High codimension links
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− | + | ====Construction of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)$==== | |
− | ====Construction of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q) | + | |
Take $x\in\pi_p(S^{m-q-1})$ | Take $x\in\pi_p(S^{m-q-1})$ | ||
Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$. | Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$. | ||
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See Figure 3.2 of \cite{Skopenkov2006}. | See Figure 3.2 of \cite{Skopenkov2006}. | ||
− | ====Definition of linking coefficient $\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$ for $m\ge q+3 | + | ====Definition of linking coefficient $\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$ for $m\ge q+3$==== |
Fix orientations of $S^p$, $S^q$, $S^m$ and $D^{m-p}$. | Fix orientations of $S^p$, $S^q$, $S^m$ and $D^{m-p}$. | ||
Take an embedding $f:S^p\sqcup S^q\to S^m$. | Take an embedding $f:S^p\sqcup S^q\to S^m$. | ||
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====$\alpha$-invariant==== | ====$\alpha$-invariant==== | ||
− | By Freudenthal Suspension Theorem | + | By Freudenthal Suspension Theorem $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for |
− | $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$. | + | $m\ge\frac p2+q+2$. |
The stable suspension of the linking coefficient can be described alternatively as follows. | The stable suspension of the linking coefficient can be described alternatively as follows. | ||
For an embedding $f:S^p\sqcup S^q\to S^m$ define a map | For an embedding $f:S^p\sqcup S^q\to S^m$ define a map | ||
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Note that $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite{Skopenkov2006}. | Note that $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite{Skopenkov2006}. | ||
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Revision as of 13:29, 13 February 2013
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
For notation and conventions throughout this page see high codimension embeddings.
`Embedded connected sum' defines a commutative group structure on for . See Figure 3.3. of [Skopenkov2006], [Haefliger1966] [Haefliger1966C].
2 General position and the Hopf linking
General Position Theorem 2.1. For each -manifold and , every two embeddings are isotopic.
The restriction in Theorem 2.1 is sharp for non-connected manifolds.
Example: the Hopf linking 2.2. For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf Linking is shown in Figure~2.1.a of [Skopenkov2006]. For arbitrary (including ) the image of the Hopf Linking is the union of two -spheres:
3 The Zeeman construction and linking coefficient
The following table was obtained by Zeeman around 1960:
1 Construction of the Zeeman map
Take Define embedding on to be the standard embedding into . Take any map . Define embedding on to be the composition
where is the equatorial inclusion and the latter inclusion is the standard. See Figure 3.2 of [Skopenkov2006].
2 Definition of linking coefficient for
Fix orientations of , , and . Take an embedding . Take an embedding such that intersects transversally at exactly one point with positive sign (see Figure 3.1 of [Skopenkov2006]). Then the restriction of to is a homotopy equivalence.
(Indeed, since , the complement is simply-connected. By Alexander duality induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.1. (a) Clearly, is indeed independent of .
(b) For there is a simpler alternative `homological' definition. That definition works for as well.
(c) Analogously one can define for .
(d) This definition works for if is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for . So is surjective and is injective.
3 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 3.2. If , then both and are isomorphisms for and for , in the PL and DIFF cases, respectively.
The surjectivity of (=the injectivity of ) follows from . The injectivity of (=the surjectivity of ) is proved in [Haefliger1962T], [Zeeman1962] (or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of [Skopenkov2006]).
4 -invariant
By Freudenthal Suspension Theorem is an isomorphism for . The stable suspension of the linking coefficient can be described alternatively as follows. For an embedding define a map
See Figure 3.1 of [Skopenkov2006]. For define the -invariant by
The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map is the quotient map. See Figure 3.4 of [Skopenkov2006]. The map is an isomorphism for .
(For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
We have by Lemma 5.1 of [Kervaire1959].
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006].
4 Invariants
5 Further discussion
6 References
- [Haefliger1962T] Template:Haefliger1962T
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966C] Template:Haefliger1966C
- [Kervaire1959] M. A. Kervaire, A note on obstructions and characteristic classes, Amer. J. Math. 81 (1959), 773–784. MR0107863 (21 #6585) Zbl 0124.16302
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
2 General position and the Hopf linking
General Position Theorem 2.1. For each -manifold and , every two embeddings are isotopic.
The restriction in Theorem 2.1 is sharp for non-connected manifolds.
Example: the Hopf linking 2.2. For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf Linking is shown in Figure~2.1.a of [Skopenkov2006]. For arbitrary (including ) the image of the Hopf Linking is the union of two -spheres:
3 The Zeeman construction and linking coefficient
The following table was obtained by Zeeman around 1960:
1 Construction of the Zeeman map
Take Define embedding on to be the standard embedding into . Take any map . Define embedding on to be the composition
where is the equatorial inclusion and the latter inclusion is the standard. See Figure 3.2 of [Skopenkov2006].
2 Definition of linking coefficient for
Fix orientations of , , and . Take an embedding . Take an embedding such that intersects transversally at exactly one point with positive sign (see Figure 3.1 of [Skopenkov2006]). Then the restriction of to is a homotopy equivalence.
(Indeed, since , the complement is simply-connected. By Alexander duality induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.1. (a) Clearly, is indeed independent of .
(b) For there is a simpler alternative `homological' definition. That definition works for as well.
(c) Analogously one can define for .
(d) This definition works for if is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for . So is surjective and is injective.
3 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 3.2. If , then both and are isomorphisms for and for , in the PL and DIFF cases, respectively.
The surjectivity of (=the injectivity of ) follows from . The injectivity of (=the surjectivity of ) is proved in [Haefliger1962T], [Zeeman1962] (or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of [Skopenkov2006]).
4 -invariant
By Freudenthal Suspension Theorem is an isomorphism for . The stable suspension of the linking coefficient can be described alternatively as follows. For an embedding define a map
See Figure 3.1 of [Skopenkov2006]. For define the -invariant by
The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map is the quotient map. See Figure 3.4 of [Skopenkov2006]. The map is an isomorphism for .
(For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
We have by Lemma 5.1 of [Kervaire1959].
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006].
4 Invariants
5 Further discussion
6 References
- [Haefliger1962T] Template:Haefliger1962T
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966C] Template:Haefliger1966C
- [Kervaire1959] M. A. Kervaire, A note on obstructions and characteristic classes, Amer. J. Math. 81 (1959), 773–784. MR0107863 (21 #6585) Zbl 0124.16302
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069