Embeddings of connected manifolds into
Author:
A. Skopenkov
Journal:
Proc. Amer. Math. Soc. 138 (2010), 33773389
MSC (2010):
Primary 57R40, 57Q37; Secondary 57R52
Published electronically:
May 4, 2010
MathSciNet review:
2653966
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We obtain estimations for isotopy classes of embeddings of closed connected manifolds into for and . This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of on the set of embeddings. The proof involves a reduction to the classification of embeddings of a punctured manifold and uses the parametric connected sum of embeddings. Corollary. Suppose that is a closed almost parallelizable connected manifold and . Then the set of isotopy classes of embeddings is in 11 correspondence with for .
 [BG71]
J.
C. Becker and H.
H. Glover, Note on the embedding of manifolds in
Euclidean space, Proc. Amer. Math. Soc. 27 (1971), 405–410.
MR
0268903 (42 #3800), http://dx.doi.org/10.1090/S00029939197102689030
 [BH70]
Jacques
Boéchat and André
Haefliger, Plongements différentiables des
variétés orientées de dimension 4 dans
𝑅⁷, Essays on Topology and Related Topics
(Mémoires dédiés à Georges de Rham), Springer,
New York, 1970, pp. 156–166 (French). MR 0270384
(42 #5273)
 [CS08]
D. Crowley and A. Skopenkov, A classification of smooth embeddings of manifolds in space, II, submitted, arXiv:math/0808.1795.
 [HCEC]
http://www.map.him.unibonn.de/index.php/ High_codimension_embeddings:_classification
 [Hu69]
J.
F. P. Hudson, Piecewise linear topology, University of Chicago
Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees,
W. A. Benjamin, Inc., New YorkAmsterdam, 1969. MR 0248844
(40 #2094)
 [PCS]
http://www.map.him.unibonn.de/index.php/Parametric_connected_sum
 [Po85]
M. M. Postnikov, Homotopy theory of CWcomplexes, Nauka, Moscow, 1985 (in Russian).
 [Pr07]
V.
V. Prasolov, Elements of homology theory, Graduate Studies in
Mathematics, vol. 81, American Mathematical Society, Providence, RI,
2007. Translated from the 2005 Russian original by Olga Sipacheva. MR 2313004
(2008d:55001)
 [Ri70]
R. D. Rigdon, Thesis, Ph.D. Thesis (1970).
 [RS99]
D.
Repovsh 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, with Russian summary); English
transl., Russian Math. Surveys 54 (1999), no. 6,
1149–1196. MR 1744658
(2001c:57028), http://dx.doi.org/10.1070/rm1999v054n06ABEH000230
 [Sa99]
Osamu
Saeki, On punctured 3manifolds in 5sphere, Hiroshima Math.
J. 29 (1999), no. 2, 255–272. MR 1704247
(2000h:57045)
 [Sk05]
A. Skopenkov, A classification of smooth embeddings of manifolds in space, submitted, arXiv:math/0512594.
 [Sk06]
A. Skopenkov, Classification of embeddings below the metastable dimension, submitted, arXiv:math/0607422.
 [Sk07]
A.
Skopenkov, A new invariant and parametric connected sum of
embeddings, Fund. Math. 197 (2007), 253–269. MR 2365891
(2008k:57044), http://dx.doi.org/10.4064/fm197012
 [Sk08]
Arkadiy
B. Skopenkov, Embedding and knotting of manifolds in Euclidean
spaces, Surveys in contemporary mathematics, London Math. Soc.
Lecture Note Ser., vol. 347, Cambridge Univ. Press, Cambridge, 2008,
pp. 248–342. MR 2388495
(2009e:57040)
 [Sk081]
Arkadiy
Skopenkov, A classification of smooth embeddings of 3manifolds in
6space, Math. Z. 260 (2008), no. 3,
647–672. MR 2434474
(2010e:57028), http://dx.doi.org/10.1007/s0020900702941
 [Sk]
A. Skopenkov, Embeddings of connected manifolds into ; arXiv:math/ 0812.0263.
 [Vr89]
Jože
Vrabec, Deforming a PL submanifold of
Euclidean space into a hyperplane, Trans. Amer.
Math. Soc. 312 (1989), no. 1, 155–178. MR 937253
(89g:57029), http://dx.doi.org/10.1090/S00029947198909372537
 [Wh50]
J.
