Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T06:53:20.310Z Has data issue: false hasContentIssue false

Classification of knotted tori

Published online by Cambridge University Press:  22 January 2019

A. Skopenkov*
Affiliation:
Moscow Institute of Physics and Technology, and Independent University of Moscow ([email protected])

Abstract

For a smooth manifold N denote by Em(N) the set of smooth isotopy classes of smooth embeddings N → ℝm. A description of the set Em(Sp × Sq) was known only for p = q = 0 or for p = 0, mq + 2 or for 2m ⩾ 2(p + q) + max{p, q} + 4. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For m ⩾ 2p + q + 3 we introduce an abelian group structure on Em(Sp × Sq) and describe this group ‘up to an extension problem’. This result has corollaries which, under stronger dimension restrictions, more explicitly describe Em(Sp × Sq). The proof is based on relations between sets Em(N) for different N and m, in particular, on a recent exact sequence of M. Skopenkov.

MSC classification

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Avvakumov, S.. The classification of certain linked 3-manifolds in 6-space. Moscow Math. J. 16 (2016), 125, arxiv:1408.3918.CrossRefGoogle Scholar
2Crowley, D., Ferry, S. C. and Skopenkov, M.. The rational classification of links of codimension > 2. Forum Math. 26 (2014), 239269, arXiv:1106.1455.CrossRefGoogle Scholar
3Cencelj, M., Repovš, D. and Skopenkov, A.. On the Browder-Haefliger-Levine-Novikov embedding theorems. Proc. Steklov Inst. Math. 247 (2004), 259268.Google Scholar
4Cencelj, M., Repovš, D. and Skopenkov, M.. Homotopy type of the complement of an immersion and classification of embeddings of tori. Russian Math. Surveys 62 (2007), 985987.CrossRefGoogle Scholar
5Cencelj, M., Repovš, D. and Skopenkov, M.. Classification of knotted tori in the 2-metastable dimension. Mat. Sbornik 203 (2012), 16541681, arXiv:0811.2745.CrossRefGoogle Scholar
6Crowley, D. and Skopenkov, A.. A classification of smooth embeddings of 4-manifolds in 7-space, II. Intern. J. Math. 22 (2011), 731757, arxiv:0808.1795.CrossRefGoogle Scholar
7Crowley, D. and Skopenkov, A.. Embeddings of non-simply connected 4-manifolds in 7-space, I. Classification modulo knots. arxiv:1611.04738.Google Scholar
8Crowley, D. and Skopenkov, A.. Embeddings of non-simply connected 4-manifolds in 7-space, II. On the smooth classification. arxiv:1612.04776.Google Scholar
9Goodwillie, T. and Weiss, M.. Embeddings from the point of view of immersion theory, II. Geom. Topol. 3 (1999), 103118.CrossRefGoogle Scholar
10Haefliger, A.. Knotted (4k − 1)-spheres in 6k-space. Ann. Math. 75 (1962), 452466.CrossRefGoogle Scholar
11Haefliger, A.. Differentiable embeddings of S n in S n + q for q > 2. Ann. Math., Ser. 3 83 (1966), 402436.CrossRefGoogle Scholar
12Haefliger, A.. Enlacements de sphères en codimension supérieure à 2. Comment. Math. Helv. 41 (1966–67), 5172.CrossRefGoogle Scholar
13Hirsch, M. W.. Differential topology (New York: Springer-Verlag: 1976).CrossRefGoogle Scholar
14Kervaire, M.. An interpretation of G. Whitehead's generalization of H. Hopf's invariant. Ann. Math. 62 (1959), 345362.CrossRefGoogle Scholar
20Milgram, R. J.. On the Haefliger knot groups. Bull. Amer. Math. Soc. 78 (1972), 861865.CrossRefGoogle Scholar
21Paechter, G.. On the groups πr(V mn), I. Quart. J. Math. Oxford, Ser.2 7 (1956), 249265.CrossRefGoogle Scholar
22Schultz, R.. On the inertia groups of a product of spheres. Trans. AMS 156 (1971), 137153.CrossRefGoogle Scholar
23Skopenkov, A.. On the deleted product criterion for embeddability of manifolds in ℝm. Comment. Math. Helv. 72 (1997), 543555.CrossRefGoogle Scholar
24Skopenkov, A.. On the Haefliger-Hirsch-Wu invariants for embeddings and immersions. Comment. Math. Helv. 77 (2002), 78124.CrossRefGoogle Scholar
25Skopenkov, A.. Classification of embeddings below the metastable dimension, arxiv:math/ 0607422v2.Google Scholar
26Skopenkov, A.. A new invariant and parametric connected sum of embeddings. Fund. Math. 197 (2007), 253269, arxiv:math/0509621.CrossRefGoogle Scholar
27Skopenkov, A.. Embedding and knotting of manifolds in Euclidean spaces. In Surveys in contemporary mathematics (eds.Young, N. and Choi, Y.). London Math. Soc. Lect. Notes,vol. 347 (Cambridge: Cambridge Univ. Press, 2008),pp. 248342, arxiv:math/0604045.Google Scholar
28Skopenkov, A.. Classification of smooth embeddings of 3-manifolds in 6-space. Math. Zeitschrift 260 (2008), 647672, arxiv:math/0603429.CrossRefGoogle Scholar
29Skopenkov, M.. A formula for the group of links in the 2-metastable dimension. Proc. AMS 137 (2009), 359369, arxiv:math/0610320.CrossRefGoogle Scholar
30Skopenkov, A.. A classification of smooth embeddings of 4-manifolds in 7-space, I. Topol. Appl. 157 (2010), 20942110, arxiv:0808.1795.CrossRefGoogle Scholar
31Skopenkov, A.. Embeddings of k-connected n-manifolds into ℝ2nk−1. Proc. AMS 138 (2010), 33773389, arxiv:0812.0263.CrossRefGoogle Scholar
32Skopenkov, M.. When is the set of embeddings finite? Intern. J. Math. 26 (2015), arxiv:1106.1878.CrossRefGoogle Scholar
33Skopenkov, A.. How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, to appear. arxiv:1402.1853.Google Scholar
34Skopenkov, A.. Classification of knotted tori, arxiv:1502.04470.Google Scholar
36Wall, C. T. C.. Surgery on compact manifolds (London: Academic Press, 1970).Google Scholar
37Zhubr, A.. On surgery of a sphere in a knotted torus, unpublished.Google Scholar