Editing Poincaré conjecture (section)

== History ==

=== Poincaré's question ===

In the 1800s, [[Bernhard Riemann]] and [[Enrico Betti]] initiated the study of [[topological invariant]]s of [[manifold]]s.<ref>{{cite thesis |last1=Riemann |first1=Bernhard |author-link1=Bernhard Riemann |year=1851 |title=Grundlagen für eine allgemeine Theorie der Functionen |publisher=[[University of Göttingen]] }} English translation: {{cite encyclopedia |author-last1=Riemann |author-first1=Bernhard |author-link1=Bernhard Riemann |title=Foundations for a general theory of functions of a complex variable |pages=1–41 |publisher=Kendrick Press|encyclopedia=Collected Papers: Bernhard Riemann |year=2004 |translator-last1=Baker |translator-last2=Christenson |translator-last3=Orde |translator-first1=Roger |translator-first2=Charles |translator-first3=Henry |zbl=1101.01013 |isbn=0-9740427-2-2 |location=Heber City, UT |mr=2121437 }}</ref><ref>{{cite journal |last1=Betti |first1=Enrico |author-link1=Enrico Betti |title=Sopra gli spazi di un numero qualunque di dimensioni |volume=4 |pages=140–158 |year=1870 |journal=[[Annali di Matematica Pura ed Applicata]] |doi=10.1007/BF02420029 |jfm=03.0301.01 |url=https://rcin.org.pl/dlibra/publication/edition/35231/content }}</ref> They introduced the [[Betti number]]s, which associate to any manifold a list of nonnegative integers. Riemann showed that a closed connected two-dimensional manifold is fully characterized by its Betti numbers. As part of his 1895 paper ''[[Analysis Situs (paper)|Analysis Situs]]'' (announced in 1892), Poincaré showed that Riemann's result does not extend to higher dimensions.<ref name="Poincare1892">{{cite journal |last1=Poincaré |first1=H. |author-link1=Henri Poincaré |year=1892 |jfm=24.0506.02 |title=Sur l'Analysis situs |journal=[[Comptes Rendus|Comptes Rendus des Séances de l'Académie des Sciences]] |url=https://gallica.bnf.fr/ark:/12148/bpt6k3071t/f633.item.zoom# }}</ref><ref name="Poincare1895">{{cite journal |last=Poincaré |first=H. |author-link=Henri Poincaré |year=1895 |title=Analysis situs |journal=Journal de l'École Polytechnique |volume=1 |jfm=26.0541.07 |series=2e Série |pages=1–121 |url=http://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7.image }}</ref><ref name="stillwell">{{cite book |last1=Poincaré |first1=Henri |title=Papers on Topology: ''Analysis Situs'' and Its Five Supplements |translator-last1=Stillwell |translator-first=John |translator-link1=John Stillwell |author-link1=Henri Poincaré |series=History of Mathematics |volume=37 |year=2010 |publisher=[[American Mathematical Society]] and [[London Mathematical Society]] |doi=10.1090/hmath/037 |isbn=978-0-8218-5234-7|mr=2723194 |zbl=1204.55002 }}</ref> To do this he introduced the [[fundamental group]] as a novel topological invariant, and was able to exhibit examples of three-dimensional manifolds which have the same Betti numbers but distinct fundamental groups. He posed the question of whether the fundamental group is sufficient to topologically characterize a manifold (of given dimension), although he made no attempt to pursue the answer, saying only that it would "demand lengthy and difficult study".<ref name="Poincare1895" /><ref name="stillwell" /><ref name="gray">{{cite book |last1=Gray |first1=Jeremy |title=Henri Poincaré: A Scientific Biography |location=Princeton, NJ |publisher=[[Princeton University Press]] |isbn=978-0-691-15271-4 |year=2013 |author-link=Jeremy Gray |jstor=j.ctt1r2fwt |mr=2986502 |zbl=1263.01002 }}</ref>

The primary purpose of Poincaré's paper was the interpretation of the Betti numbers in terms of his newly-introduced [[homology group]]s, along with the [[Poincaré duality theorem]] on the symmetry of Betti numbers. Following criticism of the completeness of his arguments, he released a number of subsequent "supplements" to enhance and correct his work. The closing remark of his second supplement, published in 1900, said:<ref>{{cite journal |last1=Poincaré |first1=H. |year=1900 |title=Second complément à l'analysis situs |journal=[[Proceedings of the London Mathematical Society]] |volume=32 |issue=1 |pages=277–308 |doi=10.1112/plms/s1-32.1.277 |author-link1=Henri Poincaré |mr=1576227 |jfm=31.0477.10 |url=https://zenodo.org/record/1447744 }}</ref><ref name="stillwell" />
<blockquote>
In order to avoid making this work too prolonged, I confine myself to stating the following theorem, the proof of which will require further developments:

Each polyhedron which has all its Betti numbers equal to 1 and all its tables {{math|''T''<sub>''q''</sub>}} orientable is simply connected, i.e., homeomorphic to a hypersphere.
</blockquote>
(In a modern language, taking note of the fact that Poincaré is using the terminology of [[simply connected|simple-connectedness]] in an unusual way,<ref name="stillcomm">cf. Stillwell's commentary in {{harvtxt|Poincaré|2010}}</ref> this says that a closed connected [[orientability|oriented]] manifold with the homology of a sphere must be homeomorphic to a sphere.<ref name="gray" />) This modified his negative generalization of Riemann's work in two ways. Firstly, he was now making use of the full homology groups and not only the Betti numbers. Secondly, he narrowed the scope of the problem from asking if an arbitrary manifold is characterized by topological invariants to asking whether the sphere can be so characterized.

However, after publication he found his announced theorem to be incorrect. In his fifth and final supplement, published in 1904, he proved this with the counterexample of the [[Poincaré homology sphere]], which is a closed connected three-dimensional manifold which has the homology of the sphere but whose fundamental group has 120 elements. This example made it clear that homology is not powerful enough to characterize the topology of a manifold. In the closing remarks of the fifth supplement, Poincaré modified his erroneous theorem to use the fundamental group instead of homology:<ref>{{cite journal |last1=Poincaré |first1=H. |year=1904 |title=Cinquième complément à l'analysis situs |journal=[[Rendiconti del Circolo Matematico di Palermo]] |volume=18 |pages=45–110 |doi=10.1007/bf03014091 |author-link1=Henri Poincaré |jfm=35.0504.13 |url=https://zenodo.org/record/1428448 }}</ref><ref name="stillwell" />
<blockquote>
One question remains to be dealt with: is it possible for the fundamental group of {{mvar|V}} to reduce to the identity without {{mvar|V}} being simply connected? [...] However, this question would carry us too far away.
</blockquote>
In this remark, as in the closing remark of the second supplement, Poincaré used the term "simply connected" in a way which is at odds with modern usage, as well as his own 1895 definition of the term.<ref name="Poincare1895" /><ref name="stillcomm" /> (According to modern usage, Poincaré's question is a [[tautology (language)|tautology]], asking if it is possible for a manifold to be simply connected without being simply connected.) However, as can be inferred from context,<ref>The opening paragraphs of {{harvtxt|Poincaré|1904}} refer to "simply connected in the true sense of the word" as the condition of being homeomorphic to a sphere.</ref> Poincaré was asking whether the triviality of the fundamental group uniquely characterizes the sphere.<ref name="gray" />

Throughout the work of Riemann, Betti, and Poincaré, the topological notions in question are not defined or used in a way that would be recognized as precise from a modern perspective. Even the key notion of a "manifold" was not used in a consistent way in Poincaré's own work, and there was frequent confusion between the notion of a [[topological manifold]], a [[PL manifold]], and a [[smooth manifold]].<ref name="stillcomm" /><ref name="dieudonne">{{cite book |last1=Dieudonné |first1=Jean |author-link1=Jean Dieudonné |title=A History of Algebraic and Differential Topology, 1900–1960 |mr=0995842 |isbn=0-8176-3388-X |location=Boston, MA |year=1989 |publisher=[[Birkhäuser|Birkhäuser Boston, Inc.]] |zbl=0673.55002 |doi=10.1007/978-0-8176-4907-4 }}</ref> For this reason, it is not possible to read Poincaré's questions unambiguously. It is only through the formalization and vocabulary of topology as developed by later mathematicians that Poincaré's closing question has been understood as the "Poincaré conjecture" as stated in the preceding section.

However, despite its usual phrasing in the form of a conjecture, proposing that all manifolds of a certain type are homeomorphic to the sphere, Poincaré only posed an open-ended question, without venturing to conjecture one way or the other. Moreover, there is no evidence as to which way he believed his question would be answered.<ref name="gray" />

=== 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>

=== Dimensions ===
{{Main|Generalized Poincaré conjecture}}
The [[Surface (topology)|classification of closed surfaces]] gives an affirmative answer to the analogous question in two dimensions. For dimensions greater than three, one can pose the Generalized Poincaré conjecture: is a [[homotopy sphere|homotopy ''n''-sphere]] homeomorphic to the ''n''-sphere? A stronger assumption than simply-connectedness is necessary; in dimensions four and higher there are simply-connected, closed manifolds which are not [[Homotopy#Homotopy equivalence|homotopy equivalent]] to an ''n''-sphere.

