

Take a vector field on S^1\times* normal to S^1\times S^{n-1}. Extend S^1\times* along this vector field to a smooth map \overline a:D^2\to S^{2n}. Since 2n>4 and n+2<2n, by general position we may assume that \overline a is an embedding and \overline a(Int D^2) misses f_0(N)\cup S^1\times S^{n-1}. Since n-1>1, we have \pi_1(V_{2n-2,n-1})=0. Hence the standard framing of S^1\times* in S^1\times S^{n-1} extends to an (n-1)-framing on \overline a(D^2) in \Rr^{2n}. Thus \overline a extends to an embedding

\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}
\displaystyle \mbox{Let}\qquad  \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.

This construction generalizes the construction of \Hud_n(a) (from \Hud_n(0)).

Clearly, W(f_a) is a or a\mod2. Thus unless n=3 and CAT=DIFF

  • all isotopy classes of embedings N\to\Rr^{2n} can be obtained (from a certain given embedding f_0) by the above construction;
  • the above construction defines an action H_1(N;\Zz_{(n-1)})\to E^{2n}(N).

4 The Whitney invariant (for either n even or N orientable)

Fix orientations on \Rr^{2n} and, if n is odd, on N. Fix an embedding f_0:N\to\Rr^{2n}. For an embedding f:N\to\Rr^{2n} the restrictions of f and f_0 to N_0 are regular homotopic [Hirsch1959]. Since N_0 has an (n-1)-dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of f and assume that f=f_0 on N_0. Take a general position homotopy F:B^n\times I\to\Rr^{2n} relative to \partial B^n between the restrictions of f and g to B^n. Then f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I) (i.e. `the intersection of this homotopy with f(N-B^n)') is a 1-manifold (possibly non-compact) without boundary. Define W(f) to be the homology class of the closure of this 1-manifold:

\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n-1)})\cong H_1(N;\Zz_{(n-1)}).

The orientation on f\cap F is defined for N orientable as follows. (This orientation is defined for each n but used only for odd n.) For each point x\in f\cap F take a vector at x tangent to f\cap F. Complete this vector to a positive base tangent to N. Since n+2(n+1)>2\cdot2n, by general position there is a unique point y\in B^n\times I such that Fy=fx. The tangent vector at x thus gives a tangent vector at y to B^n\times I. Complete this vector to a positive base tangent to B^n\times I, where the orientation on B^n comes from N. The union of the images of the constructed two bases is a base at Fy=fx of \Rr^{2n}. If this base is positive, then call the initial vector of f\cap F positive. Since a change of the orientation on f\cap F forces a change of the orientation of the latter base of \Rr^{2n}, it follows that this condition indeed defines an orientation on f\cap F.

Remark 4.1.

  • The Whitney invariant is well-defined, i.e. independent of the choice of F and of the isotopy making f=f_0 outside B^n. This is so because the above definition is clearly equivalent to an alternative one. It is for being well-defined that we need \Zz_2-coefficients when n is even.
  • Clearly, W(f_0)=0. The definition of W depends on the choice of f_0, but we write W not W_{f_0} for brevity.
  • Since a change of the orientation on N forces a change of the orientation on B^n, the class W(f) is independent of the choice of the orientation on N. For the reflection \sigma:\Rr^{2n}\to\Rr^{2n} with respect to a hyperplane we have W(\sigma\circ f)=-W(f) (because we may assume that f=f_0=\sigma\circ f on N_0 and because a change of the orientation of \Rr^{2n} forces a change of the orientation of f\cap F).
  • The above definition makes sense for each n, not only for n\ge3.
  • Clearly, W(\Hud_n(a)) is a or a\mod2 for n\ge2 for the Hudson tori.
  • W(f\#g)=W(f) for each embeddings f:N\to\Rr^{2n} and g:S^n\to\Rr^{2n}.

5 A generalization to highly-connected manifolds

Examples are Hudson tori \Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m.

5.1 Classification

Theorem 5.1. Let N be a closed orientable homologically k-connected n-manifold, k\ge0. Then the Whitney invariant

\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})

is a bijection, provided n\ge k+3 or n\ge2k+4 in the PL or DIFF categories, respectively.

Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically k-connected manifolds. The proof works for homologically k-connected manifolds.

For k=0 this is covered by Theorem 2.1; for k\ge1 it is not. The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for k+3\le n\le2k+1.

E.g. by Theorem 5.1 the Whitney invariant W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)} is bijective for 1\le p\le q-2. It is in fact a group isomorphism; the generator is the Hudson torus.

Because of the existence of knots the analogues of Theorem 5.1 for n=k+2 in the PL case, and for n\le2k+3 in the smooth case are false. So for the smooth category and n\le2k+3 a classification is much harder: for 40 years the only known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

Theorem 5.2. [Skopenkov2008] Let N be a closed homologically (2k-2)-connected (4k-1)-manifold. Then the Whitney invariant

\displaystyle W:E^{6k}_D(N)\to H_{2k-1}(N)

is surjective and for each u\in H_{2k-1}(N) the Kreck invariant

\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}

is a 1-1 correspondence, where d(u) is the divisibility of the projection of u to the free part of H_1(N).

