Knots, i.e. embeddings of spheres
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[edit] 1 Introduction
We work in a smooth category. In particular, terms embedding and smooth embedding or map and smooth map are used interchangeably. For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
[edit] 2 Examples
There are smooth embeddings which are not smoothly isotopic to the standard embedding. They are PS (piecewise smoothly) isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 of [Skopenkov2016c] and [Skopenkov2016f, Remark 1.1]).
Example 2.1. (a) Analogously to the Haefliger trefoil knot for any one constructs a smooth embedding , see [Skopenkov2016h, 5]. For even is not smoothly isotopic to the standard embedding; represents a generator of [Haefliger1962].
It would be interesting to know if for odd this embedding is a generator of . The last phrase of [Haefliger1962t] suggests that this is true for .
(b) For any let be the homotopy class of the Hopf map. Denote by the Zeeman map, see [Skopenkov2016h, Definition 2.2]. The embedded connected sum of the components of (a representative of) is not smoothly isotopic to the standard embedding; is a generator of [Skopenkov2015a, Corollary 2.13].
[edit] 3 Invariants
Let us define the Haefliger invariant . The definition is motivated by Haefliger's proof that any embedding is isotopic to the standard embedding for , and by analyzing what obstructs carrying this proof for .
By [Haefliger1962, 2.1, 2.2] any embedding has a framing extendable to a framed embedding of a -manifold whose boundary is , and whose signature is zero. For an integer -cycle in let be the linking number of with a slight shift of along the first vector of the framing. This defines a map . This map is a homomorphism (as opposed to the Arf map defined in a similar way [Pontryagin1959]). Then by Lefschetz duality there is a unique such that for any . Since has a normal framing, its intersection form is even. (Indeed, represent a class in by a closed oriented -submanifold . Then because has a normal framing.) Hence is an even integer. DefineSince the signature of is zero, there is a symplectic basis in . Then clearly
For an alternative definition via Seifert surfaces in -space, discovered in [Guillou&Marin1986], [Takase2004], see [Skopenkov2016t, the Kreck Invariant Lemma 4.5]. For a definition by Kreck, and for a generalization to 3-manifolds see [Skopenkov2016t, 4].
Sketch of a proof that is well-defined (i.e. is independent of , , and the framings), and is invariant under isotopy of . [Haefliger1962, Theorem 2.6] Analogously one defines and for a framed -submanifold of . Since is a characteristic number, it is independent of framed cobordism. So defines a homomorphism . The latter group is finite by the Serre theorem. Hence the homomorphism is trivial.
Since is a characteristic number, it is independent of framed cobordism of a framed (and hence of the isotopy of a framed ).
Therefore is a well-defined invariant of a framed cobordism class of a framed . By [Haefliger1962, 2.9] (cf. [Haefliger1962, 2.2 and 2.3]) is also independent of the framing of extendable to a framing of some -manifold having trivial signature. QED
For definition of the attaching invariant see [Haefliger1966], [Skopenkov2005, 3].
[edit] 4 Classification
Theorem 4.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. For the group is finite unless and , when is the sum of and a finite group.
Theorem 4.2 (Haefliger-Milgram). We have the following table for the group ; in the whole table ; in the fifth column ; in the last two columns :
Proof for the first four columns, and for the fifth column when is odd, are presented in [Haefliger1966, 8.15] (see also 6; some proofs are deduced from that paper using simple calculations, cf. [Skopenkov2005, 3]; there is a typo in [Haefliger1966, 8.15]: should be ). The remaining results follow from [Haefliger1966, 8.15] and [Milgram1972, Theorem F]. Alternative proofs for the cases are given in [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
Theorem 4.3 [Milgram1972, Corollary G]. We have if and only if either , or , or and , or and .
For a description of 2-components of see [Milgram1972, Theorem F]. Observe that no reliable reference (containing complete proofs) of results announced in [Milgram1972] appeared. Thus, strictly speaking, the corresponding results are conjectures.
The lowest-dimensional unknown groups are and . Hopefully application of Kreck surgery could be useful to find these groups, cf. [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
For the group has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].
Theorem 4.4 [Haefliger1966]. For there is the following exact sequence of abelian groups:
Here is the space of maps of degree . Restricting a map from to identifies as a subspace of . Define . Analogously define . Let be the stabilization homomorphism. The attaching invariant and the map are defined in [Haefliger1966], see also [Skopenkov2005, 3].
[edit] 5 Some remarks on codimension 2 knots
For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in ), a complete readily calculable classification (in the sense of Remark 1.2 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots. See e.g. the interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].
On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a] and [Ranicki1998].
[edit] 6 Proof of classification of (4k-1)-knots in 6k-space
Theorem 6.1. The Haefliger invariant is injective for .