Historically, while the conjecture in dimension three seemed plausible, the generalized conjecture was thought to be false. In 1961, [[Stephen Smale]] shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental [[h-cobordism theorem]]. In 1982, [[Michael Freedman]] proved the Poincaré conjecture in four dimensions. Freedman's work left open the possibility that there is a smooth four-manifold homeomorphic to the four-sphere which is not [[Diffeomorphism|diffeomorphic]] to the four-sphere. This so-called smooth Poincaré conjecture, in dimension four, remains open and is thought to be very difficult. [[Milnor]]'s [[exotic sphere]]s show that the smooth Poincaré conjecture is false in dimension seven, for example.

These earlier successes in higher dimensions left the case of three dimensions in limbo. The Poincaré conjecture was essentially true in both dimension four and all higher dimensions for substantially different reasons. In dimension three, the conjecture had an uncertain reputation until the [[geometrization conjecture]] put it into a framework governing all 3-manifolds. [[John Morgan (mathematician)|John Morgan]] wrote:<ref>Morgan, John W., Recent progress on the Poincaré conjecture and the classification of 3-manifolds.
Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 1, 57–78</ref>

{{Quote|It is my view that before [[William Thurston|Thurston]]'s work on [[hyperbolic 3-manifold]]s and … the Geometrization conjecture there was no consensus among the experts as to whether the Poincaré conjecture was true or false.  After Thurston's work, notwithstanding the fact that it had no direct bearing on the Poincaré conjecture, a consensus developed that the Poincaré conjecture (and the Geometrization conjecture) were true.}}

=== Hamilton's program and solution ===
[[File:Ricci flow.png|thumb|upright|200px|right|Several stages of the [[Ricci flow]] on a two-dimensional manifold]]

Hamilton's program was started in his 1982 paper in which he introduced the [[Ricci flow]] on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture.<ref>{{cite journal | last = Hamilton | first = Richard | author-link = Richard S. Hamilton | title = Three-manifolds with positive Ricci curvature | journal = Journal of Differential Geometry | volume = 17 | issue = 2 | pages = 255–306 | date = 1982 | mr = 0664497 | zbl = 0504.53034 | doi = 10.4310/jdg/1214436922 | doi-access = free }} Reprinted in: {{cite book | editor-last1 = Cao | editor-first1 = H. D. | editor1-link = Huai-Dong Cao | editor-last2 = Chow | editor-first2 = B. | editor-last3 = Chu | editor-first3 = S. C. | editor-last4 = Yau | editor-first4 = S.-T. | editor4-link = Shing-Tung Yau | title = Collected Papers on Ricci Flow | place = Somerville, MA | publisher = International Press | pages = 119–162 | series = Series in Geometry and Topology | volume = 37 | year = 2003 | isbn = 1-57146-110-8  }}</ref> In the following years, he extended this work but was unable to prove the conjecture. The actual solution was not found until [[Grigori Perelman]] published his papers.

In late 2002 and 2003, Perelman posted three papers on [[arXiv]].<ref>{{cite arXiv | last = Perelman | first = Grigori | author-link = Grigori Perelman | title = The entropy formula for the Ricci flow and its geometric applications | eprint = math.DG/0211159 | date = 2002 }}</ref><ref>{{cite arXiv | last = Perelman | first = Grigori | title = Ricci flow with surgery on three-manifolds | eprint = math.DG/0303109 | date = 2003 }}</ref><ref>{{cite arXiv | last = Perelman | first = Grigori | title = Finite extinction time for the solutions to the Ricci flow on certain three-manifolds | eprint = math.DG/0307245 | date = 2003 }}</ref> In these papers, he sketched a proof of the Poincaré conjecture and a more general conjecture, [[Thurston's geometrization conjecture]], completing the Ricci flow program outlined earlier by [[Richard S. Hamilton]].

