Embeddings just below the stable range: classification

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$1, $\S$3]. Denote $1_n:=(1,0,\ldots,0)\in S^n$.

2 Classification

For the next theorem, the Whitney invariant $W$ is defined in $\S$5 below.

Theorem 2.1. Assume that $N$ is a closed connected $n$-manifold, either $n$ is odd or $N$ is orientable, and either $n\ge4$ or $n=3$ and we are in the PL category. The Whitney invariant,

$$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$

is a 1-1 correspondence.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$ is even and $N$ is a closed connected non-orientable $n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s.$$

(We replaced the kernel $\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is the multiplication with $w_1(N)$, so $\ker Sq^1 \cong \Zz_2^{s-1}$.)

This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found e.g. in [Skopenkov2006, $\S$5]. It would be interesting to check if this description of $E^{2n}(N)$ is equivalent to different forms of description of $E^{2n}(N)$ [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$ is more complicated, see $\S$6.3 or [Skopenkov2016t].

Concerning embeddings of $n$-manifolds in $\Rr^{2n-1}$ see [Yasui1984] for $n\ge5$, [Skopenkov2016f] for $n=4$, and [Saeki1999, Skopenkov2010, Tonkonog2010] for non-closed manifolds.

Theorem 2.1 is generalized in $\S$6.2 to a description of $E^{2n-k}(N)$ for closed $k$-connected $n$-manifolds $N$.

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

Let us construct, for any $a\in\Zz$ and $n\ge3$, an embedding

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

The reader might first consider the case $n=2$, for which the constructions below work but produce embeddings which are not well-defined (without specific description of the arc involved in the definition of the embedded connected sum).

Definition 3.1. Take the natural `standard embedding' $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (defined in [Skopenkov2015a, $\S$2.1]; here $2$ means homothety with coefficient 2). Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$. Take the embeddings of the sphere and the torus,

$$\displaystyle 2\partial D^{n+1}\times(-1_{n-1})\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join the images of the embeddings by an arc whose interior misses the images. 'The Hudson torus' $\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c, $\S$5] this is a 'linked' embedded connected sum, i.e. the connected sum of two embeddings whose images are 'not' contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$. This is not the case for the next construction in Definition 3.2.

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

The original motivation for Hudson was that $\Hud_n(1)$ is not isotopic to $\Hud_n(0)$ for every $n\ge3$ (this is a particular case of Proposition 3.3 below).

One guesses that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$. And that a $\Zz$-valued invariant exists and is `realized' by the homotopy class of the map

$$\displaystyle S^n\overset g\to S^{2n}-D^{n+1}\times S^{n-1}\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$

However, this is only true for $n$ odd.

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 5.3.d below) 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, $\S$5].

Definition 3.4. Let us 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 inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ 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 [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$. The disk $D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain 'the Hudson torus'

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

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$ is identified with $D^n\times D^1$.

The embedding $\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$ instead of the `standard inclusion'.

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$. For $n\ge3$, a closed connected orientable $n$-manifold $N$, an embedding $f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$ with an $n$-sphere representing the homology Alexander dual of $a$.

More precisely, represent $a$ by an embedding $a:S^1\to N$. Since any orientable bundle over $S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Choose an identification of $\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times(-1_{n-1})\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$. Then we define $f_a$ to be the linked embedded connected sum of $f_0$ and $g$, along some arc joining their images. Since $n\ge3$, the embedding $f_a$ is independent of the choises made in the construction, except possibly of the identification $\nu_{f_0}^{-1}a(S^1)=S^1\times S^{n-1}$, for which see [Skopenkov2014].

Take a vector field on $S^1\times(-1_{n-1})$ normal to $S^1\times S^{n-1}$. Extend $S^1\times(-1_{n-1})$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$. 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 $S^1\times-1_{n-1}$ 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

$$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$ by setting

$$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

By Definition 5.1 of the Whitney invariant, $W(f_a,f_0)=a$. Thus by Theorem 2.1 all isotopy classes of embeddings $N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$ by the above construction.

This follows by Theorem 2.1 or [Skopenkov2014, Remark 18.a].

5 The Whitney invariant

Let $N$ be a closed $n$-manifold and fix an embedding $f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ of a general position homotopy $H$ between $f$ and $f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [HaefligerHirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$ is required.

Fix an orientation on $\Rr^m$. Assume that either $m-n$ is even or $N$ is oriented.

Definition 5.1. Assume that $N$ is $(2n-m)$-connected and $2m\ge3n+3$. Then restrictions of $f$ and $f_0$ to $N_0$ are regular homotopic (see [Koschorke2013, Definition 2.7], [Hirsch1959]). Since $N$ is $(2n-m)$-connected, $N_0$ retracts to an $(m-n-1)$-dimensional polyhedron. Therefore 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 $F:B^n\times I\to\Rr^m$ relative to $\partial B^n$ between the restrictions of $f$ and $f_0$ to $B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$'). Since $n+2(n+1)<2m$, by general position $Cl(f\cap F)$ is a compact $(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$. So $f\cap F$ carries a homology class with $\Zz_2$ coefficients. For $m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$ coefficients. Define $W(f)$ to be the homology class:

$$\displaystyle W(f,f_0)=W(f):=[Cl(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_{\varepsilon(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}).$$

The orientation on $f\cap F$ (extendable to $Cl(f\cap F)$) is defined for $m-n$ odd as follows. 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 $n+2(n+1)<2m$, 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 $\Rr^m$. If the latter 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 $\Rr^m$, this condition indeed defines an orientation on $f\cap F$.

Definition 5.2. Assume that $m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$.

The closure $Cl\Sigma(H)$ of the self-intersection set carries a cycle mod 2. For $m-n$ odd we shall use that the closure also carries an integer cycle. See [Hudson1967, \S11], [Skopenkov2006, $\S$2.3 `The Whitney obstruction'].

(Let us present informal explanations of these facts. For $2m\ge3n+2$ by general position the closure $Cl\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$, by general position the closure has codimension 2 singularities, see definition in $\S$7. So the closure carries a cycle mod 2. The closure also has a natural orientation, see Definition 7.1, and so carries an integer cycle.)

Define the Whitney invariant to be the homology class:

$$\displaystyle W(f,f_0)=W(f):=[Cl\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_{\varepsilon(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}).$$

Clearly, $W(f) = W(f')$ if $f$ is isotopic to $f'$. Hence the Whitney invariant defines a map

$$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

Clearly, $W(f_0)=0$ (for both definitions).

The definition of $W$ depends on the choice of $f_0$, but we write $W$ not $W_{f_0}$ for brevity.

6 A generalization to highly-connected manifolds

In this section let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Recall the unknotting theorem [Skopenkov2016c] that all embeddings $N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$ and $n\ge2k+2$. In this section we generalize Theorem 2.1 to a description of $E^{2n-k}(N)$ and further to $E^m(N)$ for $m\ge2n-2k+1$.

6.1 Examples

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

We have $W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$ (provided $n\ge k+3$ or $n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$.

The embedding $f_a$ has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].

6.2 Classification just below the stable range

Theorem 6.2. Let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Then the Whitney invariant

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

is a bijection, provided $n\ge k+3$ in the smooth category or $n\ge2k+4$ in the category.

This was proved for $k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$-connectedness assumption.

For $k=0$ Theorem 6.2 is covered by Theorem 2.1; for $k\ge1$ it is not. For $k+3\le n\le2k+1$ the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c].

An inverse to the map $W$ of Theorem 6.2 is given by Example 6.1.

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$ by Theorem 6.2 the Whitney invariants

$$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$ even $W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$ in the PL case, and for $n\le2k+3$ in the smooth case are false.

$$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$ [Haefliger1966]. The following result for $n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. Let $N$ be a closed orientable homologically $(2l-2)$-connected $(4l-1)$-manifold. Then the Whitney invariant

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

is surjective and for each $u\in H_{2l-1}(N)$ 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 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

How does one describe $E^m(N)$ when $N$ is not $(2n-m)$-connected? For general $N$ see the remarks on $E^{2n-1}(N)$ in $\S$2. We can say more as the connectivity $k$ of $N$ increases. Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ are presented in [Skopenkov2010]. For $k>1$ one can go even further:

Theorem 6.4 [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

$$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$$

For $k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$ is $(n-1)$-connected). For $k=1$ this is covered by Theorem 6.2; for $k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$ and $2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].

Observe that in Theorem 6.4 $V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$ for each $M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$ be a general position smooth map of an oriented $n$-manifold $N$. Assume that $m\ge n+2$. Then the closure $Cl\Sigma(f)$ of the self-intersection set of $f$ has codimension 2 singularities, i.e., there is $X\subset Cl\Sigma(f)$ (called a singular set) of dimension at most $\dim Cl\Sigma(f)-2$ such that $Cl\Sigma(f)-X$ is an open manifold.

We remark 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 $\Sigma(f)$.

• the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$ extends to $Cl\Sigma(f)$ if $m-n$ is odd [Hudson1969, Lemma 11.4].

We remark that a change of the orientation of $N$ forces changes of the signs of $\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)$.

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• [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.

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

• [Vrabec1977] J. Vrabec, Knotting a $k$-connected closed $\PL$ $m$-manifold in $E^{2m-k}$, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
• [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
• [Yasui1984] T. Yasui, Enumerating embeddings of $n$-manifolds in Euclidean $(2n-1)$-space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$1, $\S$3]. Denote $1_n:=(1,0,\ldots,0)\in S^n$.

2 Classification

For the next theorem, the Whitney invariant $W$ is defined in $\S$5 below.

Theorem 2.1. Assume that $N$ is a closed connected $n$-manifold, either $n$ is odd or $N$ is orientable, and either $n\ge4$ or $n=3$ and we are in the PL category. The Whitney invariant,

$$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$

is a 1-1 correspondence.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$ is even and $N$ is a closed connected non-orientable $n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s.$$

(We replaced the kernel $\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is the multiplication with $w_1(N)$, so $\ker Sq^1 \cong \Zz_2^{s-1}$.)

This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found e.g. in [Skopenkov2006, $\S$5]. It would be interesting to check if this description of $E^{2n}(N)$ is equivalent to different forms of description of $E^{2n}(N)$ [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$ is more complicated, see $\S$6.3 or [Skopenkov2016t].

Concerning embeddings of $n$-manifolds in $\Rr^{2n-1}$ see [Yasui1984] for $n\ge5$, [Skopenkov2016f] for $n=4$, and [Saeki1999, Skopenkov2010, Tonkonog2010] for non-closed manifolds.

Theorem 2.1 is generalized in $\S$6.2 to a description of $E^{2n-k}(N)$ for closed $k$-connected $n$-manifolds $N$.

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

Let us construct, for any $a\in\Zz$ and $n\ge3$, an embedding

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

The reader might first consider the case $n=2$, for which the constructions below work but produce embeddings which are not well-defined (without specific description of the arc involved in the definition of the embedded connected sum).

Definition 3.1. Take the natural `standard embedding' $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (defined in [Skopenkov2015a, $\S$2.1]; here $2$ means homothety with coefficient 2). Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$. Take the embeddings of the sphere and the torus,

$$\displaystyle 2\partial D^{n+1}\times(-1_{n-1})\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join the images of the embeddings by an arc whose interior misses the images. 'The Hudson torus' $\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c, $\S$5] this is a 'linked' embedded connected sum, i.e. the connected sum of two embeddings whose images are 'not' contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$. This is not the case for the next construction in Definition 3.2.

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

The original motivation for Hudson was that $\Hud_n(1)$ is not isotopic to $\Hud_n(0)$ for every $n\ge3$ (this is a particular case of Proposition 3.3 below).

One guesses that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$. And that a $\Zz$-valued invariant exists and is `realized' by the homotopy class of the map

$$\displaystyle S^n\overset g\to S^{2n}-D^{n+1}\times S^{n-1}\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$

However, this is only true for $n$ odd.

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 5.3.d below) 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, $\S$5].

Definition 3.4. Let us 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 inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ 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 [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$. The disk $D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain 'the Hudson torus'

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

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$ is identified with $D^n\times D^1$.

The embedding $\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$ instead of the `standard inclusion'.

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$. For $n\ge3$, a closed connected orientable $n$-manifold $N$, an embedding $f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$ with an $n$-sphere representing the homology Alexander dual of $a$.

More precisely, represent $a$ by an embedding $a:S^1\to N$. Since any orientable bundle over $S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Choose an identification of $\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times(-1_{n-1})\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$. Then we define $f_a$ to be the linked embedded connected sum of $f_0$ and $g$, along some arc joining their images. Since $n\ge3$, the embedding $f_a$ is independent of the choises made in the construction, except possibly of the identification $\nu_{f_0}^{-1}a(S^1)=S^1\times S^{n-1}$, for which see [Skopenkov2014].

Take a vector field on $S^1\times(-1_{n-1})$ normal to $S^1\times S^{n-1}$. Extend $S^1\times(-1_{n-1})$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$. 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 $S^1\times-1_{n-1}$ 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

$$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$ by setting

$$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

By Definition 5.1 of the Whitney invariant, $W(f_a,f_0)=a$. Thus by Theorem 2.1 all isotopy classes of embeddings $N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$ by the above construction.

This follows by Theorem 2.1 or [Skopenkov2014, Remark 18.a].

5 The Whitney invariant

Let $N$ be a closed $n$-manifold and fix an embedding $f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ of a general position homotopy $H$ between $f$ and $f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [HaefligerHirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$ is required.

Fix an orientation on $\Rr^m$. Assume that either $m-n$ is even or $N$ is oriented.

Definition 5.1. Assume that $N$ is $(2n-m)$-connected and $2m\ge3n+3$. Then restrictions of $f$ and $f_0$ to $N_0$ are regular homotopic (see [Koschorke2013, Definition 2.7], [Hirsch1959]). Since $N$ is $(2n-m)$-connected, $N_0$ retracts to an $(m-n-1)$-dimensional polyhedron. Therefore 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 $F:B^n\times I\to\Rr^m$ relative to $\partial B^n$ between the restrictions of $f$ and $f_0$ to $B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$'). Since $n+2(n+1)<2m$, by general position $Cl(f\cap F)$ is a compact $(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$. So $f\cap F$ carries a homology class with $\Zz_2$ coefficients. For $m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$ coefficients. Define $W(f)$ to be the homology class:

$$\displaystyle W(f,f_0)=W(f):=[Cl(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_{\varepsilon(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}).$$

The orientation on $f\cap F$ (extendable to $Cl(f\cap F)$) is defined for $m-n$ odd as follows. 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 $n+2(n+1)<2m$, 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 $\Rr^m$. If the latter 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 $\Rr^m$, this condition indeed defines an orientation on $f\cap F$.

Definition 5.2. Assume that $m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$.

The closure $Cl\Sigma(H)$ of the self-intersection set carries a cycle mod 2. For $m-n$ odd we shall use that the closure also carries an integer cycle. See [Hudson1967, \S11], [Skopenkov2006, $\S$2.3 `The Whitney obstruction'].

(Let us present informal explanations of these facts. For $2m\ge3n+2$ by general position the closure $Cl\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$, by general position the closure has codimension 2 singularities, see definition in $\S$7. So the closure carries a cycle mod 2. The closure also has a natural orientation, see Definition 7.1, and so carries an integer cycle.)

Define the Whitney invariant to be the homology class:

$$\displaystyle W(f,f_0)=W(f):=[Cl\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_{\varepsilon(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}).$$

Clearly, $W(f) = W(f')$ if $f$ is isotopic to $f'$. Hence the Whitney invariant defines a map

$$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

Clearly, $W(f_0)=0$ (for both definitions).