The proof is a certain simplification of [Haefliger1962]. We present an exposition structured to make it more accessible to non-specialists.
Lemma 6.2. Let be a framed -connected -submanifold of such that , signature of is zero, and . Then there is a submanifold such that and .
Proof of Theorem 6.1 using Lemma 6.2. By the first three paragraphs of the proof of Theorem 3.1 in [Haefliger1962], for any embedding such that there is a framed -connected -submanifold of with zero signature such that and . Then by Lemma 6.2 there is a submanifold such that and . Recall that isotopy classes of embeddings are in 1--1 correspondence with -cobordism classes of oriented submanifolds of diffeomorphic to for , , cf. [Haefliger1966, 1.8], [Kervaire1965]. Hence is isotopic to standart embedding.
To prove Lemma 6.2 we need Lemmas 6.3, 6.4 and 6.5. Below manifolds can have non-empty boundaries.
Lemma 6.3 [Whitney lemma; [Prasolov2007], 22]. Let be a map from a connected oriented -manifold to a simply connected oriented -manifold . If , then
- If , there is a homotopy such that and is an embedding.
- Suppose in addition that and there is a map with from a connected oriented -manifold such that the algebraic intersection number of and is zero. Then there is a homotopy relative to the boundary such that and does not intersect . If is an embedding, the homotopy can be chosen so that is an embedding.
Below we denote by the Hurewicz map.
Lemma 6.4. Let be a -connected -manifold, and let be homology classes such that for every . Then there are embeddings with pairwise disjoint images representing , respectively.
Proof. As is -connected, is an isomorphism. For an element , let be a representative of the homotopy class . Applying item 1 of Lemma 6.3 to , we may assume that is an embedding.
Make the following inductive procedure. At the -th step, , assume that the embeddings are already constructed, and we construct . Since and is simply connected, is simply connected for any . The algebraic intersection number of and is zero for any . Hence we can apply item 2 of Lemma 6.3 to and and as above for any . So is replaced by a homotopic embedding , and the images of are pairwise disjoint. After -th step we obtain a required set of embeddings.
Lemma 6.5. Let be an orientable -submanifold of , and be an embedding such that and over the manifold has a framing whose first vector is tangent to . Assume that has zero algebraic self-intersection in . Then extends to an embedding such that .
Proof. (A slightly different proof is presented in the proof of Proposition 3.3 in [Haefliger1962].) Since has zero algebraic self-intersection in , the Euler class of the normal bundle of in is zero. Since over the manifold has a framing, we obtain that has a framing in .
Identify all the normal spaces of with the normal space at . The normal framing of in is orthogonal to . So defines a map . Let be the homotopy class of this map. This is the obstruction to extending to a normal -framing of in (so apriori ). It suffices to prove that .
Consider the exact sequence of the bundle : . By the following well-known assertion, : if , then is the obstruction to trivialization of the orthogonal complement to the field of -frames in corresponding to a representative of .
Consider a map of the exact sequences associated to the inclusion . The composition is the boundary map . The group is in natural 1--1 correspondence with the group of -bundles over . The image is the tangent bundle of . Since the Euler class of is , the map is injective. Since is an isomorphism, the map is injective. This and imply that .
Alternatively, by [Fomenko&Fuchs2016, Corollary in 25.4] is a finite group (in [Fomenko&Fuchs2016, Corollary in 25.4] the formula for is correct, although the formula for is incorrect because ). Since , we obtain that for . This and imply that .
Lemma 6.6. Let be a -submanifold of and let be an embedding such that . Then there is a smooth submanifold homeomorphic to and such that .
Lemma 6.6 is essentialy proved in [Haefliger1962, 3.3].
Proof of Lemma 6.2 using Lemmas 6.4, 6.5. By the fourth paragraph of the proof of Theorem 3.1 in [Haefliger1962], there is a basis in such that , and for any . From Lemma 6.4 it follows that there are embeddings with pairwise disjoint images representing , respectively. [!!!such that for every ]
For denote by the homotopy class of the shift of by the first vector of the framing of on . Since , we have . Since , the complement is -connected. Hence by Hurewicz Theorem implies . Therefore there are extensions of such that .
Take such that for any . Take a tubular neighborhoods of such that for any . The algebraic intersection number of and equals . We have . So we can apply item 2 of Lemma 6.3 to , the inclusion, and . So we may assume that does not intersect . Hence we may assume that .
Apply Lemma 6.5 to one by one, and to the manifold . Denote by the resulting maps. Define manifolds for inductively. Let , and let be a manifold obtained applying Lemma 6.6 for and . By Lemma 6.6, and for . Since is a symplectic basis in , it follows that . Then from Generalized Poincare conjecture proved by Smale it follows that . Hence . Then take .
[edit] References
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