From May to July 2006, several groups presented papers that filled in the details of Perelman's proof of the Poincaré conjecture, as follows:
* [[Bruce Kleiner]] and [[John Lott (mathematician)|John W. Lott]] posted a paper on arXiv in May 2006 which filled in the details of Perelman's proof of the geometrization conjecture, following partial versions which had been publicly available since 2003.<ref>{{cite journal | first = Bruce | last = Kleiner | author-link = Bruce Kleiner |author2=John W. Lott | title = Notes on Perelman's Papers | year = 2008 | pages = 2587–2855 | volume = 12 | journal = Geometry and Topology | arxiv = math.DG/0605667 | doi=10.2140/gt.2008.12.2587 | issue = 5| s2cid = 119133773 }}</ref> Their manuscript was published in the journal ''[[Geometry and Topology]]'' in 2008. A small number of corrections were made in 2011 and 2013; for instance, the first version of their published paper made use of an incorrect version of Hamilton's compactness theorem for Ricci flow.
* [[Huai-Dong Cao]] and [[Xi-Ping Zhu]] published a paper in the June 2006 issue of the ''[[Asian Journal of Mathematics]]'' with an exposition of the complete proof of the Poincaré and geometrization conjectures.<ref>{{cite journal | first = Huai-Dong | last = Cao | author-link = Huai-Dong Cao | author2 = Xi-Ping Zhu | author2-link = Xi-Ping Zhu | title = A Complete Proof of the Poincaré and Geometrization Conjectures&nbsp;– application of the Hamilton-Perelman theory of the Ricci flow | url = http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf | journal = Asian Journal of Mathematics | volume = 10 | date = June 2006 | issue = 2 | url-status = dead | archive-url = https://web.archive.org/web/20120514194949/http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf | archive-date = 2012-05-14 }}</ref> The opening paragraph of their paper stated
{{Quote|In this paper, we shall present the Hamilton-Perelman theory of Ricci flow. Based on it, we shall give the first written account of a complete proof of the Poincaré conjecture and the geometrization conjecture of Thurston. While the complete work is an accumulated efforts of many geometric analysts, the major contributors are unquestionably Hamilton and Perelman.}}
:Some observers interpreted Cao and Zhu as taking credit for Perelman's work. They later posted a revised version, with new wording, on arXiv.<ref>{{cite arXiv |author=Cao, Huai-Dong|author2=Zhu, Xi-Ping|name-list-style=amp |eprint=math.DG/0612069 |title=Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture |date=December 3, 2006 }}</ref> In addition, a page of their exposition was essentially identical to a page in one of Kleiner and Lott's early publicly available drafts; this was also amended in the revised version, together with an apology by the journal's editorial board.
* [[John Morgan (mathematician)|John Morgan]] and [[Gang Tian]] posted a paper on arXiv in July 2006 which gave a detailed proof of just the Poincaré Conjecture (which is somewhat easier than the full geometrization conjecture)<ref>{{cite arXiv | first = John | last = Morgan | author-link = John Morgan (mathematician) | author2 = Gang Tian | author2-link = Gang Tian | title = Ricci Flow and the Poincaré Conjecture | eprint = math.DG/0607607 | date = 2006 }}</ref> and expanded this to a book.<ref>{{cite book | first = John | last = Morgan | author-link = John Morgan (mathematician) |author2=Gang Tian |author2-link=Gang Tian  | title = Ricci Flow and the Poincaré Conjecture |publisher= Clay Mathematics Institute |isbn = 978-0-8218-4328-4| date = 2007 }}</ref><ref>{{cite arXiv |last1=Morgan |first1=John |last2=Tian |first2=Gang |title=Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture |date=2015 |eprint=1512.00699 |class=math.DG }}</ref>

All three groups found that the gaps in Perelman's papers were minor and could be filled in using his own techniques.

On August 22, 2006, the [[International Congress of Mathematicians|ICM]] awarded Perelman the [[Fields Medal]] for his work on the Ricci flow, but Perelman refused the medal.<ref>{{cite magazine | first = Sylvia | last = Nasar | author-link = Sylvia Nasar |author2=David Gruber | title = Manifold destiny | magazine = [[The New Yorker]] | pages = 44–57 | date = August 28, 2006 | title-link = Manifold destiny }} [http://www.newyorker.com/archive/2006/08/28/060828fa_fact2 On-line version at the ''New Yorker'' website].</ref><ref>{{cite news | first = Kenneth | last = Chang | title = Highest Honor in Mathematics Is Refused | work = [[The New York Times]] | date = August 22, 2006 | url = https://www.nytimes.com/2006/08/22/science/22cnd-math.html?hp&ex=1156305600&en=aa3a9d418768062c&ei=5094&partner=homepage }}</ref> John Morgan spoke at the ICM on the Poincaré conjecture on August 24, 2006, declaring that "in 2003, Perelman solved the Poincaré Conjecture".<ref>A Report on the Poincaré Conjecture. Special lecture by John Morgan.</ref>

In December 2006, the journal ''[[Science (journal)|Science]]'' honored the proof of Poincaré conjecture as the [[Breakthrough of the Year]] and featured it on its cover.<ref name=science/>