Poincaré conjecture - Poincaré's conjecture

Japanese: ポアンカレの予想 - ポアンカレのよそう（英語表記）Poincaré conjecture

The question of whether a simply connected, connected closed three-dimensional manifold with a trivial fundamental group is homeomorphic to a three-sphere was proposed by H. Poincaré in 1904 and is known as the Poincaré conjecture, but it remains unsolved. The question of whether a n -dimensional closed manifold ( n ≧ 4) is homeomorphic to an n-dimensional sphere when all figures homeomorphic to an r -dimensional sphere (0≦ r ≦ n - 1) shrink to a single point (i.e., when the r- dimensional homotopy group is trivial) is homeomorphic to an n -dimensional sphere is known as the general Poincaré problem, but the case of n ≧ 5 was answered affirmatively in 1960 by J.R. Stallings and S. Smale, each with their own unique methods.

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

〈単連結，すなわち基本群が自明な連結三次元閉多様体は三次元球面に同相となるか〉という問題は，1904年H.ポアンカレにより提出され，ポアンカレの予想と呼ばれているが未解決である。〈n次元閉多様体(n≧4)において，その中のr次元球面(0≦r≦n－1)と同相な図形がすべて1点に縮む場合(すなわち，r次元ホモトピー群が自明な場合)にn次元球面と同相になるか〉という問題は，一般ポアンカレ問題といわれているが，n≧5の場合は60年にストーリングズJ.R.StallingsとスメールS.Smaleにより互いに独自の方法で肯定的に解決された。

出典　株式会社平凡社世界大百科事典第２版について　情報

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