Embeddings just below the stable range: classification
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Contents |
1 Introduction
For notation and conventions see high codimension embeddings.
Classification Theorem 1.1. Let be a closed connected -manifold. The Whitney invariant
is bijective if either or and CAT=PL [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]
Classification of smooth embeddings of 3-manifolds in the 6-space is more complicated.
2 Definition of the Whitney invariant (for either n odd or N orientable)
Fix orientations on and, if is even, on . Fix an embedding . For an embedding the restrictions of and to are regular homotopic [Hirsch1959]. Since has an -dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of and assume that on . Take a general position homotopy relative to between the restrictions of and to . Then (i.e. `the intersection of this homotopy with ') is a 1-manifold (possibly non-compact) without boundary. Define to be the homology class of the closure of this 1-manifold:
The orientation on is defined for orientable as follows. (This orientation is defined for each but used only for odd .) For each point take a vector at tangent to . Complete this vector to a positive base tangent to . Since , by general position there is a unique point such that . The tangent vector at thus gives a tangent vector at to . Complete this vector to a positive base tangent to , where the orientation on comes from . The union of the images of the constructed two bases is a base at of . If this base is positive, then call the initial vector of positive. Since a change of the orientation on forces a change of the orientation of the latter base of , it follows that this condition indeed defines an orientation on .
Remark 2.1.
- The Whitney invariant is well-defined, i.e. independent of the choice of and of the isotopy making outside . This is so because the above definition is clearly equivalent to the following: is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set of a general position homotopy between and . (For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need -coefficients when is even.
- Clearly, . The definition of depends on the choice of , but we write not for brevity.
- Since a change of the orientation on forces a change of the orientation on , the class is independent of the choice of the orientation on . For the reflection with respect to a hyperplane we have (because we may assume that on and because a change of the orientation of forces a change of the orientation of ).
- The above definition makes sense for each , not only for .
- Clearly, is or for for the Hudson tori.
- for each embeddings and .
Construction of the embedding f a:N\to\Rr^{2n} from an embedding
f 0:N\to\Rr^{2n} and a\in H 1(N) (for orientable N and n\ge3) 3 Construction of the embedding from an embedding
and (for orientable and )
Represent by an embedding . Since any orientable bundle over is trivial, . Identify with . It remains to make an embedded surgery of to obtain an -sphere , and then we set .
Take a vector field on normal to . Extend along this vector field to a smooth map . Since and , by general position we may assume that is an embedding and misses . Since , we have . Hence the standard framing of in extends to an -framing on in . Thus extends to an embedding
This construction generalizes the construction of (from ).
Clearly, is or . Thus unless and CAT=DIFF
- all isotopy classes of embedings can be obtained (from a certain given embedding ) by the above construction;
- the above construction defines an action .
4 References
- [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [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.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.
arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013
- [Vrabec1977] J. Vrabec, Knotting a -connected closed -manifold in , Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
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