×

Embeddings of non-simply-connected 4-manifolds in 7-space. I: Classification modulo knots. (English) Zbl 1484.57024

The Knotting Problem is an important classical problem in topology, which ask for classification up to isotopy of the embeddings of a given manifold into another given manifold. This paper deals with the Knotting Problem for \(4\)-manifolds in \(\mathbb{R}^7\) in the smooth setting, i.e., all manifolds, embeddings and isotopies are smooth.
Let \(N\) be a closed connected orientable \(4\)-manifold with torsion free first integral homology group \(H_1(N;\mathbb{Z})\). The main result is a complete readily calculable classification of embeddings \(N\to \mathbb{R}^7\), up to an equivalence generated by isotopies and an embedded connected sum construction with embeddings \( S^4 \to \mathbb{R}^7\). Such a classification was earlier known only for \(H_1(N;\mathbb{Z})=0\) by a result of A. Haefliger (ed.) and R. Narasimhan (ed.) [Essays on topology and related topics. Mémoires dédiés à Georges de Rham. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0192.31107)].
The paper gives a detailed overview of the Knotting Problem and a comprehensive self-contained exposition of the main notions and constructions involved.

MSC:

57R40 Embeddings in differential topology
57R52 Isotopy in differential topology
57R67 Surgery obstructions, Wall groups
57Q35 Embeddings and immersions in PL-topology
55R15 Classification of fiber spaces or bundles in algebraic topology

Citations:

