No embeddings of solenoids into surfaces

Authors:
Boju Jiang, Shicheng Wang and Hao Zheng

Journal:
Proc. Amer. Math. Soc. **136** (2008), 3697-3700

MSC (2000):
Primary 54F15, 57N35

DOI:
https://doi.org/10.1090/S0002-9939-08-09340-4

Published electronically:
May 7, 2008

MathSciNet review:
2415057

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Abstract | References | Similar Articles | Additional Information

Abstract: A quick proof of Bing's theorem indicated by the title is given. Indeed the inverse limit of a sequence of degree maps between closed oriented -manifolds can never be embedded into any closed orientable -manifold. The proof also concludes Gumerov's result on the covering of solenoids.

**[B1]**Bing, R. H.*A simple closed curve is the only homogeneous bounded plane continuum that contains an arc,*Canad. J. Math.**12**(1960), 209-230. MR**0111001 (22:1869)****[B2]**Bing, R. H.*Embedding circle-like continua in the plane,*Canad. J. Math.**14**(1962), 113-128. MR**0131865 (24:A1712)****[ES]**Eilenberg, S.; Steenrod, N.*Foundations of algebraic topology,*Princeton University Press, Princeton, New Jersey, 1952. MR**0050886 (14:398b)****[GH]**Greenberg, M. J.; Harper, J. R.*Algebraic topology. A first course,*Mathematics Lecture Note Series 58, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1981. MR**643101 (83b:55001)****[Gu]**Gumerov, R. N.*On finite-sheeted covering mappings onto solenoids,*Proc. Amer. Math. Soc.**133**(2005), 2771-2778. MR**2146226 (2006d:54024)****[KW]**Keesling, J.; Wilson, D.*Embedding -like continua in Euclidean space.*Topology Appl.**21**(1985), no. 3, 241-249. MR**812642 (87h:54033)****[Mc]**McCord, M. C.*Inverse limit sequences with covering maps,*Trans. Amer. Math. Soc.**114**(1965), no. 1, 197-209. MR**0173237 (30:3450)****[Pr]**Prajs, J.*Homogeneous continua in Euclidean -space which contain an -cube are -manifolds.*Trans. Amer. Math. Soc.**318**(1990), no. 1, 143-148. MR**943307 (90f:54055)**

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Additional Information

**Boju Jiang**

Affiliation:
Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China

Email:
bjjiang@math.pku.edu.cn

**Shicheng Wang**

Affiliation:
Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China

Email:
wangsc@math.pku.edu.cn

**Hao Zheng**

Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou 510275, People’s Republic of China

Email:
zhenghao@sysu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-08-09340-4

Received by editor(s):
November 2, 2006

Received by editor(s) in revised form:
August 9, 2007

Published electronically:
May 7, 2008

Additional Notes:
The authors were supported by an NSFC grant.

Communicated by:
Alexander N. Dranishnikov

Article copyright:
© Copyright 2008
American Mathematical Society