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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.
905:
684:
Włodzimierz
Jakobsche showed in 1978 that, if the Bing–Borsuk conjecture is true in dimension 3, then the Poincaré conjecture must also be true.
55:. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the
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There is an alternate statement of the Bing–Borsuk conjecture: suppose
860:"The Bing–Borsuk conjecture is stronger than the Poincaré conjecture"
750:
859:
733:
732:
M., Halverson, Denise; Dušan, Repovš (23 December 2008).
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and this embedding can be extended to an embedding of
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488:{\displaystyle M\times (-\varepsilon ,\varepsilon )}
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576:{\displaystyle \varphi :\partial N\rightarrow M}
734:"The Bing–Borsuk and the Busemann conjectures"
8:
772:: CS1 maint: multiple names: authors list (
647:The conjecture was first made in a paper by
792:"Decompositions and approximate fibrations"
837:. American Mathematical Soc. p. 167.
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831:Bing, R. H.; Armentrout, Steve (1998).
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515:has a mapping cylinder neighbourhood
229:(ANR) if, for every closed embedding
7:
281:is a metric space), there exists an
608:{\displaystyle \pi :N\rightarrow M}
834:The Collected Papers of R. H. Bing
561:
421:{\displaystyle \mathbb {R} ^{m+n}}
14:
796:The Michigan Mathematical Journal
583:with mapping cylinder projection
906:Unsolved problems in mathematics
254:{\displaystyle f:M\rightarrow N}
114:{\displaystyle m_{1},m_{2}\in M}
599:
567:
541:{\displaystyle N=C_{\varphi }}
482:
467:
353:
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1:
227:absolute neighborhood retract
49:absolute neighborhood retract
738:Mathematical Communications
655:in 1965, who proved it for
927:
864:Fundamenta Mathematicae
790:; Husch, L. S. (1984).
447:{\displaystyle m\geq 3}
75:if, for any two points
809:10.1307/mmj/1029003024
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22:Bing–Borsuk conjecture
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637:approximate fibration
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192:{\displaystyle m_{2}}
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165:{\displaystyle m_{1}}
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628:{\displaystyle \pi }
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359:{\displaystyle f(M)}
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326:{\displaystyle f(M)}
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53:topological manifold
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689:Busemann conjecture
674:{\displaystyle n=1}
57:Poincaré conjecture
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691:states that every
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283:open neighbourhood
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24:states that every
706:{\displaystyle G}
508:{\displaystyle M}
382:{\displaystyle M}
297:{\displaystyle U}
274:{\displaystyle N}
218:{\displaystyle M}
138:{\displaystyle M}
69:topological space
37:{\displaystyle n}
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788:Daverman, R. J.
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858:Jakobsche, W.
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802:(2): 197–214.
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304:of the image
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123:homeomorphism
121:, there is a
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44:-dimensional
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799:
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768:cite journal
741:
737:
727:
686:
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653:Karol Borsuk
646:
548:of some map
368:
204:metric space
201:
145:which takes
72:
66:
21:
15:
901:Conjectures
73:homogeneous
63:Definitions
51:space is a
46:homogeneous
18:mathematics
890:Categories
719:References
649:R. H. Bing
911:Manifolds
876:0016-2736
818:0026-2285
760:1331-0623
751:0811.0886
693:Busemann
623:π
600:→
591:π
568:→
562:∂
556:φ
534:φ
480:ε
474:ε
471:−
465:×
439:≥
428:for some
246:→
106:∈
896:Topology
391:embedded
335:retracts
681:and 2.
643:History
615:, then
261:(where
874:
841:
816:
758:
713:-space
635:is an
333:which
225:is an
20:, the
870:(2).
746:arXiv
744:(2).
495:. If
872:ISSN
839:ISBN
814:ISSN
774:link
756:ISSN
687:The
651:and
868:106
804:doi
393:in
389:is
337:to
172:to
125:of
71:is
16:In
892::
866:.
862:.
812:.
800:31
798:.
794:.
770:}}
766:{{
754:.
742:13
740:.
736:.
639:.
366:.
202:A
199:.
67:A
59:.
878:.
847:.
820:.
806::
776:)
762:.
748::
701:G
669:1
666:=
663:n
603:M
597:N
594::
571:M
565:N
559::
530:C
526:=
523:N
503:M
483:)
477:,
468:(
462:M
442:3
436:m
414:n
411:+
408:m
403:R
377:M
354:)
351:M
348:(
345:f
321:)
318:M
315:(
312:f
292:U
269:N
249:N
243:M
240::
237:f
213:M
185:2
181:m
158:1
154:m
133:M
109:M
101:2
97:m
93:,
88:1
84:m
32:n
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