Embeddings just below the stable range: classification

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Contents

1 Introduction

Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.

Recall the Whitney-Wu Unknotting Theorem: if N is a connected manifold of dimension n>1, and m \ge2n+1, then every two embeddings N \to\Rr^m are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for m=2n\ge6 and N is a connected, as well as in some more general situations. For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S3]. Denote 1_n:=(1,0,\ldots,0)\in S^n.

2 Classification

For the next theorem, the Whitney invariant W is defined in \S5 below.

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

(a) If N is oriented, the Whitney invariant,

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

is a 1-1 correspondence.

(b) If N is non-orientable, then there is a 1-1 correspondence

\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\  \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.

Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in \S2], [Hudson1969, \S11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].

Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in \S2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.

In Part (b) 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 given by multiplication with the first Stefel-Whitney class w_1(N) (which equals to the first Wu class v_1(N) [Milnor&Stasheff1974, Theorem 11.4]). Since N is non-orientable, w_1(N)\neq 0. So by Poincaré duality, \ker Sq^1 \cong \Zz_2^{s-1}.

The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, \S5]. It would be interesting to check if part (b) 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 Theorem 6.3 below for l=1 or [Skopenkov2016t].

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

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

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t, Example 2.1], [Skopenkov2006, Example 3.4], the examples of Hudson tori presented below were the first examples of non-isotopic embeddings in codimension greater than 2. (Hudson's construction [Hudson1963] was not as explicit as those below.) Abbreviate
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to just
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.

Example 3.1. Let us construct, for any a\in\Zz and n\ge2, a smooth embedding

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

We start with the cases a=0,1.

Take the standard inclusion \partial D^2\subset\partial D^{n+1}. The 'standard embedding' \Hud_n(0) is given by the standard inclusions

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Define the `standard embedding'
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analogously to
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, where 2 means homothety with coefficient 2.

Take the embedding g_1 given by

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The segment
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joins the images of \Hud_n(0) and g_1; the interior of this segment misses the images. Let \Hud_n(1) be the linked embedded connected sum of \Hud_n(0) and g_1 along this segment, compatible with the orientation, cf. [Avvakumov2017, \S1.5]. (Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the unlinked embedded connected sum [Skopenkov2016c, \S5].)

For a\in\Zz we repeat the above construction of g_1 replacing 2\partial D^{n+1}\times1_{n-1} by |a| copies (1+\frac1k)\partial D^{n+1}\times1_{n-1} of S^n, k=1,\ldots,|a|. The copies are outside D^{n+1}\times S^{n-1} and are `parallel' to \partial D^{n+1}\times1_{n-1}. The copies have the standard orientation for a>0 or the opposite orientation for a<0. Then we make embedded connected sum along natural segments joining every k-th copy to the (k+1)-th copy. We obtain an embedding g_a:S^n\to\Rr^{2n} which has disjoint images with \Hud_n(0). Let \Hud_n(a) be the linked embedded connected sum of \Hud_n(0) and g_a.

The original motivation for Hudson was that \Hud_n(1) is not isotopic to \Hud_n(0) for any n\ge3 (this is a particular case of Proposition 3.2 below). One might guess that \Hud_n(a) is not isotopic to \Hud_n(a') for a\ne a' and that a \Zz-valued invariant of E^{2n}(S^1 \times S^{n-1}) can be defined by the homotopy class of the map

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However, this is only true for n odd.

Proposition 3.2. 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'\text{ mod}2.

Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for n even, by Theorem 2.1. This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of \Hud_n(a) for the PL category). It would be interesting to find an explicit construction of an isotopy between \Hud_{2k}(a) and \Hud_{2k}(a+2), cf. [Vrabec1977, \S5]. Analogously, \Hud_2(a) is not isotopic to \Hud_2(a') if a\not\equiv a'\text{ mod}2. It would be interesting to know if the converse holds, e.g. is \Hud_2(0) (PS or smoothly) isotopic to \Hud_2(2).

