Bing–Borsuk conjecture

In mathematics, the Bing–Borsuk conjecture states that every $n$ -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the Poincaré conjecture.

Definitions

A topological space is homogeneous if, for any two points $m_{1},m_{2}\in M$ , there is a homeomorphism of $M$ which takes $m_{1}$ to $m_{2}$ .

A metric space $M$ is an absolute neighborhood retract (ANR) if, for every closed embedding $f:M\rightarrow N$ (where $N$ is a metric space), there exists an open neighbourhood $U$ of the image $f(M)$ which retracts to $f(M)$ .^[1]

There is an alternate statement of the Bing–Borsuk conjecture: suppose $M$ is embedded in $\mathbb {R} ^{m+n}$ for some $m\geq 3$ and this embedding can be extended to an embedding of $M\times (-\varepsilon ,\varepsilon )$ . If $M$ has a mapping cyclinder neighbourhood $N=C_{\varphi }$ of some map $\varphi :\partial N\rightarrow M$ with mapping cyclinder projection $\pi :N\rightarrow M$ , then $\pi$ is an approximate fibration.^[2]

History

The conjecture was first made in a paper by R. H. Bing and Karol Borsuk in 1965, who proved it for $n=1$ and 2.^[3]

Włodzimierz Jakobsche showed in 1978 that, if the Bing–Borsuk conjecture is true in dimension 3, then the Poincaré conjecture must also be true.^[4]

The Busemann conjecture states that every Busemann $G$ -space is a topological manifold. It is a special case of the Bing–Borsuk conjecture. The Busemann conjecture is known to be true for dimensions 1 to 4.

References

↑ M., Halverson, Denise; Dušan, Repovš, (23 December 2008). "The Bing–Borsuk and the Busemann conjectures". Mathematical Communications. 13 (2). ISSN 1331-0623.
↑ Daverman, R. J.; Husch, L. S. (1984). "Decompositions and approximate fibrations". The Michigan Mathematical Journal. 31 (2): 197–214. doi:10.1307/mmj/1029003024. ISSN 0026-2285.
↑ Bing, R. H.; Armentrout, Steve (1998). The Collected Papers of R. H. Bing. American Mathematical Soc. p. 167. ISBN 9780821810477.
↑ Jakobsche, W. "The Bing–Borsuk conjecture is stronger than the Poincaré conjecture". Fundamenta Mathematicae. 106 (2). ISSN 0016-2736.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] M., Halverson, Denise; Dušan, Repovš, (23 December 2008). "The Bing–Borsuk and the Busemann conjectures". Mathematical Communications. 13 (2). ISSN 1331-0623.

[2] Daverman, R. J.; Husch, L. S. (1984). "Decompositions and approximate fibrations". The Michigan Mathematical Journal. 31 (2): 197–214. doi:10.1307/mmj/1029003024. ISSN 0026-2285.

[3] Bing, R. H.; Armentrout, Steve (1998). The Collected Papers of R. H. Bing. American Mathematical Soc. p. 167. ISBN 9780821810477.

[4] Jakobsche, W. "The Bing–Borsuk conjecture is stronger than the Poincaré conjecture". Fundamenta Mathematicae. 106 (2). ISSN 0016-2736.