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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the Deleted Product Criterion
For Embeddability in $\mathbb{R}^{m}$


Author: A. Skopenkov
Journal: Proc. Amer. Math. Soc. 126 (1998), 2467-2476
MSC (1991): Primary 57Q35, 54C25; Secondary 55S15, 57Q30, 57Q65, 57Q40
MathSciNet review: 1423334
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Abstract: For a space $K$ let $\tilde K=\{(x,y)\in K\times K| x\not =y\}$. Let $\mathbb{Z}_{2}$ act on $\tilde K$ and on $S^{m-1}$ by exchanging factors and antipodes respectively. We present a new short proof of the following theorem by Weber: For an $n$-polyhedron $K$ and $m\geqslant \frac{3(n+1)}{2}$, if there exists an equivariant map $F:\nobreak \tilde K\rightarrow S^{m-1}$, then $K$ is embeddable in $\mathbb{R}^{m}$. We also prove this theorem for a peanian continuum $K$ and $m=2$. We prove that the theorem is not true for the 3-adic solenoid $K$ and $m=2$.


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Additional Information

A. Skopenkov
Affiliation: Chair of Differential Geometry, Department of Mechanics and Mathematics, Moscow State University, Moscow,119899, Russia
Email: skopenko@nw.math.msu.su

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04142-2
PII: S 0002-9939(98)04142-2
Keywords: Embedding, deleted product, engulfing, quasi-embedding, metastable case, peanian continua, 3-adic solenoid, relative regular neighborhood
Received by editor(s): April 12, 1995
Received by editor(s) in revised form: January 3, 1997
Additional Notes: Supported by the Russian Fundamental Research Foundation, Grant No 96-01-01166A
Communicated by: James West
Article copyright: © Copyright 1998 American Mathematical Society