Example 3.3. Take any a\in\Zz. Take a map \overline a:S^{n-1}\to S^{n-1} of degree a (so we can take \overline 1=\id). Recall that D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}. Define the smooth embedding \Hud_n'(a) to be the composition

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Figure 1: The Hudson torus

Let us present a geometric description of this embedding. Define a map \widetilde a:S^0\times S^{n-1}\to S^{n-1} by \widetilde a(s,t):=s\overline a(t). This map gives an embedding

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See Figure 1. The image of
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is the union of the graphs of the maps \overline a and -\overline a. For any t\in S^{n-1} the disk
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intersects the image at two points lying in
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, i.e., at the image of an embedding
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. The embedding \Hud_n'(a) is obtained by extending the latter embeddings to embeddings
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for all t. See Figure 2.
Figure 2: To the construction of the Hudson torus

Remark 3.4. (a) The analogue of Proposition 3.2 for \Hud_n replaced to \Hud_n' holds, with an analogous proof.

(b) The embeddings \Hud_n(a) and \Hud_n'(a) are smoothly isotopic for n\ge4 and are PS isotopic for n\ge3 [Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]). This follows by calculation of the Whitney invariant (Remark 5.3.d below). It would be interesting to know if they are smoothly isotopic for n=3. It would be interesting to know if they are piecewise smoothly isotopic for n=2.

(c) For n=2 Example 3.3 gives what we call the 'left' Hudson torus. The 'right' Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of S^1\times S^1. The right and the left Hudson tori are not isotopic by Remark 5.3.d below.

(d) Analogously one constructs the Hudson torus \Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1} for a\in\Zz and n>p\ge0 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 m>n>p\ge0. There are versions \Hud'_{m,n,p}(a) of these constructions corresponding to Definition 3.3. For p=0 this corresponds to the Zeeman map [Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a, \S2.3]. It would be interesting to know if the links \Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3 are isotopic, cf. [Skopenkov2015a, Remark 2.7.b]. These constructions can be further generalized [Skopenkov2016k].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus \Hud(a). Let N be a closed connected oriented n-manifold. We work in the smooth category which we omit. Apparently analogous results hold for n\ge3 in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).

Example 4.1. For any n\ge4, an embedding f_0:N\to\Rr^{2n} and a\in H_1(N;\Zz), we shall 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' A:=\widehat{A_{f_0}}a\in H_n(C_{f_0}) of a (defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).

Represent a by an embedding a:S^1\to N. By definition, the class A is represented by properly oriented \nu_{f_0}^{-1}a(S^1). Since any orientable bundle over S^1 is trivial, \nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}. Take an embedding g:S^1\times S^{n-1}\to C_{f_0} whose image is \nu_{f_0}^{-1}a(S^1) and which represents A. By embedded surgery on S^1\times1_{n-1}\subset S^1\times S^{n-1} we obtain an embedding g_1:S^n\to C_{f_0} representing A (see details in Proposition 4.2 below). Define f_a to be the linked embedded connected sum of f_0 and g_1, along some arc joining their images.

Proposition 4.2 (Embedded surgery). For any n\ge3, a neighborhood U of a codimension at least 3 subpolyhedron in \Rr^{2n} and an embedding g:S^1\times S^{n-1}\to\Rr^{2n}-U there is an embedding g_1:S^n\to\Rr^{2n}-U homologous to g.

Proof. Take a vector field on g(S^1\times1_{n-1}) normal to g(S^1\times S^{n-1}). Extend g|_{S^1\times1_{n-1}} along this vector field to a map \overline b:D^2\to\Rr^{2n}.

Since 2n>4 and U is a neighborhood U of a codimension at least 3 subpolyhedron, by general position we may assume that \overline b is an embedding and that
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misses U\cup g(S^1\times S^{n-1}).

Since n-1>1, we have \pi_1(V_{2n-2,n-1})=0. Hence the standard (n-1)-framing of S^1\times1_{n-1} in S^1\times S^{n-1} extends to an (n-1)-framing on \overline b(D^2) in \Rr^{2n}. Thus \overline b extends to an embedding

\displaystyle  \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).

Take an embedding g_1:S^n\to\R^{2n}-U such that

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with proper orientation so that g_1 is homologous to g. QED

The isotopy class of the embedding f_a is independent of the choises in the construction. The independence of the arc and of the maps g,g_1 follows by n\ge3 and by Proposition 4.3 below, respectively.

By Definition 5.1 of the Whitney invariant, W(f_a,f_0) is a for n\ge3 odd and a \mod2 for n\ge4 even. Thus by Theorem 2.1.a for n\ge4 all isotopy classes of embeddings N\to\Rr^{2n} can be obtained from any chosen embedding f_0 by the above construction.

Proposition 4.3. For any n\ge4 both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of H_1(N;\Zz_{\varepsilon(n-1)}) on E^{2n}(N).

