Embeddings just below the stable range: classification

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This page is being refereed under the supervision of the editorial board. Hence the page may not be edited at present. As always, the discussion page remains open for observations and comments.


1 Introduction

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]. In this page we summarize the situation for m=2n\ge6 and some more general situations.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S3].

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, 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

(because Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2) is given by multiplication with w_1(N)). 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 in [Skopenkov2006, \S5]. It would be interesting to check if this description of E^{2n}(N) is equivalent to its different forms [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in \Rr^6 is more complicated, see \S6.3 or [Skopenkov2016t].

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

Theorem 2.1 is generalized in \S6.2 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.)

For m\ge n+2 we define the standard embedding S^p\times S^{n-p}\to\Rr^m as the composition of the standard embeddings S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m. (Note that in [Skopenkov2015], [Skopenkov2015a] a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.)

Let 1_n:=(1,0,\ldots,0)\in S^n.

Let us construct, for each a\in\Zz and n\ge2, an embedding

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

The reader might first consider the case n=2.

Definition 3.1. 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}. Take the embedded sphere and embedded 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 them by an arc whose interior misses the two embedded manifolds. 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] 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.

Definition 3.2. For a\in\Zz we repeat the construction of Definition 3.1 replacing 2\partial D^{n+1}\times-1_{n-1} by |a| copies (1+\frac1k)\partial D^{n+1}\times-1_{n-1} (k=1,\dots,|a|) of S^n. The copies are outside D^{n+1}\times S^{n-1} and are `parallel' to \partial D^{n+1}\times-1_{n-1}. The copies have the standard orientation for a>0 or the opposite orientation for a<0. Then we make embedded connected sum by tubes joining each k-th copy to the (k+1)-th copy. We obtain an embedding g:S^n\to\Rr^{2n}. Let \Hud_n(a) be the linked embedded connected sum of g with the embedding \partial D^2\times S^{n-1}\subset\Rr^{2n} from Definition 3.1.

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 each n (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. 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 3.3 follows by calculation of the Whitney invariant (Remark 5.3.e below) and, for n even, by Theorem 2.1.

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) isotopic to \Hud_2(2)? It would also be interesting to find an explicit construction of an isotopy between \Hud_{2k}(a) and \Hud_{2k}(a+2), cf. [Vrabec1977, \S5].

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'.

Remark 3.5. (a) The analogue of Proposition 3.3 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 PL isotopic for n\ge3 [Skopenkov2006a]. It would be interesting to know if they are isotopic for n=2, or if they are smoothly isotopic for n=3.

(c) For n=2 these construction give 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.e 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 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}). There are versions of these constructions corresponding to Definition 3.4. For p=0 this corresponds to the Zeeman construction [Skopenkov2016h] and its composition with the second unframed Kirby move. 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 2.9.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). 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 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}. Identify \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*\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 a certain arc joining their images).

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.

Proposition 4.1. Linked embedded connected sum, or parametric connected sum, define free transitive actions of H_1(N;\Zz_{\varepsilon(n-1)}) on E^{2n}(N), unless n=3 in the smooth category.

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]. The definition in the PL category is analogous [Hudson1969, \S12], [Vrabec1977, p. 145], [Skopenkov2006, \S2.4]. 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 [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):=[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. Since m\ge n+2, by general position the closure Cl\Sigma(H) of the self-intersection set has codimension 2 singularities. So the closure 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 (\S7) and so carries a homology class with \Zz coefficients. Define the Whitney invariant to be the homology class:

\displaystyle  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)}).

(The orientation is defined for each m,n but used only for odd m-n. When m-n is even, for W(f) being well-defined we need \Zz_2-coefficients.)

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.

Remark 5.3. (a) The Whitney invariant is well-defined by Definition 5.2, i.e. is independent of the choice of H, which is proved analogously to [Skopenkov2006, \S2.4].

(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if f=f_0 on N_0, we can take F to be fixed on N_0.) 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) 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^m\to\Rr^m 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^m forces a change of the orientation of f\cap F).

(d) For the Hudson tori W(\Hud_n(a))=W(\Hud'_n(a)) is a or a\mod2 for n\ge3, is (a\mod2,0) for n=2.

(e) W(f\#g)=W(f) for each pair of embeddings f:N\to\Rr^m and g:S^n\to\Rr^m.

(f) For m=2n+1 the Whitney invariant can be recovered from the collection of pairwise linking coefficients of the components of N, cf. Remark 3.2.b of [Skopenkov2016h].

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.

Example 6.1 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If N is k-connected, then for an embedding f_0:N\to S^{2n-k} and a class a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)}) one can construct an embedding f_a:N\to S^{2n-k} by linked connected sum analogously to the case k=0.

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 categoriy.

This was proved for k-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, \S11], [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.

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.
\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 \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]. 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 so that the closure Cl\Sigma(f) of the self-intersection set of f has codimension 2 singularities.

Definition 7.1 (a natural orientation on \Sigma(f)). Take points x,y\in N away from the 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.

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].

Definition 7.2 (a natural orientation on f\Sigma(f) for m-n even). Take a (2n-m)-base \xi at a point x\in f\Sigma(f) away from the 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). 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 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_-).

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).

8 References

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