The definition of $W$ depends on the choice of $f_0$, but we write $W$ not $W_{f_0}$ for brevity.

6 A generalization to highly-connected manifolds

In this section let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Recall the unknotting theorem [Skopenkov2016c] that all embeddings $N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$ and $n\ge2k+2$. In this section we generalize Theorem 2.1 to a description of $E^{2n-k}(N)$ and further to $E^m(N)$ for $m\ge2n-2k+1$.

6.1 Examples

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

We have $W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$ (provided $n\ge k+3$ or $n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$.

The embedding $f_a$ has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].

6.2 Classification just below the stable range

Theorem 6.2. Let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Then the Whitney invariant

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

is a bijection, provided $n\ge k+3$ in the smooth category or $n\ge2k+4$ in the category.

This was proved for $k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$-connectedness assumption.

For $k=0$ Theorem 6.2 is covered by Theorem 2.1; for $k\ge1$ it is not. For $k+3\le n\le2k+1$ the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c].

An inverse to the map $W$ of Theorem 6.2 is given by Example 6.1.

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$ by Theorem 6.2 the Whitney invariants

$$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$ even $W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$ in the PL case, and for $n\le2k+3$ in the smooth case are false.

$$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$ [Haefliger1966]. The following result for $n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. Let $N$ be a closed orientable homologically $(2l-2)$-connected $(4l-1)$-manifold. Then the Whitney invariant

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

is surjective and for each $u\in H_{2l-1}(N)$ 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 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

How does one describe $E^m(N)$ when $N$ is not $(2n-m)$-connected? For general $N$ see the remarks on $E^{2n-1}(N)$ in $\S$2. We can say more as the connectivity $k$ of $N$ increases. Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ are presented in [Skopenkov2010]. For $k>1$ one can go even further:

Theorem 6.4 [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

$$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$$

For $k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$ is $(n-1)$-connected). For $k=1$ this is covered by Theorem 6.2; for $k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$ and $2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].

Observe that in Theorem 6.4 $V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$ for each $M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$ be a general position smooth map of an oriented $n$-manifold $N$. Assume that $m\ge n+2$. Then the closure $Cl\Sigma(f)$ of the self-intersection set of $f$ has codimension 2 singularities, i.e., there is $X\subset Cl\Sigma(f)$ (called a singular set) of dimension at most $\dim Cl\Sigma(f)-2$ such that $Cl\Sigma(f)-X$ is an open manifold.

We remark 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 $\Sigma(f)$.

• the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$ extends to $Cl\Sigma(f)$ if $m-n$ is odd [Hudson1969, Lemma 11.4].

We remark that a change of the orientation of $N$ forces changes of the signs of $\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)$.

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• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
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• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
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arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013

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For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$1, $\S$3]. Denote $1_n:=(1,0,\ldots,0)\in S^n$.

2 Classification

For the next theorem, the Whitney invariant $W$ is defined in $\S$5 below.

Theorem 2.1. Assume that $N$ is a closed connected $n$-manifold, either $n$ is odd or $N$ is orientable, and either $n\ge4$ or $n=3$ and we are in the PL category. The Whitney invariant,

$$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$

is a 1-1 correspondence.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$ is even and $N$ is a closed connected non-orientable $n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s.$$

(We replaced the kernel $\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is the multiplication with $w_1(N)$, so $\ker Sq^1 \cong \Zz_2^{s-1}$.)

This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found e.g. in [Skopenkov2006, $\S$5]. It would be interesting to check if this description of $E^{2n}(N)$ is equivalent to different forms of description of $E^{2n}(N)$ [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$ is more complicated, see $\S$6.3 or [Skopenkov2016t].

Concerning embeddings of $n$-manifolds in $\Rr^{2n-1}$ see [Yasui1984] for $n\ge5$, [Skopenkov2016f] for $n=4$, and [Saeki1999, Skopenkov2010, Tonkonog2010] for non-closed manifolds.