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

5 The Whitney invariant

Let N be a closed n-manifold. Take an embedding f_0:N\to\Rr^m. Fix an orientation on \Rr^m. For any other embedding f \colon N \to \Rr^m we define the Whitney invariant

\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).

Here the coefficients \Zz_N are \Zz if N is oriented and m-n is odd, and are \Zz_2 otherwise.

Roughly speaking,
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is defined as the homology class of the closure of the self-intersection set 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, \S11], [Vrabec1977, p. 145], [Skopenkov2006, \S2.4 `The Whitney invariant']. We begin by presenting a simpler definition, Definition 5.1, for a particular case.

For Theorem 2.1 only the case m=2n is required.

Definition 5.1. Assume that N is (2n-m)-connected and 2m\ge3n+3. Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of f and f_0 to N_0 are isotopic, cf. [Takase2006, Lemma 2.2]. (Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of f and f_0 to N_0 are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since N is (2n-m)-connected, N_0 retracts to an (m-n-1)-dimensional polyhedron. Therefore these restrictions are isotopic.)

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
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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. If m-n is odd and N is oriented, then f\cap F has a natural orientation defined below, and so carries a homology class with \Zz coefficients. Define W(f) to be the homology class:

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The orientation on f\cap F (extendable to
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) is defined (for m-n odd and N is oriented) as follows (cf. Remark 7.3). For any point x_f\in f\cap F take a base \xi_f at x_f tangent to f\cap F. Complete this base to a positive base (\xi_f,\eta_f) tangent to N. Since n+2(n+1)<2m, by general position there is a unique point x_F\in B^n\times I such that Fx_F=fx_f. The tangent base \xi_f at x_f thus gives a base \xi_F at x_F tangent to B^n\times I such that df(x_f)\xi_f=dF(x_F)\xi_F. Complete this base \xi_F to a positive base (\xi_F,\eta_F) tangent to B^n\times I, where the orientation on B^n comes from N. The union \zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F) of the images of the constructed bases is a base at fx_f=Fx_F of \Rr^m. If \zeta is positive, then call the tangent base \xi_f of f\cap F `positive'. Since a change of the orientation on f\cap F forces a change of the orientation of \zeta, 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
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of the self-intersection set carries a cycle mod 2. If N is oriented and m-n is odd, the closure also carries an integer cycle. See [Hudson1967, \S11], [Skopenkov2006, \S2.3 `The Whitney obstruction'].


Let us informally explain these facts. For 2m\ge3n+2 by general position the closure
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can be assumed to be a submanifold. In general, since m\ge n+2, by general position the closure has codimension 2 singularities, as defined in \S7. So the closure carries a cycle mod 2. When m-n is odd the closure also has a canonical orientation (see Definition 7.1 and Remark 7.2), so the closure carries an integer cycle.

Define the Whitney invariant to be the homology class:

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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_N),\quad [f] \mapsto W(f).

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.

Remark 5.3. (a) The Whitney invariant is well-defined by Definition 5.2, i.e. is independent of the choice of a general position homotopy H:N\times I\to\Rr^m\times I from f_0 to f.

This follows from the equality
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for a general position homotopy H_{01}:N\times I\times I\to\Rr^m\times I\times I between general position homotopies H_0,H_1:N\times I\to\Rr^m\times I from f_0 to f. See details in [Hudson1969, \S11].

(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if f=f_0 on N_0, we can take H to be fixed on N_0. See details in [Skopenkov2010, Difference Lemma 2.4].) Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice of F and of the isotopy making f=f_0 outside B^n.

(c) The class W(f) is independent of the choice of the orientation on N (because a change of the orientation on N forces a change of the orientation on f\cap F or on
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). For the reflection \sigma:\Rr^m\to\Rr^m with respect to a hyperplane we have W(\sigma\circ f)=-W(f) (because a change of the orientation on \Rr^m forces a change of the orientation on f\cap F or on
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; for Definition 5.1 also observe that we may assume that f=f_0=\sigma\circ f on N_0).

(d) For the Hudson tori W(\Hud_n(a))=W(\Hud'_n(a)) is a or a\mod2 for n\ge3, and W(\Hud'_2(a))=(a\mod2,0).

For \Hud_n(a) this is clear by Definition 5.1. For \Hud'_n(a) and n\ge3 this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For \Hud'_2(a) the proof is analogous.

(e) W(f\#g)=W(f) for any pair of embeddings f:N\to\Rr^m and g:S^n\to\Rr^m. This is clear by Definition 5.1 because W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0. Let us prove the latter equality. Take the identical isotopy H_f of f and a general position homotopy H_g between g and the standard embedding. Then the boundary connected sum H_f\sharp H_g is a general position homotopy between f\#g and an embedding isotopic to f. The cycle
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is null-homologous in S^n and hence in N\cong N\#S^n; cf. [Skopenkov2008, Addendum to the Classification Theorem].

(f) For m=2n+1 and N=S^n\sqcup S^n the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h, \S3].

(g) The Whitney invariant need not be a bijection for m<2n. This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking n even, N non-orientable, m=2n and applying by Theorem 2.1.b.

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, Theorem 2.4] 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.

Example 6.1 (cf. [Skopenkov2010, Definition 1.4]). Assume that N is k-connected and n-k\ge3. Then for an embedding f_0:N\to\Rr^{2n-k} and a class a\in H_{k+1}(N;\Zz) one can construct an embedding f_a:N\to\Rr^{2n-k} by linked embedded connected sum analogously to the case k=0 presented in Example 4.1.

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.

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

6.2 Classification

Theorem 6.2. Let N be a closed oriented 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\ge2k+4 in the smooth category or n\ge k+3 in the PL category.

This was proved for k-connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in [Weber1967], [Hudson1969, \S11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993, \S7]. The proof actually used the homological k-connectedness assumption (basically because the k-connectedness was used to ensure high enough connectedness of the complement in \Rr^m to the image of N, by Alexander duality and simple connectedness of the complement, so homological k-connectedness of N is sufficient).

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, Theorem 2.3]. For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].

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

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. So for the smooth category, n\le2k+3 and N closed connected, a classification of E^{2n-k}(N) is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres N. E.g. E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)} for any s\ge2 [Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2]. The following result for n=2k+3 was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008, \S4].

Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. 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 any 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\}.

How does one describe E^m(N) when N is not (2n-m)-connected? For general N see the sentence on E^{2n-1}(N) at the end of \S2. 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, Corollary 1.3]. 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}].

The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, \S5]. For k=0 Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because V_{2n+1,n+1} is (n-1)-connected). For k=1 Theorem 6.4 is covered by Theorem 6.2; for k\ge2 it is not. For application to knotted tori see [Skopenkov2016k, Theorem 5.1]. For generalization to arbitrary manifolds see survey [Skopenkov2006, \S5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4 V_{m,n+1} can be replaced by V_{M,M+n-m+1} for any M>n.

7 An orientation on the self-intersection set

Let f:N\to\Rr^m be a smooth map from an oriented n-manifold N where m\ge n+2. We assume that the closure
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of the self-intersection set of f has codimension 2 singularities, i.e., there is
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such that
  • both P and
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    are subpolyhedra of some triangulation of N,
  • we have \dim P\le\dim\Sigma(f)-2 and
  • \Sigma(f)-P is an open manifold consisting of self-transverse double points of f.

Definition 7.1 (A canonical orientation on \Sigma(f)-P). Take points x,y\in N away from P and such that fx=fy. Then a (2n-m)-base \xi_x tangent to \Sigma(f)-P at x gives a (2n-m)-base \xi_y:=df_y^{-1}df_x(\xi_x) tangent to \Sigma(f)-P at y. Since N is oriented, 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.

Remark 7.2 (Properties of the orientation just defined on \Sigma(f) - P)..

  1. 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)-P.
  2. The orientation on \Sigma(f)-P need not extend to
    Tex syntax error
    : take the smooth cone f:D^3\to\Rr^5 over a general position map g:S^2\to\Rr^4 having only two transverse self-intersection points, where the smooth cone is defined by f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2)), for x\in S^2 and t\in[0,1].
  3. The orientation on \Sigma(f)-P extends to
    Tex syntax error
    if m-n is odd [Hudson1969, Lemma 11.4].

Remark 7.3 (A canonical orientation on f\Sigma(f)-fP for m-n even). This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.

Take a (2n-m)-base \xi at a point x\in f\Sigma(f)-fP. Since N is oriented, we can take a positive (m-n)-base \eta_+ normal to f\Sigma(f) in one sheet of fN. Analogously construct an (m-n)-base \eta_- for the other sheet of fN. Since m-n is even, the orientation of the base (\xi,\eta_+,\eta_-) of \Rr^m does not depend on choosing the first and the other sheet of fN at x. 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_-).

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)-fP.

8 References

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

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