Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On embedding all $ n$-manifolds into a single $ (n+1)$-manifold


Authors: Fan Ding, Shicheng Wang and Jiangang Yao
Journal: Trans. Amer. Math. Soc. 360 (2008), 6017-6030
MSC (2000): Primary 57N35
DOI: https://doi.org/10.1090/S0002-9947-08-04439-5
Published electronically: June 13, 2008
MathSciNet review: 2425700
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For each composite number $ n\ne 2^k$, there does not exist a single connected closed $ (n+1)$-manifold such that any smooth, simply-connected, closed $ n$-manifold can be topologically flatly embedded into it. There is a single connected closed $ 5$-manifold $ W$ such that any simply-connected, $ 4$-manifold $ M$ can be topologically flatly embedded into $ W$ if $ M$ is either closed and indefinite, or compact and with non-empty boundary.


References [Enhancements On Off] (What's this?)

  • [Di] F. Ding, Smooth structures on some open $ 4$-manifolds, Topology 36 (1997), no. 1, 203-207. MR 1410471 (97g:57040)
  • [EL] A. L. Edmonds, C. Livingston, Embedding punctured lens spaces in four-manifolds, Comment. Math. Helv. 71 (1996), no. 2, 169-191. MR 1396671 (97h:57049)
  • [Fa1] F. Fang, Embedding $ 3$-manifolds and smooth structures of $ 4$-manifolds, Topology Appl. 76 (1997), no. 3, 249-259. MR 1441755 (98b:57037)
  • [Fa2] F. Fang, Smooth structures on $ \Sigma\times {\mathbb{R}}$, Topology Appl. 99 (1999), no. 1, 123-131. MR 1720366 (2001a:57044)
  • [Fr] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357-453. MR 679066 (84b:57006)
  • [FQ] M. H. Freedman, F. Quinn, Topology of $ 4$-manifolds, Princeton University Press, Princeton, NJ, 1990. MR 1201584 (94b:57021)
  • [GS] R. E. Gompf, A. I. Stipsicz, $ 4$-manifolds and Kirby calculus, Graduate Studies in Math. 20, Amer. Math. Soc., Providence, RI, 1999. MR 1707327 (2000h:57038)
  • [Hi] M. W. Hirsch, The imbedding of bounding manifolds in Euclidean space, Ann. of Math. (2) 74 (1961), 494-497. MR 0133136 (24:A2970)
  • [HJ] R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990. MR 1084815 (91i:15001)
  • [Ka] A. Kawauchi, The imbedding problem of $ 3$-manifolds into $ 4$-manifolds, Osaka J. Math. 25 (1988), no. 1, 171-183. MR 937194 (89g:57042)
  • [Ki] R. C. Kirby, The topology of $ 4$-manifolds, Lecture Notes in Math. 1374, Springer-Verlag, Berlin, 1989. MR 1001966 (90j:57012)
  • [KS] R. C. Kirby, L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Ann. of Math. Studies 88, Princeton University Press, Princeton, NJ, 1977. MR 0645390 (58:31082)
  • [La] S. Lang, Algebra, Graduate Texts in Math. 211, Springer-Verlag, New York, 2002. MR 1878556 (2003e:00003)
  • [Ro] V. A. Rohlin, The embedding of non-orientable three-manifolds into five-dimensional Euclidean space, Soviet Math. Dokl. 6 (1965), 153-156. MR 0184246 (32:1719)
  • [Sh] R. Shaw, Linear algebra and group representations, Vol. II, Academic Press, London-New York, 1983. MR 701854 (84m:15003)
  • [Shi] T. Shiomi, On imbedding $ 3$-manifolds into $ 4$-manifolds, Osaka J. Math. 28 (1991), no. 3, 649-661. MR 1144478 (92m:57028)
  • [Sp] E. H. Spanier, Algebraic topology, Springer-Verlag, New York, 1966. MR 666554 (83i:55001)
  • [Su] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269-331 (1978). MR 0646078 (58:31119)
  • [Wa] C. T. C. Wall, All $ 3$-manifolds imbed in $ 5$-space, Bull. Amer. Math. Soc. 71 (1965), 564-567. MR 0175139 (30:5324)
  • [WZ] S. Wang, Q. Zhou, How to embed $ 3$-manifolds into $ 5$-space, Adv. in Math. (China) 24 (1995), no. 4, 309-312. MR 1358889

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57N35

Retrieve articles in all journals with MSC (2000): 57N35


Additional Information

Fan Ding
Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: dingfan@math.pku.edu.cn

Shicheng Wang
Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: wangsc@math.pku.edu.cn

Jiangang Yao
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Email: jgyao@math.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04439-5
Received by editor(s): September 1, 2005
Received by editor(s) in revised form: May 12, 2006, and October 31, 2006
Published electronically: June 13, 2008
Additional Notes: The authors would like to thank Jianzhong Pan for informing them of Sullivan’s work \cite{Su}
The first two authors were partially supported by grant No. 10201003 of NSFC and a grant of MSTC
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society