High codimension links

From Manifold Atlas
Jump to: navigation, search
This page has been accepted for publication in the Bulletin of the Manifold Atlas.

The user responsible for this page is Askopenkov. No other user may edit this page at present.

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.

On this page we describe readily calculable classifications of embeddings of closed disconnected manifolds into \Rr^m up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

For an s-tuple (n):=(n_1,\ldots,n_s) denote S^{(n)}:=S^{n_1}\sqcup\ldots\sqcup S^{n_s}. Although S^{(n)} is not a manifold when n_1,\ldots,n_s are not all equal, embeddings S^{(n)}\to S^m and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by E^m(S^{(n)}) the set of embeddings S^{(n)}\to S^m up to isotopy.

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

The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):

\displaystyle \begin{array}{c|c|c|c|c|c|c|c}  m                   &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1    &\infty      &2      &2         &24        &1         &1  \end{array}
Figure 1: Component-wise embedded connected sum

A component-wise version of embedded connected sum [Skopenkov2016c, \S5] defines a commutative group structure on the set E^m(S^{(n)}) for m-3\ge n_i [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, \S1], [Avvakumov2017, \S1.4], see Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

2 Examples

Recall that for any q-manifold N and m\ge2q+2, every two embeddings N\to\Rr^m are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction m\ge2q+2 is sharp for non-connected manifolds.

Example 2.1 (The Hopf Link). (a) For every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \partial D^{q+1}\times0 and 0\times\partial D^{q+1} in \partial(D^{q+1}\times D^{q+1})\cong S^{2q+1};
  • or they are given as the sets of points in \Rr^{2q+1} satisfying the following equations:
\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right.  \qquad\text{and}\qquad  \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.

This embedding is distinguished from the standard embedding by the linking number (cf. \S3).

(b) For any p,q there is an embedding S^p\sqcup S^q\to\Rr^{p+q+1} which is not isotopic to the standard embedding.

Analogously to (a), the image is the union of two spheres which can be described as follows:

  • either the spheres are \partial D^{p+1}\times0 and 0\times\partial D^{q+1} in \partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}.
  • or they are given as the points in \Rr^{p+q+1} satisfying the following equations:
\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right.  \qquad\text{and}\qquad  \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.

This embedding is also distinguished from the standard embedding by the linking number (cf. \S3).

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with prescribed linking coefficient
\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

Tex syntax error
see Figure 3. We have
Tex syntax error
. Let
Tex syntax error
.

One can easily check that \zeta is well-defined and is a homomorphism.

3 The linking coefficient

Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.

Definition 3.1 (The linking coefficient). We define a map

\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.

Take an embedding f:S^p\sqcup S^q\to S^m representing an element [f]\in E^m(S^p\sqcup S^q). Take an embedding g:D^{m-q}\to S^m such that g(D^{m-q}) intersects f(S^q) transversely at exactly one point with positive sign; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

Then the restriction h':S^{m-q-1}\to S^m-fS^q of g to \partial D^{m-q} is a homotopy equivalence.

(Indeed, since m\ge q+3, by general position the complement S^m-fS^q is simply-connected. By Alexander duality, h' induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems h' is a homotopy equivalence.)

Let h be a homotopy inverse of h'. Define

\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).

Remark 3.2. (a) Clearly, \lambda[f] is well-defined, i.e. is independent of the choices of g,h',h and of the representative f of [f]. One can check that \lambda is a homomorphism.

(b) Analogously one can define \lambda_{21}[f]\in\pi_q(S^{m-p-1}) for m\ge p+3, by exchanging p and q in the above definition.

(c) Clearly, \lambda\zeta=\id\pi_p(S^{m-q-1}) for the Zeeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].

Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism \Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m} is an isomorphism for m\ge\frac p2+q+2. The stable suspension of the linking coefficient can be described as follows.

Definition 3.3 (The \alpha-invariant). We define a map E^m(S^p\sqcup S^q)\to\pi^S_{p+q+1-m} for p,q\le m-2. Take an embedding f:S^p\sqcup S^q\to S^m representing an element [f]\in E^m(S^p\sqcup S^q). Define the Gauss map (see Figure 4)

\displaystyle \widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.

For p,q\le m-2 define the \alpha-invariant by

\displaystyle \alpha[f]=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.

The second isomorphism in this formula is the suspension isomorphism. The map v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q} is the quotient map, see Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence for m\ge q+2. (For m\ge q+3 this follows by general position and for m=q+2 by the cofibration Barratt-Puppe exact sequence of pair (S^p\times S^q,S^p\vee S^q) and by the existence of a retraction \Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q).)

One can easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in Example 5.2.a,c.

Note that the \alpha-invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, \S5].

4 Classification in the metastable range

The Haefliger-Zeeman Theorem 4.1. If 1\le p\le q, then both \lambda and \zeta are isomorphisms for m\ge\frac{3q}2+2 in the smooth category, and for m\ge\frac p2+q+2 in the PL category.

The surjectivity of \lambda (or the injectivity of \zeta) follows from \lambda\zeta=\id. The injectivity of \lambda (or the surjectivity of \zeta) is proved in [Haefliger1962t, Theorem in \S5], [Zeeman1962] (or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).