Theorem 2.1 is generalized in $\S$6.2 to a description of $E^{2n-k}(N)$ for closed $k$-connected $n$-manifolds $N$.

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

Let us construct, for any $a\in\Zz$ and $n\ge3$, an embedding

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

The reader might first consider the case $n=2$, for which the constructions below work but produce embeddings which are not well-defined (without specific description of the arc involved in the definition of the embedded connected sum).

Definition 3.1. Take the natural `standard embedding' $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (defined in [Skopenkov2015a, $\S$2.1]; here $2$ means homothety with coefficient 2). Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$. Take the embeddings of the sphere and the torus,

$$\displaystyle 2\partial D^{n+1}\times(-1_{n-1})\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join the images of the embeddings by an arc whose interior misses the images. 'The Hudson torus' $\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c, $\S$5] this is a 'linked' embedded connected sum, i.e. the connected sum of two embeddings whose images are 'not' contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$. This is not the case for the next construction in Definition 3.2.

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

The original motivation for Hudson was that $\Hud_n(1)$ is not isotopic to $\Hud_n(0)$ for every $n\ge3$ (this is a particular case of Proposition 3.3 below).

One guesses that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$. And that a $\Zz$-valued invariant exists and is `realized' by the homotopy class of the map

$$\displaystyle S^n\overset g\to S^{2n}-D^{n+1}\times S^{n-1}\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$

However, this is only true for $n$ odd.

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 5.3.d below) 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, $\S$5].

Definition 3.4. Let us 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 inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ 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 [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$. The disk $D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain 'the Hudson torus'

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

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$ is identified with $D^n\times D^1$.

The embedding $\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$ instead of the `standard inclusion'.

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$. For $n\ge3$, a closed connected orientable $n$-manifold $N$, an embedding $f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$ with an $n$-sphere representing the homology Alexander dual of $a$.

More precisely, represent $a$ by an embedding $a:S^1\to N$. Since any orientable bundle over $S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Choose an identification of $\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times(-1_{n-1})\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$. Then we define $f_a$ to be the linked embedded connected sum of $f_0$ and $g$, along some arc joining their images. Since $n\ge3$, the embedding $f_a$ is independent of the choises made in the construction, except possibly of the identification $\nu_{f_0}^{-1}a(S^1)=S^1\times S^{n-1}$, for which see [Skopenkov2014].

Take a vector field on $S^1\times(-1_{n-1})$ normal to $S^1\times S^{n-1}$. Extend $S^1\times(-1_{n-1})$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$. 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 $S^1\times-1_{n-1}$ 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

$$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$ by setting

$$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

By Definition 5.1 of the Whitney invariant, $W(f_a,f_0)=a$. Thus by Theorem 2.1 all isotopy classes of embeddings $N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$ by the above construction.

This follows by Theorem 2.1 or [Skopenkov2014, Remark 18.a].

5 The Whitney invariant

Let $N$ be a closed $n$-manifold and fix an embedding $f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ of a general position homotopy $H$ between $f$ and $f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [HaefligerHirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$ is required.

Fix an orientation on $\Rr^m$. Assume that either $m-n$ is even or $N$ is oriented.

Definition 5.1. Assume that $N$ is $(2n-m)$-connected and $2m\ge3n+3$. Then restrictions of $f$ and $f_0$ to $N_0$ are regular homotopic (see [Koschorke2013, Definition 2.7], [Hirsch1959]). Since $N$ is $(2n-m)$-connected, $N_0$ retracts to an $(m-n-1)$-dimensional polyhedron. Therefore 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 $F:B^n\times I\to\Rr^m$ relative to $\partial B^n$ between the restrictions of $f$ and $f_0$ to $B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$'). Since $n+2(n+1)<2m$, by general position $Cl(f\cap F)$ is a compact $(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$. So $f\cap F$ carries a homology class with $\Zz_2$ coefficients. For $m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$ coefficients. Define $W(f)$ to be the homology class:

$$\displaystyle W(f,f_0)=W(f):=[Cl(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_{\varepsilon(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}).$$

The orientation on $f\cap F$ (extendable to $Cl(f\cap F)$) is defined for $m-n$ odd as follows. 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 $n+2(n+1)<2m$, 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 $\Rr^m$. If the latter 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 $\Rr^m$, this condition indeed defines an orientation on $f\cap F$.