H. C. Whitehead, A certain exact sequence, Ann. of Math. (2)
52 (1950), 51–110. MR 0035997
(12,43c)
 [Ya83]
Tsutomu
Yasui, On the map defined by regarding embeddings as
immersions, Hiroshima Math. J. 13 (1983), no. 3,
457–476. MR
725959 (85g:57016)
 [BG71]
 J. C. Becker and H. H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. of the Amer. Math. Soc. 27:2 (1971), 405410. MR 0268903 (42:3800)
 [BH70]
 J. Boéchat and A. Haefliger, Plongements différentiables de variétés orientées de dimension dans , Essays on topology and related topics, Springer, 1970, 156166. MR 0270384 (42:5273)
 [CS08]
 D. Crowley and A. Skopenkov, A classification of smooth embeddings of manifolds in space, II, submitted, arXiv:math/0808.1795.
 [HCEC]
 http://www.map.him.unibonn.de/index.php/ High_codimension_embeddings:_classification
 [Hu69]
 J. F. P. Hudson, PiecewiseLinear Topology, Benjamin, New York, Amsterdam, 1969. MR 0248844 (40:2094)
 [PCS]
 http://www.map.him.unibonn.de/index.php/Parametric_connected_sum
 [Po85]
 M. M. Postnikov, Homotopy theory of CWcomplexes, Nauka, Moscow, 1985 (in Russian).
 [Pr07]
 V. V. Prasolov, Elements of Homology Theory, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI, 2007. Earlier Russian version available at http://www.mccme.ru/prasolov. MR 2313004 (2008d:55001)
 [Ri70]
 R. D. Rigdon, Thesis, Ph.D. Thesis (1970).
 [RS99]
 D. Repovš and A. Skopenkov, New results on embeddings of polyhedra and manifolds into Euclidean spaces, Uspekhi Mat. Nauk 54:6 (1999), 61109 (in Russian); English transl., Russ. Math. Surv. 54:6 (1999), 11491196. MR 1744658 (2001c:57028)
 [Sa99]
 O. Saeki, On punctured manifolds in sphere, Hiroshima Math. J. 29 (1999), 255272. MR 1704247 (2000h:57045)
 [Sk05]
 A. Skopenkov, A classification of smooth embeddings of manifolds in space, submitted, arXiv:math/0512594.
 [Sk06]
 A. Skopenkov, Classification of embeddings below the metastable dimension, submitted, arXiv:math/0607422.
 [Sk07]
 A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253269. arXiv:math/0509621. MR 2365891 (2008k:57044)
 [Sk08]
 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, Cambridge Univ. Press, Cambridge, 2008, 248342. arXiv:math/0604045. MR 2388495 (2009e:57040)
 [Sk081]
 A. Skopenkov, A classification of smooth embeddings of manifolds in space, Math. Zeitschrift 260:3 (2008), 647672. arXiv:math/0603429. MR 2434474
 [Sk]
 A. Skopenkov, Embeddings of connected manifolds into ; arXiv:math/ 0812.0263.
 [Vr89]
 J. Vrabec, Deforming a PL submanifold of Euclidean space into a hyperplane, Trans. Amer. Math. Soc. 312:1 (1989), 155178. MR 937253 (89g:57029)
 [Wh50]
 J.H.C. Whitehead, A certain exact sequence, Annals of Math. (2) 52 (1950), 51110. MR 0035997 (12:43c)
 [Ya83]
 T. Yasui, On the map defined by regarding embeddings as immersions, Hiroshima Math. J. 13 (1983), 457476. MR 725959 (85g:57016)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
57R40,
57Q37,
57R52
Retrieve articles in all journals
with MSC (2010):
57R40,
57Q37,
57R52
Additional Information
A. Skopenkov
Affiliation:
Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia 119992 – and – Independent University of Moscow, B. Vlasyevskiy, 11, 119002, Moscow, Russia
Email:
skopenko@mccme.ru
DOI:
http://dx.doi.org/10.1090/S0002993910104250
PII:
S 00029939(10)104250
Keywords:
Embedding,
selfintersection,
isotopy,
parametric connected sum
Received by editor(s):
December 16, 2008
Received by editor(s) in revised form:
December 31, 2009
Published electronically:
May 4, 2010
Communicated by:
Alexander N. Dranishnikov
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
