Editing Poincaré conjecture (section)

=== 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" />