Definition 5.2. Assume that $m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$.

The closure $Cl\Sigma(H)$ of the self-intersection set carries a cycle mod 2. For $m-n$ odd we shall use that the closure also carries an integer cycle. See [Hudson1967, \S11], [Skopenkov2006, $\S$2.3 `The Whitney obstruction'].

(Let us present informal explanations of these facts. For $2m\ge3n+2$ by general position the closure $Cl\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$, by general position the closure has codimension 2 singularities, see definition in $\S$7. So the closure carries a cycle mod 2. The closure also has a natural orientation, see Definition 7.1, and so carries an integer cycle.)

Define the Whitney invariant to be the homology class:

$$\displaystyle W(f,f_0)=W(f):=[Cl\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_{\varepsilon(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}).$$

Clearly, $W(f) = W(f')$ if $f$ is isotopic to $f'$. Hence the Whitney invariant defines a map

$$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

Clearly, $W(f_0)=0$ (for both definitions).

The definition of $W$ depends on the choice of $f_0$, but we write $W$ not $W_{f_0}$ for brevity.

6 A generalization to highly-connected manifolds

In this section let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Recall the unknotting theorem [Skopenkov2016c] that all embeddings $N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$ and $n\ge2k+2$. In this section we generalize Theorem 2.1 to a description of $E^{2n-k}(N)$ and further to $E^m(N)$ for $m\ge2n-2k+1$.

6.1 Examples

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

We have $W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$ (provided $n\ge k+3$ or $n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$.

The embedding $f_a$ has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].

6.2 Classification just below the stable range

Theorem 6.2. Let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Then the Whitney invariant

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

is a bijection, provided $n\ge k+3$ in the smooth category or $n\ge2k+4$ in the category.

This was proved for $k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$-connectedness assumption.

For $k=0$ Theorem 6.2 is covered by Theorem 2.1; for $k\ge1$ it is not. For $k+3\le n\le2k+1$ the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c].

An inverse to the map $W$ of Theorem 6.2 is given by Example 6.1.

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$ by Theorem 6.2 the Whitney invariants

$$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$ even $W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$ in the PL case, and for $n\le2k+3$ in the smooth case are false.

$$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$ [Haefliger1966]. The following result for $n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. Let $N$ be a closed orientable homologically $(2l-2)$-connected $(4l-1)$-manifold. Then the Whitney invariant

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

is surjective and for each $u\in H_{2l-1}(N)$ 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 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

How does one describe $E^m(N)$ when $N$ is not $(2n-m)$-connected? For general $N$ see the remarks on $E^{2n-1}(N)$ in $\S$2. We can say more as the connectivity $k$ of $N$ increases. Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ are presented in [Skopenkov2010]. For $k>1$ one can go even further:

Theorem 6.4 [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

$$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$$

For $k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$ is $(n-1)$-connected). For $k=1$ this is covered by Theorem 6.2; for $k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$ and $2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].

Observe that in Theorem 6.4 $V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$ for each $M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$ be a general position smooth map of an oriented $n$-manifold $N$. Assume that $m\ge n+2$. Then the closure $Cl\Sigma(f)$ of the self-intersection set of $f$ has codimension 2 singularities, i.e., there is $X\subset Cl\Sigma(f)$ (called a singular set) of dimension at most $\dim Cl\Sigma(f)-2$ such that $Cl\Sigma(f)-X$ is an open manifold.

We remark 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 $\Sigma(f)$.

• the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$ extends to $Cl\Sigma(f)$ if $m-n$ is odd [Hudson1969, Lemma 11.4].