Differential Manifolds by Antoni A. Kosinski

Author:Antoni A. Kosinski
Language: eng
Format: epub, mobi, pdf
Publisher: Dover Publications
Published: 2000-01-01T16:00:00+00:00

The two circles, both marked β1, are identified as the arrows indicate as are the circles marked β2. There results a surface of a handlebody B of genus 2, and the continuous and dotted lines represent simple closed curves alt α1, α2 on it. This is a Heegaard diagram of a manifold M; that is, there is a homeomorphism of δB onto itself mapping αi, onto βi. Poincaré gives two proofs of this: The short and ingenious one consists in noticing that if we cut the surface along α1, α2 then the resulting diagram is exactly the same, with the roles of α1, α2 and β1, β2 interchanged. (A more pedestrian argument would consist in showing that after attaching 2-handles to B along α1, α2 the boundary becomes a 2-sphere.)

ExerciseShow that M is a homology sphere.

Now, the fundamental group of M has generators g, h and relations

,.

Poincaré shows that it is non-trivial by showing that after adding the relation g−1 hg−1 h = 1 it becomes the icosahedral group. Thus M is not homeomorphic to the 3-sphere.

Poincaré concludes this computation by asking the question: “Is it possible for the fundamental group of M to reduce to the identity element and M not being homeomorphic to the 3-sphere?” After rephrasing this slightly, he ends the paper, his last paper on topology, by saying that “this question would lead us too far.”

Indeed, this question, known as the Poincaré conjecture, led to a good part of topology created in the 80 years since then. In the next chapter we will present a solution due to S. Smale of a generalization of the Poincaré conjecture in dimensions larger than 4. Recently, M. Freedman [F] solved the 4-dimensional case. But the original question of Poincaré remains unanswered.

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