Zbl 0192.31107
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Mosc. Math. J. 16 (2016), no. 1, 1-25. MR 3470574. Preprint version arXiv:1408.3918 [math.GT]. · Zbl 1347.57026
[2] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365-385. MR 184241 · Zbl 0136.20602
[3] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension 4 dans R 7 , Essays on Topology and Related Topics (Mémoires dédiésà Georges de Rham), Springer, New York, 1970, pp. 156-166. MR 0270384 · Zbl 0199.27003
[4] M.Čadek, M. Crabb, and J. Vanžura, Obstruction theory on 8-manifolds, Manuscripta Math. 127 (2008), no. 2, 167-186. MR 2442894 · Zbl 1157.55011
[5] M. Cencelj, D. Repovš, and A. Skopenkov, On the Browder-Levin-Novikov embedding theo-rems, Tr. Mat. Inst. Steklova 247 (2004), Geom. Topol. i Teor. Mnozh., 280-290 (Russian). MR 2168178. English translaton: Proc. of the Steklov Inst. Math. 247 (2004), 259-268. · Zbl 1108.57021
[6] M. Cencelj, D. Repovš, and M. Skopenkov, Homotopy type of complement to the immersion and classification of embeddings of tori, Uspekhi Mat. Nauk 62 (2007), no. 5(377), 165-166 (Russian). MR 2373765. English translation: Russian Math. Surveys 62 (2007), no. 5, 985-987. · Zbl 1141.57009
[7] M. Cencelj, D. Repovš, and M. Skopenkov, Classification of embeddings of tori in the 2-metastable dimension, Mat. Sb. 203 (2012), no. 11, 129-158 (Russian). MR 3053230. English translation: Sbornik: Math. 203 (2012), no. 11, 1654-1681. · Zbl 1263.57019
[8] D. Crowley, 5-manifolds: 1-connected, Bull. Man. Atl. (2011), http://www.boma.mpim-bonn. mpg.de/articles/33.
[9] D. Crowley, S. C. Ferry, and M. Skopenkov, The rational classification of links of codimension > 2, Forum Math. 26 (2014), no. 1, 239-269. MR 3176630. Preprint version arXiv:1106.1455 [math.AT]. · Zbl 1300.57032
[10] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knots, preprint arXiv:1611.04738 [math.GT].
[11] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification, published online by Proc. Royal Soc. of Edinburgh Sec. A, http://dx.doi.org/10.1017/prm.2020.103. Preprint version: arXiv:1612.04776 [math.GT]. · Zbl 1490.57036 · doi:10.1017/prm.2020.103
[12] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space. III. Piecewise-linear classification, unpublished.
[13] D. Crowley and A. Skopenkov, A classification of smooth embeddings of four-manifolds in seven-space, II, Internat. J. Math. 22 (2011), no. 6, 731-757. MR 2812086. Preprint version arXiv:0808.1795 [math.GT]. · Zbl 1230.57026
[14] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern geometry -methods and appli-cations. Part II: the geometry and topology of manifolds, Springer Verlag, 2012.
[15] T. G. Goodwillie and M. Weiss, Embeddings from the point of view of immersion theory. II, Geom. Topol. 3 (1999), 103-118. MR 1694808 · Zbl 0927.57028
[16] A. Haefliger, Differential embeddings of S n in S n+q for q > 2, Ann. of Math. (2) 83 (1966), 402-436. MR 202151 · Zbl 0151.32502
[17] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no. 3, 707-754. MR 1709301 · Zbl 0935.57039
[18] M. Kreck and W. Lück, The Novikov conjecture, Oberwolfach Seminars, vol. 33, Birkhäuser Verlag, Basel, 2005. MR 2117411. Geometry and algebra. · Zbl 1058.19001
[19] M. Kreck and S. Stolz, Some nondiffeomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature, J. Differential Geom. 33 (1991), no. 2, 465-486. MR 1094466 · Zbl 0733.53025
[20] Embeddings in euclidean space: an introduction to their classification, http://www.map.mpim-bonn.mpg.de/Embeddings_in_Euclidean_space:_an_introduction_to_their_classification, Manifold Atlas Project.
[21] 4-manifolds in 7-space, www.map.mpim-bonn.mpg.de/4-manifolds_in_7-space, Manifold At-las Project.
[22] J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64 (1956), 399-405. MR 82103 · Zbl 0072.18402
[23] J. Milnor, Lectures on the h-cobordism theorem, Notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, N.J., 1965. MR 0190942 · Zbl 0161.20302
[24] G. F. Paechter, The groups πr(Vn,m). I, Quart. J. Math. Oxford Ser. (2) 7 (1956), 249-268. MR 131878 · Zbl 0073.18402
[25] D. Repovš and A. Skopenkov, New results on embeddings of polyhedra and manifolds into Euclidean spaces, Uspekhi Mat. Nauk 54 (1999), no. 6(330), 61-108 (Russian). MR 1744658. English translation: Russian Math. Surveys 54 (1999), no. 6, 1149-1196. · Zbl 0958.57025
[26] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999), no. 2, 255-272. MR 1704247 · Zbl 0931.57023
[27] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions, Com-ment. Math. Helv. 77 (2002), no. 1, 78-124. MR 1898394 · Zbl 1012.57035
[28] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253-269. MR 2365891. Preprint version: arXiv:math/0509621 [math.GT]. · Zbl 1145.57019
[29] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, Surveys in con-temporary mathematics, London Math. Soc. Lecture Note Ser., vol. 347, Cambridge Univ. Press, Cambridge, 2008, pp. 248-342. MR 2388495. Preprint version: arXiv:math/0604045 [math.GT]. · Zbl 1154.57019
[30] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no. 3, 647-672. MR 2434474. Preprint version: arXiv:math/0603429 [math.GT]. · Zbl 1167.57013
[31] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, I, Topology Appl. 157 (2010), no. 13, 2094-2110. MR 2665233. Preprint version: arXiv:math/0512594 [math.GT]. · Zbl 1202.57026
[32] A. Skopenkov, Embeddings of k-connected n-manifolds into R 2n−k−1 , Proc. Amer. Math. Soc. 138 (2010), no. 9, 3377-3389. MR 2653966. Preprint version: arXiv:0812.0263 [math.GT]. · Zbl 1206.57034
[33] A. Skopenkov, How do autodiffeomorphisms act on embeddings?, Proc. Roy. Soc. Edin-burgh Sect. A 148 (2018), no. 4, 835-848. MR 3841501. Preprint version: arXiv:1402.1853 [math.GT]. · Zbl 1409.57030
[34] A. Skopenkov, Classification of knotted tori, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 2, 549-567. MR 4080451. Preprint version: arXiv:1502.04470 [math.GT]. · Zbl 1439.57048
[35] M. Skopenkov, When is the set of embeddings finite up to isotopy?, Internat. J. Math. 26 (2015), no. 7, 1550051, 28. MR 3357040. Preprint version: arXiv:1106.1878 [math.GT]. · Zbl 1325.57014
[36] S. Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. (2) 69 (1959), 327-344. MR 105117 · Zbl 0089.18201
[37] S. Smale, On the structure of 5-manifolds, Ann. of Math. (2) 75 (1962), 38-46. MR 141133 · Zbl 0101.16103
[38] N. Steenrod, The topology of fibre bundles, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. MR 1688579 · Zbl 0942.55002
[39] D. Tonkonog, Embedding punctured n-manifolds in Euclidean (2n − 1)-space, preprint arXiv:1010.4271 [math.GT].
[40] C. T. C. Wall, Classification problems in differential topology. V. On certain 6-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid. 2 (1966), 306. MR 215313 · Zbl 0149.20601
[41] C. T. C. Wall, Surgery on compact manifolds, London Math. Soc. Monographs, no. 1, Aca-demic Press, London-New York, 1970. MR 0431216 · Zbl 0219.57024
[42] G. W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508 · Zbl 0406.55001
[43] T. Yasui, Enumerating embeddings of n-manifolds in Euclidean (2n − 1)-space, J. Math. Soc. Japan 36 (1984), no. 4, 555-576. MR 759414 · Zbl 0557.57019
[44] E. C. Zeeman, A brief history of topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch’s 60th birthday, http://zakuski.utsa.edu/ gokhman/ecz/hirsch60.pdf.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.