Recall that the divisibility of zero is zero and the divisibility of x\in G-\{0\} is \max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}.

E.g. by Theorem 5.2 the Whitney invariant W:E^{6k}_D(S^{2k-1}\times S^{2k})\to\Zz is surjective and for each u\in\Zz there is a 1-1 correspondence W^{-1}u\to\Zz_u.

Theorem 5.3. [Becker&Glover1971] Let N be a closed k-connected n-manifold embeddable into \Rr^m, m\ge2n-2k+1 and 2m\ge 3n+4. Then there is a 1-1 correspondence

\displaystyle E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.

For k=0,1 this is covered by Theorem 5.1; for k\ge2 it is not.

E.g. by Theorem 5.3 there is a 1-1 correspondence E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X), X:=V_{M,M+p+q-m+1}, for M>p+q, m\ge2q+3 and 2m\ge3q+3p+4. For a generalization see Knotted tori [Skopenkov2002].

5.2 The Whitney invariant

Let N be an n-manifold and f,f_0:N\to\Rr^m embeddings. Roughly speaking, W(f,f_0)=W_{f_0}(f) is defined as the homology class of the self-intersection set \Sigma(H) of a general position homotopy H between f and f_0. We present an accurate definition in the smooth category for m\ge n+2 when either m-n is even or N is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.

Fix orientations on N and on \Rr^m. Take embeddings f,f_0:N\to\Rr^m. Take a general position homotopy H:N\times I\to\Rr^m\times I between f_0 and f. By general position the closure Cl\Sigma(H) of the self-intersection set has codimension 2 singularities and so carries a homology class with \Zz_2 coefficients. (Note that Cl\Sigma(H) can be assumed to be a submanifold for 2m\ge3n+2.) For m-n odd it has a natural orientation and so carries a homology class with \Zz coefficients. Define the Whitney invariant

\displaystyle W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})

by W(f)=W_{f_0}(f):=[Cl\Sigma(H)]. Analogously to [Skopenkov2006], \S2.4, this is well-defined.

6 An orientation on the self-intersection set

Let f:N\to\Rr^m be a general position smooth map of an orientable n-manifold N. Assume that m\ge n+2 so that the closure Cl\Sigma(f) of the self-intersection set of f has codimension 2 singularities. Then

  • (1) \Sigma(f) has a natural orientation.
  • (2) the natural orientation on \Sigma(f) need not extend to Cl\Sigma(f).
  • (3) the natural orientation on \Sigma(f) extend to Cl\Sigma(f) if m-n is odd [Hudson1969], Lemma 11.4.
  • (4) f\Sigma(f) has a natural orientation if m-n is even.

Fix an orientation on N and on \Rr^m.

Let us prove (1). Take points x,y\in N outside singularities of \Sigma(f) and such that fx=fy. Then a (2n-m)-base \xi_x tangent to \Sigma(f) at x gives a (2n-m)-base \xi_y:=df_y^{-1}df_x(\xi_x) tangent to \Sigma(f) at y. Since N is orientable, we can take positive (m-n)-bases \eta_x and \eta_y at x and y normal to \xi_x and to \xi_y. If the base (df_x(\xi_x),df_x(\eta_x),df_y(\eta_y)) of \Rr^m is positive, then call the base \xi_x positive. This is well-defined because a change of the sign of \xi_x forces changes of the signs of \xi_y,\eta_x and \eta_y. (Note that a change of the orientation of N forces changes of the signs of \eta_x and \eta_y and so does not change the orientation of \Sigma(f).)

We can see that (2) holds by considering the cone D^3\to\Rr^5 over a general position map S^2\to\Rr^4 having only one self-intersection point.

Let us prove (4). Take a (2n-m)-base \xi at a point x\in f\Sigma(f) outside singularities of f\Sigma(f). Since N is orientable, we can take a positive (m-n)-base \eta_+ normal to f\Sigma(f) in one sheet of f(N). Analogously construct an (m-n)-base \eta_- for the other sheet of f(N). If m-n is even, then the orientation of the base (\xi,\eta_+,\eta_-) of \Rr^m does not depend on choosing the first and the other sheet of f(N). If the base (\xi,\eta_+,\eta_-) is positive, then call the base \xi positive. This is well-defined because a change of the sign of \xi forces changes of the signs of \eta_+,\eta_- and so of (\xi,\eta_+,\eta_-). (Note that a change of the orientation of N forces changes of the signs of \eta_+,\eta_- and so does not change the orientation of f\Sigma(f).)

7 References

arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013

This page has not been refereed. The information given here might be incomplete or provisional.

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