Manifolds are the central geometric objects in modern mathematics.
An attempt to understand the nature of manifolds leads to many
interesting questions. One of the most obvious questions is the
following. Let M and N be manifolds: how can we decide whether M
and N are ho- topy equivalent or homeomorphic or di?eomorphic (if
the manifolds are smooth)? The prototype of a beautiful answer is
given by the Poincar e Conjecture. If n N is S, the n-dimensional
sphere, and M is an arbitrary closed manifold, then n it is easy to
decide whether M is homotopy equivalent to S . Thisisthecaseif and
only if M is simply connected (assumingn> 1, the case n = 1 is
trivial since 1 every closed connected 1-dimensional manifold is
di?eomorphic toS ) and has the n homology of S . The
PoincareConjecture states that this is also su?cient for the n
existenceof ahomeomorphism fromM toS . For n = 2this
followsfromthewe- known classi?cation of surfaces. Forn> 4 this
was proved by Smale and Newman in the 1960s, Freedman solved the
case in n = 4 in 1982 and recently Perelman announced a proof for n
= 3, but this proof has still to be checked thoroughly by the
experts. In the smooth category it is not true that manifolds
homotopy n equivalent to S are di?eomorphic. The ?rst examples were
published by Milnor in 1956 and together with Kervaire he analyzed
the situation systematically in the 1960s."
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