Editing Poincaré conjecture (section)

=== Solutions ===
In the 1930s, [[J. H. C. Whitehead]] claimed a proof but then retracted it. In the process, he discovered some examples of simply-connected (indeed contractible, i.e. homotopically equivalent to a point) non-compact 3-manifolds not homeomorphic to <math>\R^3</math>, the prototype of which is now called the [[Whitehead manifold]].

In the 1950s and 1960s, other mathematicians attempted proofs of the conjecture only to discover that they contained flaws. Influential mathematicians such as [[Georges de Rham]], [[R. H. Bing]], [[Wolfgang Haken]], [[Edwin E. Moise]], and [[Christos Papakyriakopoulos]] attempted to prove the conjecture. In 1958, R. H. Bing proved a weak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere.<ref>{{cite journal | last = Bing | first = R. H. | author-link = R. H. Bing | title = Necessary and sufficient conditions that a 3-manifold be S<sup>3</sup> | journal = [[Annals of Mathematics]] |series=Second Series | volume = 68 | issue = 1 | pages = 17–37 | date = 1958 | doi = 10.2307/1970041 | jstor=1970041}}</ref> Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.<ref>{{cite conference | last = Bing | first = R. H. | title = Some aspects of the topology of 3-manifolds related to the Poincaré conjecture | book-title=Lectures on Modern Mathematics | volume=II | pages = 93–128 | publisher = Wiley | date = 1964 | location = New York }}</ref>

Włodzimierz Jakobsche showed in 1978 that, if the [[Bing–Borsuk conjecture]] is true in dimension 3, then the Poincaré conjecture must also be true.<ref>{{cite journal |last1=Halverson |first1=Denise M. |last2=Dušan |first2=Repovš |title=The Bing–Borsuk and the Busemann conjectures |journal=Mathematical Communications |date=23 December 2008 |volume=13 |issue=2 |url=https://hrcak.srce.hr/30884 |language=en |arxiv=0811.0886 }}</ref>

Over time, the conjecture gained the reputation of being particularly tricky to tackle. [[J. W. Milnor|John Milnor]] commented that sometimes the errors in false proofs can be "rather subtle and difficult to detect".<ref>{{cite web | url = http://www.math.sunysb.edu/~jack/PREPRINTS/poiproof.pdf | title = The Poincaré Conjecture 99 Years Later: A Progress Report | access-date=2007-05-05 | last = Milnor |first = John |author-link = John Milnor | date = 2004 }}</ref> Work on the conjecture improved understanding of 3-manifolds. Experts in the field were often reluctant to announce proofs and tended to view any such announcement with skepticism. The 1980s and 1990s witnessed some well-publicized fallacious proofs (which were not actually published in [[peer review|peer-reviewed]] form).<ref>{{cite journal | last = Taubes | first = Gary | title = What happens when hubris meets nemesis | journal = Discover | volume = 8 | pages = 66–77 | date = July 1987 }}</ref><ref>{{cite news | first = Robert | last = Matthews | title = $1 million mathematical mystery "solved" | url = https://www.newscientist.com/article.ns?id=dn2143 | work = NewScientist.com | date = 9 April 2002 |access-date = 2007-05-05 }}</ref>

An exposition of attempts to prove this conjecture can be found in the non-technical book ''Poincaré's Prize'' by [[George Szpiro]].<ref>{{cite book |last=Szpiro |first=George |title=Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles |year=2008 |publisher=[[Plume (publisher)|Plume]] |isbn=978-0-452-28964-2}}</ref>