An analogue of this result holds for links with many components, each of the same dimension [Haefliger1962t, Theorem in \S5]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

Theorem 4.2. The collection of pairwise linking coefficients

\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}

is a 1-1 correspondence for m\ge\frac{3q}2+2.

5 Examples beyond the metastable range

For l=1 the results of this section are parts of low-dimensional link theory, so they were known well before given references.

First we present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.

Example 5.1 (Borromean rings). (a) There is a non-trivial embedding S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, \S6].

In order to construct such an embedding, denote coordinates in \Rr^{3l} by (x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l). The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:

\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1,\end{array}\right. \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1. \end{array}\right.
Figure 6: The Borromean rings

See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.

(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) 2l-disks disjoint from the remaining sphere.

(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint 2l-disks (intersecting the remaining sphere). Moreover, we can take these 2l-disks so that

  • each one of them intersects the remaining sphere transversely by an (l-1)-sphere;
  • the obtained two disjoint (l-1)-spheres in the remaining sphere have linking number \pm1, i.e. one of them spans an l-disk (in the remaining sphere) itersecting the other transversely at exacly one point.

(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding g:S^{2l-1}_1\sqcup S^{2l-1}_2\sqcup S^{2l-1}_3\to S^{3l}. Assume that g is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let D_2,D_3\subset S^{3l} be disjoint oriented embedded 2l-disks in general position to g_1:=g|_{S^{2l-1}_1}, and such that g(S^{2l-1}_i)=\partial D_i for i=2,3. Then for j=2,3 the preimage g_1^{-1}D_j is an oriented (l-1)-submanifold of S^{2l-1}_1 missing g_1^{-1}D_{5-j}. Let \mu(g) be the linking number of g_1^{-1}D_2 and g_1^{-1}D_3 in S^{2l-1}_1.

(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of D_2,D_3, and of the isotopy of g. The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, \S4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, \S7] by the well-known `linking number' definition of the Whitehead invariant \pi_{2l-1}(S^l\vee S^l)\to\Z [Skopenkov2020e, \S2, Sketch of a proof of (b1)]. If g is pairwise unlinked, then the number \mu is independent of permutation of the components, up to multiplication by \pm1 [HaefligerSteer1965] (this can be easily proved directly).

(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, \S4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].

Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.

Example 5.2 (Whitehead link). (a) For every positive integer l there is a non-trivial embedding w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient \lambda_{21}(w) is trivial).

Figure 7: The Borromean rings and Whitehead link for l = 1

Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)

(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.

(c) For l\ne1,3,7 the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, \lambda_{12}(w) equals to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l) [Haefliger1962t, end of \S6] (I do not know a written proof of this except [Skopenkov2015a, the Whitehead link Lemma 2.14] for l even). For l=1,3,7 the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].

(d) This example (in higher dimensions, i.e. for l>1) seems to have been discovered by Whitehead, in connection with Whitehead product.

(e) For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link see [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no high-connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed connected manifolds and \frac{m-1}2=\dim N_i\ge2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 otherwise. Then the collection of pairwise linking coefficients

\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} :  E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. Let N_1 and N_2 be closed n-dimensional homologically (2n-m+1)-connected orientable manifolds. For an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016, \S1].

Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

In this section we assume that (n) is an s-tuple such that m\ge n_i+3.

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)}) [Haefliger1966a, \S\S 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, \S1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-tuple of the isotopy classes of its components:

\displaystyle  r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. Define

\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u is defined to be the homomorphism

\displaystyle  u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a, Theorem 2.4] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If p\le q\le m-3 and 3m\ge2p+2q+6, then there is a homomorphism \beta for which the following map is an isomorphism

\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).

The map \beta and its right inverse \pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12} are constructed in [Haefliger1966a, \S10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 implies that for any l\ge2 we have an isomorphism

\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{\varepsilon(l)}.

(b) For any l\not\in\{1,3,7\}, the map

\displaystyle \lambda_{12}\oplus\lambda_{21}:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\pi_{2l-1}(S^l)

is injective and its image is \{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977, Main Theorem in \S1] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any l\in\{1,3,7\} the map \lambda_{12}\oplus\lambda_{21} in (b) above is not injective [Haefliger1962t, \S6].

Theorem 8.3. For any l>1 there an isomorphism

\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{\varepsilon(l)}^3\oplus\Z

which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (\S5).

This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, \S6].

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and \S1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
Tex syntax error
.

(b) determine whether E^m_{PL}(S^{(n)}) is finite.

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups

\displaystyle  \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the s-tuples m-n_1,\ldots,m-n_s are the same for different terms. Denote W :=\vee_{i=1}^s S^{m-n_i-1}. For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, define \lambda:= \oplus_{j=1}^s \lambda_j.

Taking the Whitehead product with the class of the identity in \pi_{n_j}(S^{n_j}) defines a homomorphism w_j \colon \ker p_{n_j,j}\to \Pi_{m-2}^{(n-1)}. Define w:=\oplus_{j=1}^s w_j.

The definition of the homomorphism \mu is given in [Haefliger1966a, 1.5].

Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

(b) The map \lambda\otimes\Qq \colon E^m_{PL}(S^{(n)})\otimes\Qq \to \ker(w\otimes\Qq) is an isomorphism.

Part (b) follows because \mu \otimes \Qq = 0 [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers \Qq splits into short exact sequences.

In general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox