文档介绍：Towards the Poincaré
Conjecture and the
Classification of
3-Manifolds
John Milnor
he Poincaré Conjecture was posed y- space SO(3)/I60 . Here SO(3) is the group of rota-
nine years ago and may possibly have tions of Euclidean 3-space, and I60 is the subgroup
been proved in the last few months. This consisting of those rotations which carry a regu-
note will be an account of some of the lar icosahedron or dodecahedron onto itself (the
Tmajor results over the past hundred unique simple group of order 60). This manifold
years which have paved the way towards a proof has the homology of the 3-sphere, but its funda-
and towards the even more ambitious project of mental group π1(SO(3)/I60) is a perfect group of
classifying pact 3-dimensional manifolds. order 120. He concluded the discussion by asking,
The final paragraph provides a brief description of again translated into modern language:
the latest developments, due to Grigory Perelman.
A more serious discussion of Perelman’s work If a closed 3-dimensional manifold has
will be provided in a subsequent note by Michael trivial fundamental group, must it be
Anderson. homeomorphic to the 3-sphere?
The conjecture that this is indeed the case has
Poincaré’s Question
come to be known as the Poincaré Conjecture. It
At the very beginning of the twentieth century, has turned out to be an extraordinaril